manim/active_projects/quaternions.py

from big_ol_pile_of_manim_imports import *
# from pprint import pprint


# Helpers
def get_three_d_scene_config(high_quality=True):
    hq_config = {
        "camera_config": {
            "should_apply_shading": True,
            "exponential_projection": True,
        },
        "three_d_axes_config": {
            "num_axis_pieces": 1,
            "number_line_config": {
                "unit_size": 2,
                # "tick_frequency": 0.5,
                "tick_frequency": 1,
                "numbers_with_elongated_ticks": [0, 1, 2],
                "stroke_width": 2,
            }
        },
        "sphere_config": {
            "radius": 2,
            "resolution": (24, 48),
        }
    }
    lq_added_config = {
        "camera_config": {
            "should_apply_shading": False,
        },
        "three_d_axes_config": {
            "num_axis_pieces": 1,
        },
        "sphere_config": {
            # "resolution": (4, 12),
        }
    }
    if high_quality:
        return hq_config
    else:
        return merge_config([
            lq_added_config,
            hq_config
        ])


def q_mult(q1, q2):
    w1, x1, y1, z1 = q1
    w2, x2, y2, z2 = q2
    w = w1 * w2 - x1 * x2 - y1 * y2 - z1 * z2
    x = w1 * x2 + x1 * w2 + y1 * z2 - z1 * y2
    y = w1 * y2 + y1 * w2 + z1 * x2 - x1 * z2
    z = w1 * z2 + z1 * w2 + x1 * y2 - y1 * x2
    return np.array([w, x, y, z])


def stereo_project_point(point, axis=0, r=1, max_norm=10000):
    point = fdiv(point * r, point[axis] + r)
    point[axis] = 0
    norm = get_norm(point)
    if norm > max_norm:
        point *= max_norm / norm
    return point


def stereo_project(mobject, axis=0, r=1, outer_r=10, **kwargs):
    epsilon = 1
    for submob in mobject.family_members_with_points():
        points = submob.points
        n = len(points)
        for i in range(n):
            if points[i, axis] == -r:
                js = it.chain(
                    range(i + 1, n),
                    range(i - 1, -1, -1)
                )
                for j in js:
                    if points[j, axis] == -r:
                        continue
                    else:
                        vect = points[j] - points[i]
                        points[i] += epsilon * vect
                        break
        submob.apply_function(
            lambda p: stereo_project_point(p, axis, r, **kwargs)
        )

        # If all points are outside a certain range, this
        # shouldn't be displayed
        norms = np.apply_along_axis(get_norm, 1, submob.points)
        if np.all(norms > outer_r):
            # TODO, instead set opacity?
            # submob.points[:, :] = 0
            submob.set_fill(opacity=0)
            submob.set_stroke(opacity=0)

    return mobject


class Linus(VGroup):
    CONFIG = {
        "body_config": {
            "stroke_width": 15,
            "stroke_color": LIGHT_GREY,
            "sheen": 0.4,
        },
        "height": 2,
    }

    def __init__(self, **kwargs):
        VGroup.__init__(self, **kwargs)
        self.body = self.get_body_line()
        self.eyes = Eyes(self.body)

        self.add(self.body, self.eyes)
        self.set_height(self.height)
        self.center()

    def change_mode(self, mode, thing_to_look_at=None):
        self.eyes.change_mode(mode, thing_to_look_at)
        if mode == "sad":
            self.become_squiggle()
        elif mode == "confused":
            self.become_squiggle(factor=-0.1)
        elif mode == "pleading":
            self.become_squiggle(factor=0.3)
        else:
            self.become_line()
        return self

    def change(self, *args, **kwargs):
        self.change_mode(*args, **kwargs)
        return self

    def look_at(self, thing_to_look_at=None):
        self.eyes.look_at(thing_to_look_at)
        return self

    def blink(self):
        self.eyes.blink()
        return self

    def get_squiggle(self, factor=0.2):
        sine_curve = FunctionGraph(
            lambda x: factor * np.sin(x),
            x_min=0, x_max=TAU,
        )
        sine_curve.rotate(TAU / 4)
        sine_curve.match_style(self.body)
        sine_curve.match_height(self.body)
        sine_curve.move_to(self.body, UP)
        return sine_curve

    def get_body_line(self, **kwargs):
        config = dict(self.body_config)
        config.update(kwargs)
        line = Line(ORIGIN, 1.5 * UP, **config)
        if hasattr(self, "body"):
            line.match_style(self.body)
            line.match_height(self.body)
            line.move_to(self.body, UP)
        return line

    def become_squiggle(self, **kwargs):
        self.body.become(self.get_squiggle(**kwargs))
        return self

    def become_line(self, **kwargs):
        self.body.become(self.get_body_line(**kwargs))
        return self

    def copy(self):
        return self.deepcopy()


class Felix(PiCreature):
    CONFIG = {
        "color": GREEN_D
    }


class PushPin(SVGMobject):
    CONFIG = {
        "file_name": "push_pin",
        "height": 0.5,
        "sheen": 0.7,
        "fill_color": GREY,
    }

    def __init__(self, **kwargs):
        SVGMobject.__init__(self, **kwargs)
        self.rotate(20 * DEGREES)

    def pin_to(self, point):
        self.move_to(point, DR)


class Hand(SVGMobject):
    CONFIG = {
        "file_name": "pinch_hand",
        "height": 0.4,
        "sheen": 0.2,
        "fill_color": GREY,
    }

    def __init__(self, **kwargs):
        SVGMobject.__init__(self, **kwargs)
        self.add(VectorizedPoint().next_to(self, UP, buff=0.15))


class CheckeredCircle(Circle):
    CONFIG = {
        "n_pieces": 16,
        "colors": [BLUE_E, BLUE_C],
        "stroke_width": 5,
    }

    def __init__(self, **kwargs):
        Circle.__init__(self, **kwargs)
        pieces = self.get_pieces(self.n_pieces)
        self.points = np.zeros((0, 3))
        self.add(*pieces)
        n_colors = len(self.colors)
        for i, color in enumerate(self.colors):
            self[i::n_colors].set_color(color)


class StereoProjectedSphere(Sphere):
    CONFIG = {
        "stereo_project_config": {
            "axis": 2,
        },
        "max_r": 32,
        "max_width": FRAME_WIDTH,
        "max_height": FRAME_WIDTH,
        "max_depth": FRAME_WIDTH,
        "radius": 1,
    }

    def __init__(self, rotation_matrix=None, **kwargs):
        digest_config(self, kwargs)
        if rotation_matrix is None:
            rotation_matrix = np.identity(3)
        self.rotation_matrix = rotation_matrix

        self.stereo_project_config["r"] = self.radius
        ParametricSurface.__init__(
            self, self.post_projection_func, **kwargs
        )
        self.submobjects.sort(
            key=lambda m: -m.get_width()
        )
        self.fade_far_out_submobjects()

    def post_projection_func(self, u, v):
        point = self.radius * Sphere.func(self, u, v)
        rot_point = np.dot(point, self.rotation_matrix.T)
        result = stereo_project_point(
            rot_point, **self.stereo_project_config
        )
        epsilon = 1e-4
        if np.any(np.abs(result) == np.inf) or np.any(np.isnan(result)):
            return self.func(u + epsilon, v)
        return result

    def fade_far_out_submobjects(self, **kwargs):
        max_r = kwargs.get("max_r", self.max_r)
        max_width = kwargs.get("max_width", self.max_width)
        max_height = kwargs.get("max_height", self.max_height)
        max_depth = kwargs.get("max_depth", self.max_depth)
        for submob in self.submobjects:
            violations = [
                np.any(np.apply_along_axis(get_norm, 1, submob.get_anchors()) > max_r),
                submob.get_width() > max_width,
                submob.get_height() > max_height,
                submob.get_depth() > max_depth
            ]
            if any(violations):
                # self.remove(submob)
                submob.fade(1)
        return self


class StereoProjectedSphereFromHypersphere(StereoProjectedSphere):
    CONFIG = {
        "stereo_project_config": {
            "axis": 0,
        },
        "radius": 2,
    }

    def __init__(self, quaternion=None, null_axis=0, **kwargs):
        if quaternion is None:
            quaternion = np.array([1, 0, 0, 0])
        self.quaternion = quaternion
        self.null_axis = null_axis
        ParametricSurface.__init__(self, self.q_mult_projection_func, **kwargs)
        self.fade_far_out_submobjects()

    def q_mult_projection_func(self, u, v):
        point = list(Sphere.func(self, u, v))
        point.insert(self.null_axis, 0)
        post_q_mult = q_mult(self.quaternion, point)
        projected = list(self.radius * stereo_project_point(
            post_q_mult, **self.stereo_project_config
        ))
        if np.any(np.abs(projected) == np.inf):
            return self.func(u + 0.001, v)
        ignored_axis = self.stereo_project_config["axis"]
        projected.pop(ignored_axis)
        return np.array(projected)


class StereoProjectedCircleFromHypersphere(CheckeredCircle):
    CONFIG = {
        "n_pieces": 48,
        "radius": 2,
        "max_length": 2 * FRAME_WIDTH,
        "max_r": 20,
        "basis_vectors": [
            [1, 0, 0, 0],
            [0, 1, 0, 0],
        ]
    }

    def __init__(self, quaternion=None, **kwargs):
        CheckeredCircle.__init__(self, **kwargs)
        if quaternion is None:
            quaternion = [1, 0, 0, 0]
        self.quaternion = quaternion
        self.pre_positioning_matrix = self.get_pre_positioning_matrix()
        self.apply_function(self.projection)
        self.remove_large_pieces()
        self.set_shade_in_3d(True)

    def get_pre_positioning_matrix(self):
        v1, v2 = [np.array(v) for v in self.basis_vectors]
        v1 = normalize(v1)
        v2 = v2 - np.dot(v1, v2) * v1
        v2 = normalize(v2)
        return np.array([v1, v2]).T

    def projection(self, point):
        q1 = self.quaternion
        q2 = np.dot(self.pre_positioning_matrix, point[:2]).flatten()
        new_q = q_mult(q1, q2)
        projected = stereo_project_point(
            new_q, axis=0, r=self.radius,
        )
        if np.any(projected == np.inf) or np.any(np.isnan(projected)):
            epsilon = 1e-6
            return self.projection(rotate_vector(point, epsilon))
        return projected[1:]

    def remove_large_pieces(self):
        for piece in self:
            length = get_norm(piece.points[0] - piece.points[-1])
            violations = [
                length > self.max_length,
                get_norm(piece.get_center()) > self.max_r,
            ]
            if any(violations):
                piece.fade(1)


class QuaternionTracker(ValueTracker):
    CONFIG = {
        "force_unit": True,
        "dim": 4,
    }

    def __init__(self, four_vector=None, **kwargs):
        Mobject.__init__(self, **kwargs)
        if four_vector is None:
            four_vector = np.array([1, 0, 0, 0])
        self.set_value(four_vector)
        if self.force_unit:
            self.add_updater(lambda q: q.normalize())

    def set_value(self, vector):
        self.points = np.array(vector).reshape((1, 4))
        return self

    def get_value(self):
        return self.points[0]

    def normalize(self):
        self.set_value(normalize(
            self.get_value(),
            fall_back=np.array([1, 0, 0, 0])
        ))
        return self


class RubiksCube(VGroup):
    CONFIG = {
        "colors": [
            "#C41E3A", "#009E60", "#0051BA",
            "#FF5800", "#FFD500", "#FFFFFF"
        ],
    }

    def __init__(self, **kwargs):
        digest_config(self, kwargs)
        vectors = [OUT, RIGHT, UP, LEFT, DOWN, IN]
        faces = [
            self.create_face(color, vector)
            for color, vector in zip(self.colors, vectors)
        ]
        VGroup.__init__(self, *it.chain(*faces), **kwargs)
        self.set_shade_in_3d(True)

    def create_face(self, color, vector):
        squares = VGroup(*[
            self.create_square(color)
            for x in range(9)
        ])
        squares.arrange_submobjects_in_grid(
            3, 3,
            buff=0
        )
        squares.set_width(2)
        squares.move_to(OUT, OUT)
        squares.apply_matrix(z_to_vector(vector))
        return squares

    def create_square(self, color):
        square = Square(
            stroke_width=3,
            stroke_color=BLACK,
            fill_color=color,
            fill_opacity=1,
            side_length=1,
        )
        square.flip()
        return square
        # back = square.copy()
        # back.set_fill(BLACK, 0.85)
        # back.set_stroke(width=0)
        # back.shift(0.5 * IN)
        # return VGroup(square, back)

    def get_face(self, vect):
        self.sort_submobjects(lambda p: np.dot(p, vect))
        return self[-(12 + 9):]


# Abstract scenes
class SpecialThreeDScene(ThreeDScene):
    CONFIG = {
        "cut_axes_at_radius": True,
    }

    def __init__(self, **kwargs):
        digest_config(self, kwargs)
        if self.frame_duration == PRODUCTION_QUALITY_FRAME_DURATION:
            high_quality = True
        else:
            high_quality = False
        config = get_three_d_scene_config(high_quality)
        ThreeDScene.__init__(self, **config)

    def get_axes(self):
        axes = ThreeDAxes(**self.three_d_axes_config)
        for axis in axes:
            if self.cut_axes_at_radius:
                p0 = axis.main_line.get_start()
                p1 = axis.number_to_point(-1)
                p2 = axis.number_to_point(1)
                p3 = axis.main_line.get_end()
                new_pieces = VGroup(
                    Line(p0, p1), Line(p1, p2), Line(p2, p3),
                )
                for piece in new_pieces:
                    piece.shade_in_3d = True
                new_pieces.match_style(axis.pieces)
                axis.pieces.submobjects = new_pieces.submobjects
            for tick in axis.tick_marks:
                tick.add(VectorizedPoint(
                    1.5 * tick.get_center(),
                ))
        return axes

    def get_sphere(self):
        return Sphere(**self.sphere_config)

    def get_default_camera_position(self):
        return {
            "phi": 70 * DEGREES,
            "theta": -110 * DEGREES,
        }


# Animated scenes

class ManyNumberSystems(Scene):
    def construct(self):
        # Too much dumb manually positioning in here...
        title = Title("Number systems")
        name_location_color_example_tuples = [
            ("Reals", [-4, 2, 0], YELLOW, "1.414"),
            ("Complex numbers", [5, 0, 0], BLUE, "2 + i"),
            ("Quaternions", [3, 2, 0], PINK, "2 + 7i + 1j + 8k"),
            ("Rationals", [3, -2, 0], RED, "1 \\over 3"),
            ("p-adic numbers", [-2, -2, 0], GREEN, "\\overline{142857}2"),
            ("Octonions", [-3, 0, 0], LIGHT_GREY, "3e_1 - 2.3e_2 + \\dots + 1.6e_8"),
        ]
        systems = VGroup()
        for name, location, color, ex in name_location_color_example_tuples:
            system = TextMobject(name)
            system.set_color(color)
            system.move_to(location)
            example = TexMobject(ex)
            example.next_to(system, DOWN)
            system.add(example)
            systems.add(system)
        R_label, C_label, H_label = systems[:3]

        number_line = NumberLine(x_min=-3, x_max=3)
        number_line.add_numbers()
        number_line.shift(0.25 * FRAME_WIDTH * LEFT)
        number_line.shift(0.5 * DOWN)
        R_example_dot = Dot(number_line.number_to_point(1.414))
        plane = ComplexPlane(x_radius=3.5, y_radius=2.5)
        plane.add_coordinates()
        plane.shift(0.25 * FRAME_WIDTH * RIGHT)
        plane.shift(0.5 * DOWN)
        C_example_dot = Dot(plane.coords_to_point(2, 1))

        self.add(title)
        self.play(LaggedStart(
            FadeInFromLarge, systems,
            lambda m: (m, 4)
        ))
        self.wait()
        self.add(number_line, plane, systems)
        self.play(
            R_label.move_to, 0.25 * FRAME_WIDTH * LEFT + 2 * UP,
            C_label.move_to, 0.25 * FRAME_WIDTH * RIGHT + 2 * UP,
            H_label.move_to, 0.75 * FRAME_WIDTH * RIGHT + 2 * UP,
            FadeOutAndShift(systems[3:], 2 * DOWN),
            Write(number_line),
            Write(plane),
            GrowFromCenter(R_example_dot),
            R_label[-1].next_to, R_example_dot, UP,
            GrowFromCenter(C_example_dot),
            C_label[-1].next_to, C_example_dot, UR, SMALL_BUFF,
            C_label[-1].shift, 0.4 * LEFT,
        )
        number_line.add(R_example_dot)
        plane.add(C_example_dot)
        self.wait(2)
        self.play(LaggedStart(
            ApplyMethod,
            VGroup(
                H_label,
                VGroup(plane, C_label),
                VGroup(number_line, R_label),
            ),
            lambda m: (m.shift, 0.5 * FRAME_WIDTH * LEFT),
            lag_ratio=0.8,
        ))
        randy = Randolph()
        randy.next_to(plane, RIGHT)
        self.play(
            randy.change, "maybe", H_label,
            VFadeIn(randy),
        )
        self.play(Blink(randy))
        self.play(randy.change, "confused", H_label.get_top())
        self.wait()


class RotationsIn3d(SpecialThreeDScene):
    def construct(self):
        self.set_camera_orientation(**self.get_default_camera_position())
        self.begin_ambient_camera_rotation(rate=0.02)
        sphere = self.get_sphere()
        vectors = VGroup(*[
            Vector(u * v, color=color).next_to(sphere, u * v, buff=0)
            for v, color in zip(
                [RIGHT, UP, OUT],
                [GREEN, RED, BLUE],
            )
            for u in [-1, 1]
        ])
        vectors.set_shade_in_3d(True)
        sphere.add(vectors)

        self.add(self.get_axes())
        self.add(sphere)
        angle_axis_pairs = [
            (90 * DEGREES, RIGHT),
            (120 * DEGREES, UR),
            (-45 * DEGREES, OUT),
            (60 * DEGREES, IN + DOWN),
            (90 * DEGREES, UP),
            (30 * DEGREES, UP + OUT + RIGHT),
        ]
        for angle, axis in angle_axis_pairs:
            self.play(Rotate(
                sphere, angle,
                axis=axis,
                run_time=2,
            ))
            self.wait()


class TODOInsertIntroduceThreeDNumbers(TODOStub):
    CONFIG = {"message": "Insert IntroduceThreeDNumbers"}


class IntroduceHamilton(Scene):
    def construct(self):
        hamilton = ImageMobject("Hamilton", height=6)
        hamilton.to_corner(UL)
        shamrock = SVGMobject(file_name="shamrock")
        shamrock.set_height(1)
        shamrock.set_color("#009a49")
        shamrock.set_fill(opacity=0.25)
        shamrock.next_to(hamilton.get_corner(UL), DR)
        shamrock.align_to(hamilton, UP)
        hamilton_name = TextMobject(
            "William Rowan Hamilton"
        )
        hamilton_name.match_width(hamilton)
        hamilton_name.next_to(hamilton, DOWN)

        quote = TextMobject(
            """\\huge ...Every morning in the early part of the above-cited
            month, on my coming down to breakfast, your (then)
            little brother William Edwin, and yourself, used to
            ask me,""",
            "``Well, Papa, can you multiply triplets''?",
            """Whereto I was always obliged to reply, with a sad
            shake of the head: ``No, I can only add and subtract
            them.''...""",
            alignment=""
        )
        quote.set_color_by_tex("Papa", YELLOW)
        quote = VGroup(*it.chain(*quote))
        quote.set_width(FRAME_WIDTH - hamilton.get_width() - 2)
        quote.to_edge(RIGHT)
        quote_rect = SurroundingRectangle(quote, buff=MED_SMALL_BUFF)
        quote_rect.set_stroke(WHITE, 2)
        quote_rect.stretch(1.1, 1)
        quote_label = TextMobject(
            "August 5, 1865 Letter\\\\from Hamilton to his son"
        )
        quote_label.next_to(quote_rect, UP)
        quote_label.set_color(BLUE)
        VGroup(quote, quote_rect, quote_label).to_edge(UP)

        plaque = ImageMobject("BroomBridgePlaque")
        plaque.set_width(FRAME_WIDTH / 2)
        plaque.to_edge(LEFT)
        plaque.shift(UP)
        equation = TexMobject(
            "i^2 = j^2 = k^2 = ijk = -1",
            tex_to_color_map={"i": GREEN, "j": RED, "k": BLUE}
        )
        equation_rect = Rectangle(width=3.25, height=0.7)
        equation_rect.move_to(3.15 * LEFT + 0.25 * DOWN)
        equation_rect.set_color(WHITE)
        equation_arrow = Vector(DOWN)
        equation_arrow.match_color(equation_rect)
        equation_arrow.next_to(equation_rect, DOWN)
        equation.next_to(equation_arrow, DOWN)

        self.play(
            FadeInFromDown(hamilton),
            Write(hamilton_name),
        )
        self.play(DrawBorderThenFill(shamrock))
        self.wait()

        self.play(
            LaggedStart(
                FadeIn, quote,
                lag_ratio=0.2,
                run_time=4
            ),
            FadeInFromDown(quote_label),
            ShowCreation(quote_rect)
        )
        self.wait(3)
        self.play(
            ApplyMethod(
                VGroup(hamilton, shamrock, hamilton_name).to_edge, RIGHT,
                run_time=2,
                rate_func=squish_rate_func(smooth, 0.5, 1),
            ),
            LaggedStart(
                FadeOutAndShiftDown, VGroup(*it.chain(
                    quote, quote_rect, quote_label
                ))
            )
        )
        self.wait()
        self.play(FadeIn(plaque))
        self.play(
            ShowCreation(equation_rect),
            GrowArrow(equation_arrow)
        )
        self.play(ReplacementTransform(
            equation.copy().replace(equation_rect).fade(1),
            equation
        ))
        self.wait()


class QuaternionRotationOverlay(Scene):
    def construct(self):
        equations = VGroup(
            TexMobject(
                "p", "\\rightarrow",
                "{}",
                "{}",
                "\\left(q_1",
                "p",
                "q_1^{-1}\\right)",
                "{}",
                "{}",
            ),
            TexMobject(
                "p", "\\rightarrow",
                "{}",
                "\\left(q_2",
                "\\left(q_1",
                "p",
                "q_1^{-1}\\right)",
                "q_2^{-1}\\right)",
                "{}",
            ),
            TexMobject(
                "p", "\\rightarrow",
                "\\left(q_3",
                "\\left(q_2",
                "\\left(q_1",
                "p",
                "q_1^{-1}\\right)",
                "q_2^{-1}\\right)",
                "q_3^{-1}\\right)",
            ),
        )
        for equation in equations:
            equation.set_color_by_tex_to_color_map({
                "1": GREEN, "2": RED, "3": BLUE,
            })
            equation.set_color_by_tex("rightarrow", WHITE)
            equation.to_corner(UL)

        equation = equations[0].copy()
        self.play(Write(equation))
        self.wait()
        for new_equation in equations[1:]:
            self.play(
                Transform(equation, new_equation)
            )
            self.wait(2)


class RotateCubeThreeTimes(SpecialThreeDScene):
    def construct(self):
        cube = RubiksCube()
        cube.set_fill(opacity=0.8)
        cube.set_stroke(width=1)
        randy = Randolph(mode="pondering")
        randy.set_height(cube.get_height() - 2 * SMALL_BUFF)
        randy.move_to(cube.get_edge_center(OUT))
        randy.set_fill(opacity=0.8)
        # randy.set_shade_in_3d(True)
        cube.add(randy)
        axes = self.get_axes()

        self.add(axes, cube)
        self.move_camera(
            phi=70 * DEGREES,
            theta=-140 * DEGREES,
        )
        self.begin_ambient_camera_rotation(rate=0.02)
        self.wait(2)
        self.play(Rotate(cube, TAU / 4, RIGHT, run_time=3))
        self.wait(2)
        self.play(Rotate(cube, TAU / 4, UP, run_time=3))
        self.wait(2)
        self.play(Rotate(cube, -TAU / 3, np.ones(3), run_time=3))
        self.wait(7)


class HereWeTackle4d(TeacherStudentsScene):
    def construct(self):
        titles = VGroup(
            TextMobject(
                "This video:\\\\",
                "Quaternions in 4d"
            ),
            TextMobject(
                "Next video:\\\\",
                "Quaternions acting on 3d"
            )
        )
        for title in titles:
            title.move_to(self.hold_up_spot, DOWN)
        titles[0].set_color(YELLOW)

        self.play(
            self.teacher.change, "raise_right_hand",
            FadeInFromDown(titles[0]),
            self.get_student_changes("confused", "horrified", "sad")
        )
        self.look_at(self.screen)
        self.wait()
        self.change_student_modes(
            "erm", "thinking", "pondering",
            look_at_arg=self.screen
        )
        self.wait(3)
        self.change_student_modes(
            "pondering", "confused", "happy"
        )
        self.look_at(self.screen)
        self.wait(3)
        self.play(
            self.teacher.change, "hooray",
            FadeInFrom(titles[1]),
            ApplyMethod(
                titles[0].shift, 2 * UP,
                rate_func=squish_rate_func(smooth, 0.2, 1)
            )
        )
        self.change_all_student_modes("hooray")
        self.play(self.teacher.change, "happy")
        self.look_at(self.screen)
        self.wait(3)


class TableOfContents(Scene):
    def construct(self):
        chapters = VGroup(
            TextMobject(
                "\\underline{Chapter 1}\\\\", "Linus the Linelander"
            ),
            TextMobject(
                "\\underline{Chapter 2}\\\\", "Felix the Flatlander"
            ),
            TextMobject(
                "\\underline{Chapter 3}\\\\", " You, the 3d-lander"
            ),
        )
        for chapter in chapters:
            chapter.space_out_submobjects(1.5)
        chapters.arrange_submobjects(
            DOWN, buff=1.5, aligned_edge=LEFT
        )
        chapters.to_edge(LEFT)

        for chapter in chapters:
            self.play(FadeInFromDown(chapter))
            self.wait(2)
        for chapter in chapters:
            chapters.save_state()
            other_chapters = VGroup(*[
                c for c in chapters if c is not chapter
            ])
            self.play(
                chapter.set_width, 0.5 * FRAME_WIDTH,
                chapter.center,
                other_chapters.fade, 1
            )
            self.wait(3)
            self.play(chapters.restore)


class IntroduceLinusTheLinelander(Scene):
    def construct(self):
        self.introduce_linus()
        self.show_real_number_line()
        self.look_at_complex_plane()

    def introduce_linus(self):
        linus = Linus()
        linus.move_to(3 * LEFT)
        name = TextMobject("Linus the Linelander")
        name.next_to(linus, DR, buff=MED_LARGE_BUFF)
        arrow = Arrow(name.get_top(), linus.get_right())

        self.play(FadeInFromDown(linus))
        self.play(
            Write(name),
            GrowArrow(arrow),
            linus.change, "gracious", name,
        )
        self.play(
            linus.become_squiggle, {"factor": -0.1},
        )
        self.play(Blink(linus))
        self.wait()

        self.name_text = name
        self.name_arrow = arrow
        self.linus = linus

    def show_real_number_line(self):
        linus = self.linus
        number_line = NumberLine()
        number_line.add_numbers()
        number_line.to_edge(UP)

        algebra = VGroup(
            TexMobject("3 \\cdot 4 = 12"),
            TexMobject("3 + 4 = 7"),
            TexMobject("(-2) \\cdot 3 = -6"),
        )
        algebra.arrange_submobjects(DOWN)
        algebra.next_to(number_line, DOWN, LARGE_BUFF)
        algebra.shift(3 * RIGHT)

        self.play(
            ShowCreation(number_line.main_line),
            linus.look_at, number_line
        )
        self.play(
            LaggedStart(FadeInFromDown, number_line.numbers),
            LaggedStart(ShowCreation, number_line.tick_marks),
            linus.change, "happy"
        )
        self.play(
            LaggedStart(FadeInFromDown, algebra),
            linus.look_at, algebra
        )
        self.play(Blink(linus))
        self.wait()

        self.algebra = algebra

    def look_at_complex_plane(self):
        linus = self.linus
        to_fade = VGroup(
            self.name_text,
            self.name_arrow,
            self.algebra,
        )
        frame = ScreenRectangle()
        frame.set_width(8)
        frame.to_corner(DR)

        q_marks = VGroup(*[
            TexMobject("?").shift(
                random.random() * RIGHT,
                random.random() * UP,
            )
            for x in range(50)
        ])
        q_marks.next_to(linus.body, UP, buff=0)
        q_marks.set_color_by_gradient(BLUE, GREEN, YELLOW)
        random.shuffle(q_marks.submobjects)
        q_marks_anim = LaggedStart(
            FadeIn, q_marks,
            run_time=15,
            rate_func=there_and_back,
            lag_ratio=0.1
        )
        q_marks_continual = NormalAnimationAsContinualAnimation(q_marks_anim)

        self.play(
            FadeOut(to_fade),
            ShowCreation(frame),
            linus.look_at, frame
        )
        self.add(q_marks_continual)
        self.play(linus.change_mode, "confused")
        self.wait()
        self.play(Blink(linus))
        self.play(linus.change, "confused", frame.get_bottom())
        self.wait()
        self.play(linus.change, "sad", frame.get_center())
        self.wait(10)


class ShowComplexMultiplicationExamples(Scene):
    CONFIG = {
        "plane_config": {
            "x_radius": 9,
            "y_radius": 9,
            "stroke_width": 3,
        },
        "background_plane_config": {
            "color": LIGHT_GREY,
            "secondary_color": DARK_GREY,
            "stroke_width": 0.5,
            "stroke_opacity": 0.5,
            "secondary_line_ratio": 0,
        }
    }

    def construct(self):
        self.add_planes()
        z_tuples = [
            (complex(2, 1), "2 + i", UP),
            (complex(5, 2), "5 + 2i", LEFT),
            (
                complex(-np.sqrt(2) / 2, np.sqrt(2) / 2),
                "-\\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{2}}{2} i",
                LEFT,
            ),
            (complex(-4, 1.5), "-4 + 1.5i", RIGHT),
            (complex(3, 0), "3 + 0i", UP),
            (complex(4, -3), "4 + -3i", UP),
        ]

        for z, z_tex, label_vect in z_tuples:
            self.show_multiplication(z, z_tex, label_vect)

    def add_planes(self, include_title=True):
        plane = ComplexPlane(**self.plane_config)
        self.plane = plane
        background_plane = ComplexPlane(**self.background_plane_config)
        background_plane.add_coordinates()
        self.background_plane = background_plane

        self.add(background_plane)
        self.add(plane)

        if include_title:
            title = TextMobject("Complex plane")
            title.scale(1.5)
            title.to_corner(UL, buff=MED_LARGE_BUFF)
            title.shift(SMALL_BUFF * UR)
            self.title = title
            self.add_foreground_mobjects(title)

    def show_multiplication(self, z, z_tex, label_vect):
        z_color = WHITE
        plane = self.plane
        new_plane = plane.deepcopy()
        real_tex, imag_tex = z_tex.split("+")
        label = TexMobject(
            "\\text{Multiply by}\\\\",
            real_tex, "+", imag_tex,
            alignment="",
        )
        label[1].set_color(GREEN)
        label[3].set_color(RED)
        label.scale(1.2)

        h_line = Line(
            plane.number_to_point(0),
            plane.number_to_point(z.real),
            color=GREEN,
            stroke_width=5,
        )
        v_line = Line(
            plane.number_to_point(z.real),
            plane.number_to_point(z),
            color=RED,
            stroke_width=5,
        )
        lines = VGroup(h_line, v_line)

        z_point = plane.number_to_point(z)
        z_dot = Dot(z_point)
        z_dot.set_color(z_color)
        label[1:].next_to(z_dot, label_vect)
        label[0].next_to(label[1:], UP)
        for mob in label:
            label.add_to_back(BackgroundRectangle(mob))

        one_dot = Dot(plane.number_to_point(1))
        one_dot.set_color(YELLOW)
        for dot in z_dot, one_dot:
            dot.save_state()
            dot.scale(5)
            dot.set_fill(opacity=0)
            dot.set_stroke(width=1, opacity=0.5)
        to_fade_out = VGroup(
            plane, label, lines, z_dot, one_dot
        )

        self.play(
            ShowCreation(lines),
            FadeInFromDown(label),
            Restore(z_dot),
        )
        self.play(Restore(one_dot))
        angle = np.log(z).imag
        self.play(
            one_dot.move_to, z_dot,
            plane.apply_complex_function, lambda w: z * w,
            path_arc=angle,
            run_time=3
        )
        self.wait()
        self.play(
            FadeOut(to_fade_out),
            FadeIn(new_plane),
        )
        self.plane = new_plane


class DefineComplexNumbersPurelyAlgebraically(Scene):
    def construct(self):
        self.add_linus()
        self.add_title()
        self.show_example_number()
        self.show_multiplication()
        self.emphsize_result_has_same_form()

    def add_linus(self):
        linus = self.linus = Linus()
        linus.move_to(3 * LEFT)

    def add_title(self):
        title = self.title = Title(
            "No spatial reasoning, just symbols"
        )
        self.play(
            FadeInFromDown(title[:-1]),
            ShowCreation(title[-1]),
            self.linus.look_at, title
        )

    def show_example_number(self):
        linus = self.linus

        number = TexMobject("2.35", "+", "3.14", "i")
        number.next_to(self.title, DOWN, buff=1.5)
        number.shift(3 * RIGHT)
        real, imag = number[0], number[2]
        real_brace = Brace(real, UP)
        imag_brace = Brace(imag, DOWN)
        real_label = real_brace.get_text("Some real number")
        imag_label = imag_brace.get_text("Some other real number")
        VGroup(real, real_label).set_color(GREEN)
        VGroup(imag, imag_label).set_color(RED)

        i = number[-1]
        i_def = TexMobject("i", "^2", "=", "-1")
        i_def.next_to(
            self.title, DOWN,
            buff=MED_LARGE_BUFF,
            aligned_edge=LEFT,
        )
        i_def_rect = SurroundingRectangle(i_def, color=YELLOW, buff=MED_SMALL_BUFF)
        definition_label = TextMobject("Definition")
        definition_label.next_to(i_def_rect, DOWN)
        definition_label.match_color(i_def_rect)

        self.play(Write(number, run_time=1))
        self.play(
            GrowFromCenter(real_brace),
            LaggedStart(FadeIn, real_label),
            linus.change, "confused", number,
            run_time=1
        )
        self.wait()
        self.play(
            GrowFromCenter(imag_brace),
            LaggedStart(FadeIn, imag_label),
            run_time=1
        )
        self.play(Blink(linus))
        self.play(
            linus.change, "erm", i_def,
            ReplacementTransform(
                i.copy(), i_def[0],
                path_arc=-30 * DEGREES
            ),
            FadeIn(i_def_rect),
            FadeIn(definition_label),
        )
        self.play(Write(i_def[1:]))
        self.wait()
        self.play(Blink(linus))

        self.to_fade = VGroup(
            real_brace, real_label,
            imag_brace, imag_label,
        )
        self.number = number

    def show_multiplication(self):
        linus = self.linus
        to_fade = self.to_fade
        z1 = self.number

        z2 = TexMobject("4", "+", "5", "i")
        z2.match_style(z1)

        for z in z1, z2:
            lp, rp = z.parens = TexMobject("()")
            lp.next_to(z, LEFT, SMALL_BUFF)
            rp.next_to(z, RIGHT, SMALL_BUFF)
            z.real = z[0]
            z.imag = z[2:]
            for part in z.real, z.imag:
                part.targets = [part.copy(), part.copy()]

        z1.generate_target()
        product = VGroup(
            VGroup(z1.target, z1.parens),
            VGroup(z2, z2.parens),
        )
        product.arrange_submobjects(RIGHT, SMALL_BUFF)
        product.move_to(2 * RIGHT + 2 * UP)

        foil = VGroup(*map(TextMobject, [
            "First", "Outside", "Inside", "Last",
        ]))
        foil.arrange_submobjects(
            DOWN, buff=MED_SMALL_BUFF,
            aligned_edge=LEFT
        )
        foil.scale(1.25)
        for word in foil:
            word[0].set_color(BLUE)
        foil.move_to(product).to_edge(DOWN, LARGE_BUFF)

        def get_cdot():
            return TexMobject("\\cdot")

        def get_lp():
            return TexMobject("(")

        def get_rp():
            return TexMobject(")")

        def get_plus():
            return TexMobject("+")

        expansion = VGroup(
            z1.real.targets[0], get_cdot(), z2.real.targets[0], get_plus(),
            z1.real.targets[1], get_cdot(), z2.imag.targets[0], get_plus(),
            z1.imag.targets[0], get_cdot(), z2.real.targets[1], get_plus(),
            z1.imag.targets[1], get_cdot(), z2.imag.targets[1],
        )
        expansion.arrange_submobjects(RIGHT, buff=0.15)
        expansion.next_to(product, DOWN, buff=LARGE_BUFF)
        expansion_parts = VGroup(*[
            expansion[4 * i: 4 * i + 3]
            for i in range(4)
        ])
        expansion_part_braces = VGroup(*[
            Brace(part, DOWN) for part in expansion_parts
        ])
        for word, brace in zip(foil, expansion_part_braces):
            word.next_to(brace, DOWN)

        final_prouct = VGroup(
            get_lp(),
            z1[0].copy(), get_cdot(), z2[0].copy(),
            TexMobject("-"),
            z1[2].copy(), get_cdot(), z2[2].copy(),
            get_rp(), get_plus(),
            get_lp(),
            z1[0].copy(), get_cdot(), z2[2].copy(),
            get_plus(),
            z1[2].copy(), get_cdot(), z2[0].copy(),
            get_rp(), TexMobject("i")
        )
        final_prouct.arrange_submobjects(RIGHT, buff=0.15)
        final_prouct.next_to(expansion, DOWN, buff=2)
        final_arrows = VGroup()
        for i, brace in zip([1, 11, 15, 5], expansion_part_braces):
            target = final_prouct[i:i + 3]
            if i == 5:
                arrow = Line(
                    brace.get_bottom() + SMALL_BUFF * DOWN,
                    target.get_top() + MED_SMALL_BUFF * UP,
                )
                arrow.points[1] = arrow.points[0] + DOWN
                arrow.points[2] = arrow.points[3] + UP
                tip = RegularPolygon(3, start_angle=-100 * DEGREES)
                tip.set_height(0.2)
                tip.set_stroke(width=0)
                tip.set_fill(WHITE, opacity=1)
                tip.move_to(arrow.get_end())
                arrow.add(tip)
            else:
                arrow = Arrow(
                    brace.get_bottom(),
                    target.get_top(),
                    use_rectangular_stem=False,
                )
            final_arrows.add(arrow)
        final_arrows.set_stroke(BLACK, width=6, background=True)

        # Move example number into product
        self.play(
            FadeOut(to_fade),
            MoveToTarget(z1),
            FadeIn(z1.parens),
            FadeInFromDown(z2),
            FadeIn(z2.parens),
            linus.change, "happy", product,
        )
        self.wait()

        # Show distribution
        pairs = list(it.product([z1.real, z1.imag], [z2.real, z2.imag]))
        for i in range(4):
            left, right = pair = VGroup(*pairs[i])
            word = foil[i]
            dot = expansion[4 * i + 1]
            plus = expansion[4 * i + 3] if i < 3 else VMobject()
            brace = expansion_part_braces[i]

            self.play(pair.shift, 0.5 * DOWN)
            self.play(
                FadeIn(dot),
                GrowFromCenter(brace),
                FadeInFromDown(word),
                linus.move_to, 4 * LEFT + DOWN,
                *[
                    ReplacementTransform(
                        part.copy(),
                        part.targets.pop(0)
                    )
                    for part in pair
                ]
            )
            self.play(
                pair.shift, 0.5 * UP,
                FadeIn(plus)
            )
        self.play(Blink(linus))

        self.play(
            FadeOut(foil),
            FadeInFromDown(final_prouct),
            linus.look_at, final_prouct,
        )
        self.play(
            LaggedStart(ShowCreation, final_arrows),
            run_time=3,
        )
        self.play(linus.change, "confused")
        self.wait()

        self.final_prouct = final_prouct

    def emphsize_result_has_same_form(self):
        final_product = self.final_prouct
        real = final_product[1:1 + 7]
        imag = final_product[11:11 + 7]

        real_brace = Brace(real, DOWN)
        real_label = real_brace.get_text("Some real number")
        real_label.set_color(GREEN)
        imag_brace = Brace(imag, DOWN)
        imag_label = imag_brace.get_text(
            "Some other \\\\ real number"
        )
        imag_label.set_color(RED)
        braces = VGroup(real_brace, imag_brace)
        labels = VGroup(real_label, imag_label)
        self.play(
            LaggedStart(GrowFromCenter, braces),
            LaggedStart(Write, labels),
        )
        self.wait()


class TextbookQuaternionDefinition(TeacherStudentsScene):
    CONFIG = {
        "random_seed": 1,
    }

    def construct(self):
        equation = TexMobject(
            """
            (w_1 + x_1 i + y_1 j + z_1 k)
            (w_2 + x_2 i + y_2 j + z_2 k) =
            &(w_1 w_2 - x_1 x_2 - y_1 y_2 - z_1 z_2) \\, +\\\\
            &(w_1 x_2 + x_1 w_2 + y_1 z_2 - z_1 y_2)i \\, +\\\\
            &(w_1 y_2 + y_1 w_2 + z_1 x_2 - x_1 z_2)j \\, +\\\\
            &(w_1 z_2 + z_1 w_2 + x_1 y_2 - y_1 x_2)k \\\\
            """,
            tex_to_color_map={
                "w_1": YELLOW,
                "w_2": YELLOW,
                "x_1": GREEN,
                "x_2": GREEN,
                "y_1": RED,
                "y_2": RED,
                "z_1": BLUE,
                "z_2": BLUE,
            }
        )
        equation.set_width(FRAME_WIDTH - 1)
        equation.to_edge(UP)

        defining_products = VGroup(*[
            TexMobject(
                tex,
                tex_to_color_map={
                    "i": GREEN,
                    "j": RED,
                    "k": BLUE,
                }
            )
            for tex in [
                "i^2 = j^2 = k^2 = -1",
                "ij = -ji = k",
                "ki = -ik = j",
                "jk = -kj = i",
            ]
        ])
        defining_products.arrange_submobjects(DOWN)
        defining_products.next_to(self.students, UP, LARGE_BUFF)
        def_rect = SurroundingRectangle(defining_products)

        self.play(
            LaggedStart(FadeInFromDown, defining_products),
            self.get_student_changes(*3 * ["confused"]),
            self.teacher.change, "raise_right_hand",
        )
        self.play(ShowCreation(def_rect))
        self.play(
            Write(equation, run_time=4, lag_ratio=0.2),
            self.get_student_changes(
                "horrified", "pleading", "sick",
                equation
            ),
            self.teacher.change, "erm", equation,
        )
        self.blink()
        self.look_at(equation.get_corner(UL))
        self.blink()
        self.look_at(equation.get_corner(UR))
        self.wait(2)


class ProblemsWhereComplexNumbersArise(Scene):
    def construct(self):
        text = "Problems where complex numbers are surprisingly useful"
        title = TextMobject(*text.split(" "))
        title.to_edge(UP)
        title.set_color(BLUE)
        underline = Line(LEFT, RIGHT)
        underline.set_width(title.get_width() + 0.5)
        underline.next_to(title, DOWN)

        problems = VGroup(
            TextMobject(
                "Integer solutions to\\\\ $a^2 + b^2 = c^2$",
                alignment=""
            ),
            TextMobject(
                "Understanding\\\\",
                "$1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\cdots" +
                "=\\frac{\\pi}{4}$",
                alignment="",
            ),
            TextMobject("Frequency analysis")
        )
        problems.arrange_submobjects(
            DOWN, buff=LARGE_BUFF, aligned_edge=LEFT
        )
        for problem in problems:
            problems.add(Dot().next_to(problem[0], LEFT))
        problems.next_to(underline, DOWN, buff=MED_LARGE_BUFF)
        problems.to_edge(LEFT)
        v_dots = TexMobject("\\vdots")
        v_dots.scale(2)
        v_dots.next_to(problems, DOWN, aligned_edge=LEFT)

        self.add(problems, v_dots)
        self.play(
            ShowCreation(underline),
            LaggedStart(FadeInFromDown, title, lag_ratio=0.5),
            run_time=3
        )
        self.wait()


class WalkThroughComplexMultiplication(ShowComplexMultiplicationExamples):
    CONFIG = {
        "z": complex(2, 3),
        "w": complex(1, -1),
        "z_color": YELLOW,
        "w_color": PINK,
        "product_color": RED,
    }

    def construct(self):
        self.add_planes(include_title=False)
        self.introduce_z_and_w()
        self.show_action_on_all_complex_numbers()

    def introduce_z_and_w(self):
        # Tolerating code repetition here in case I want
        # to specialize behavior for z or w...
        plane = self.plane
        origin = plane.number_to_point(0)

        z_point = plane.number_to_point(self.z)
        z_dot = Dot(z_point)
        z_line = Line(origin, z_point)
        z_label = VGroup(
            TexMobject("z="),
            DecimalNumber(self.z, num_decimal_places=0)
        )
        z_label.arrange_submobjects(
            RIGHT, buff=SMALL_BUFF,
        )
        z_label.next_to(z_dot, UP, buff=SMALL_BUFF)
        z_label.set_color(self.z_color)
        z_label.add_background_rectangle()
        VGroup(z_line, z_dot).set_color(self.z_color)

        w_point = plane.number_to_point(self.w)
        w_dot = Dot(w_point)
        w_line = Line(origin, w_point)
        w_label = VGroup(
            TexMobject("w="),
            DecimalNumber(self.w, num_decimal_places=0)
        )
        w_label.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
        w_label.next_to(w_dot, DOWN, buff=SMALL_BUFF)
        w_label.set_color(self.w_color)
        w_label.add_background_rectangle()
        VGroup(w_line, w_dot).set_color(self.w_color)

        VGroup(
            z_label[1], w_label[1]
        ).shift(0.25 * SMALL_BUFF * DOWN)

        product = TexMobject("z", "\\cdot", "w")
        z_sym, w_sym = product[0], product[2]
        z_sym.set_color(self.z_color)
        w_sym.set_color(self.w_color)
        product.scale(2)
        product.to_corner(UL)
        product.add_background_rectangle()

        self.add(
            z_line, z_label,
            w_line, w_label,
        )
        self.play(LaggedStart(
            FadeInFromLarge, VGroup(z_dot, w_dot),
            lambda m: (m, 5),
            lag_ratio=0.8,
            run_time=1.5
        ))
        self.play(
            ReplacementTransform(z_label[1][0].copy(), z_sym)
        )
        self.add(product[:-1])
        self.play(
            ReplacementTransform(w_label[1][0].copy(), w_sym),
            FadeInAndShiftFromDirection(product[2], LEFT),
            FadeIn(product[0]),
        )
        self.wait()

        self.set_variables_as_attrs(
            product,
            z_point, w_point,
            z_dot, w_dot,
            z_line, w_line,
        )

    def show_action_on_all_complex_numbers(self):
        plane = self.plane
        plane.save_state()
        origin = plane.number_to_point(0)
        z = self.z
        angle = np.log(z).imag
        product_tex = self.product[1:]
        z_sym, cdot, w_sym = product_tex

        product = self.z * self.w
        product_point = plane.number_to_point(product)
        product_dot = Dot(product_point)
        product_line = Line(origin, product_point)
        for mob in product_line, product_dot:
            mob.set_color(self.product_color)

        rect = SurroundingRectangle(VGroup(z_sym, cdot))
        rect.set_fill(BLACK, opacity=1)
        func_words = TextMobject("Function on the plane")
        func_words.next_to(
            rect, DOWN,
            buff=MED_SMALL_BUFF,
            aligned_edge=LEFT,
        )
        func_words.set_color(self.z_color)

        sparkly_plane = VGroup()
        for line in plane.family_members_with_points():
            if self.camera.is_in_frame(line):
                for piece in line.get_pieces(10):
                    p1, p2 = piece.get_pieces(2)
                    p1.rotate(PI)
                    pair = VGroup(p1, p2)
                    pair.scale(0.3)
                    sparkly_plane.add(pair)
        sparkly_plane.sort_submobjects(
            lambda p: 0.1 * get_norm(p) + random.random()
        )
        sparkly_plane.set_color_by_gradient(YELLOW, RED, PINK, BLUE)
        sparkly_plane.set_stroke(width=4)

        pin = PushPin()
        pin.move_to(origin, DR)

        one_dot = Dot(plane.number_to_point(1))
        one_dot.set_fill(WHITE)
        one_dot.set_stroke(BLACK, 1)
        hand = Hand()
        hand.move_to(plane.number_to_point(1), LEFT)

        zero_eq = TexMobject("z \\cdot 0 = 0")
        one_eq = TexMobject("z \\cdot 1 = z")
        equations = VGroup(zero_eq, one_eq)
        equations.arrange_submobjects(DOWN)
        equations.scale(1.5)
        for eq in equations:
            eq.add_background_rectangle()
        equations.next_to(func_words, DOWN)
        equations.to_edge(LEFT)

        product_label = VGroup(
            TexMobject("z \\cdot w ="),
            DecimalNumber(product, num_decimal_places=0)
        )
        product_label.arrange_submobjects(RIGHT)
        product_label[0].shift(0.025 * DOWN)
        product_label.next_to(product_dot, UP, SMALL_BUFF)
        product_label.add_background_rectangle()

        big_rect = Rectangle(
            height=4,
            width=6,
            fill_color=BLACK,
            fill_opacity=0.9,
            stroke_width=0,
        )
        big_rect.to_corner(UL, buff=0)

        self.add(big_rect, product_tex, rect, z_sym, cdot)
        self.play(
            FadeIn(big_rect),
            ShowCreation(rect),
            Write(func_words),
            run_time=1
        )
        self.play(
            ReplacementTransform(
                self.w_line.copy(), product_line,
            ),
            ReplacementTransform(
                self.w_dot.copy(), product_dot,
            ),
            path_arc=angle,
            run_time=3
        )
        self.wait()
        self.play(FadeOut(VGroup(product_line, product_dot)))
        self.play(LaggedStart(
            ShowCreationThenDestruction, sparkly_plane,
            lag_ratio=0.5,
            run_time=2
        ))
        self.play(
            plane.apply_complex_function, lambda w: z * w,
            Transform(self.w_line, product_line),
            Transform(self.w_dot, product_dot),
            path_arc=angle,
            run_time=9,
            rate_func=lambda t: there_and_back_with_pause(t, 2 / 9)
        )
        self.wait()
        self.play(FadeInFrom(pin, UL))
        self.play(Write(zero_eq))
        self.play(
            FadeInFromLarge(one_dot),
            FadeInFrom(hand, UR)
        )
        self.play(Write(one_eq))
        self.wait()
        self.play(
            plane.apply_complex_function, lambda w: z * w,
            ReplacementTransform(self.w_line.copy(), product_line),
            ReplacementTransform(self.w_dot.copy(), product_dot),
            one_dot.move_to, self.z_point,
            hand.move_to, self.z_point, LEFT,
            path_arc=angle,
            run_time=4,
        )
        self.play(Write(product_label))


class ShowUnitCircleActions(ShowComplexMultiplicationExamples):
    CONFIG = {
        "random_seed": 0,
        "plane_config": {
            "secondary_line_ratio": 0,
        }
    }

    def construct(self):
        self.add_planes(include_title=False)
        self.show_unit_circle_actions()

    def show_unit_circle_actions(self):
        plane = self.plane
        origin = plane.number_to_point(0)
        one_point = plane.number_to_point(1)
        one_dot = Dot(one_point)
        one_dot.set_fill(WHITE)
        one_dot.set_stroke(BLACK, 1)
        plane.add(one_dot)

        pin = PushPin()
        pin.move_to(origin, DR)
        hand = Hand()
        update_hand = UpdateFromFunc(
            hand, lambda m: m.move_to(one_dot.get_center(), LEFT)
        )

        circle = Circle(
            color=YELLOW,
            radius=get_norm(one_point - origin)
        )

        self.add(circle)
        self.add(pin, one_dot)
        self.add_foreground_mobjects(hand)

        title = TextMobject(
            "Numbers on the unit circle",
            "$\\rightarrow$", "pure rotation."
        )
        title.set_width(FRAME_WIDTH - 1)
        title.to_edge(UP, buff=MED_SMALL_BUFF)
        title.add_background_rectangle(buff=SMALL_BUFF)
        self.add_foreground_mobjects(title)
        self.background_plane.coordinate_labels.submobjects.pop(-1)

        n_angles = 12
        angles = list(np.linspace(-PI, PI, n_angles + 2)[1:-1])
        random.shuffle(angles)

        for angle in angles:
            plane.save_state()
            temp_plane = plane.copy()

            z = np.exp(complex(0, angle))
            if angle is angles[0]:
                z_label = DecimalNumber(
                    z, num_decimal_places=2,
                )
                z_label.set_stroke(BLACK, width=11, background=True)
                z_label_rect = BackgroundRectangle(z_label)
                z_label_rect.set_fill(opacity=0)
            z_point = plane.number_to_point(z)
            z_arrow = Arrow(2.5 * z_point, z_point, buff=SMALL_BUFF)
            z_dot = Dot(z_point)
            z_label_start_center = z_label.get_center()
            z_label.next_to(
                z_arrow.get_start(),
                np.sign(z_arrow.get_start()[1]) * UP,
            )
            z_label_end_center = z_label.get_center()
            z_group = VGroup(z_arrow, z_dot, z_label)
            z_group.set_color(GREEN)
            z_group.add_to_back(z_label_rect)
            z_arrow.set_stroke(BLACK, 1)
            z_dot.set_stroke(BLACK, 1)

            if angle is angles[0]:
                self.play(
                    FadeInFromDown(z_label_rect),
                    FadeInFromDown(z_label),
                    GrowArrow(z_arrow),
                    FadeInFromLarge(z_dot),
                )
            else:
                alpha_tracker = ValueTracker(0)
                self.play(
                    ReplacementTransform(old_z_dot, z_dot),
                    ReplacementTransform(old_z_arrow, z_arrow),
                    UpdateFromFunc(
                        z_label_rect,
                        lambda m: m.replace(z_label)
                    ),
                    ChangeDecimalToValue(
                        z_label, z,
                        position_update_func=lambda m: m.move_to(
                            interpolate(
                                z_label_start_center,
                                z_label_end_center,
                                alpha_tracker.get_value()
                            )
                        )
                    ),
                    alpha_tracker.set_value, 1,
                    # hand.move_to, one_point, LEFT
                )
            old_z_dot = z_dot
            old_z_arrow = z_arrow
            VGroup(old_z_arrow, old_z_dot)
            self.play(
                Rotate(plane, angle, run_time=2),
                update_hand,
                Animation(z_group),
            )
            self.wait()
            self.add(temp_plane, z_group)
            self.play(
                FadeOut(plane),
                FadeOut(hand),
                FadeIn(temp_plane),
            )
            plane.restore()
            self.remove(temp_plane)
            self.add(plane, *z_group)


class IfYouNeedAWarmUp(TeacherStudentsScene):
    def construct(self):
        screen = self.screen
        screen.set_height(4)
        screen.to_corner(UL)
        self.add(screen)

        self.teacher_says(
            "If you need \\\\ a warm up",
            bubble_kwargs={"width": 3.5, "height": 3},
        )
        self.change_all_student_modes(
            "pondering", look_at_arg=screen,
        )
        self.wait(3)
        self.play(RemovePiCreatureBubble(self.teacher))
        self.wait(3)


class LinusThinksAboutStretching(Scene):
    def construct(self):
        linus = Linus()
        top_line = NumberLine(color=GREY)
        top_line.to_edge(UP)
        top_line.add_numbers()

        linus.move_to(3 * LEFT + DOWN)

        self.add(linus, top_line)

        scalars = [3, 0.5, 2]

        for scalar in scalars:
            lower_line = NumberLine(
                x_min=-14,
                x_max=14,
                color=BLUE
            )
            lower_line.next_to(top_line, DOWN, MED_LARGE_BUFF)
            lower_line.numbers = lower_line.get_number_mobjects()
            for number in lower_line.numbers:
                number.add_updater(lambda m: m.next_to(
                    lower_line.number_to_point(m.get_value()),
                    DOWN, MED_SMALL_BUFF,
                ))
            lower_line.save_state()
            lower_line.numbers.save_state()
            self.add(lower_line, *lower_line.numbers)

            words = TextMobject("Multiply by {}".format(scalar))
            words.next_to(lower_line.numbers, DOWN)

            self.play(
                ApplyMethod(
                    lower_line.stretch, scalar, 0,
                    run_time=2
                ),
                # LaggedStart(FadeIn, words, run_time=1),
                FadeInFromLarge(words, 1.0 / scalar),
                linus.look_at, top_line.number_to_point(scalar)
            )
            self.play(Blink(linus))
            self.play(
                FadeOut(lower_line),
                FadeOut(lower_line.numbers),
                FadeOut(words),
                FadeIn(lower_line.saved_state, remover=True),
                FadeIn(lower_line.numbers.saved_state, remover=True),
                linus.look_at, top_line.number_to_point(0)
            )
        self.play(linus.change, "confused", DOWN + RIGHT)
        self.wait(2)
        self.play(Blink(linus))
        self.wait(2)


class LinusReactions(Scene):
    def construct(self):
        linus = Linus()
        for mode in "confused", "sad", "erm", "angry", "pleading":
            self.play(linus.change, mode, 2 * RIGHT)
            self.wait()
            self.play(Blink(linus))
            self.wait()


class OneDegreeOfFreedomForRotation(Scene):
    def construct(self):
        circle = CheckeredCircle(radius=2, stroke_width=10)
        r_line = Line(circle.get_center(), circle.get_right())
        moving_r_line = r_line.copy()
        right_dot = Dot(color=WHITE)
        right_dot.move_to(circle.get_right())
        circle.add(moving_r_line, right_dot)

        def get_angle():
            return moving_r_line.get_angle() % TAU

        angle_label = Integer(0, unit="^\\circ")
        angle_label.scale(2)
        angle_label.add_updater(
            lambda m: m.set_value(get_angle() / DEGREES)
        )
        angle_label.add_updater(
            lambda m: m.next_to(circle, UP, MED_LARGE_BUFF)
        )

        def get_arc():
            return Arc(
                angle=get_angle(),
                radius=0.5,
                color=LIGHT_GREY,
            )

        arc = get_arc()
        arc.add_updater(lambda m: m.become(get_arc()))

        self.add(circle, r_line, angle_label, arc)
        angles = np.random.uniform(-TAU, TAU, 10)
        for angle in angles:
            self.play(Rotate(circle, angle, run_time=2))


class StereographicProjectionTitle(Scene):
    def construct(self):
        title = TextMobject("Stereographic projection")
        final_title = title.copy()
        final_title.set_width(10)
        final_title.to_edge(UP)

        title.rotate(-90 * DEGREES)
        title.next_to(RIGHT, RIGHT, SMALL_BUFF)
        title.apply_complex_function(np.exp)
        title.rotate(90 * DEGREES)
        title.set_height(6)
        title.to_edge(UP)

        self.play(Write(title))
        self.play(Transform(title, final_title, run_time=2))
        self.wait()


class IntroduceStereographicProjection(MovingCameraScene):
    CONFIG = {
        "n_sample_points": 16,
        "circle_config": {
            "n_pieces": 16,
            "radius": 2,
            "stroke_width": 7,
        },
        "example_angles": [
            30 * DEGREES,
            120 * DEGREES,
            240 * DEGREES,
            80 * DEGREES,
            -60 * DEGREES,
            135 * DEGREES,
        ]
    }

    def construct(self):
        self.setup_plane()
        self.draw_lines()
        self.describe_individual_points()
        self.remind_that_most_points_are_not_projected()

    def setup_plane(self):
        self.plane = self.get_plane()
        self.circle = self.get_circle()
        self.circle_shadow = self.get_circle_shadow()

        self.add(self.plane)
        self.add(self.circle_shadow)
        self.add(self.circle)

    def draw_lines(self):
        plane = self.plane
        circle = self.circle
        circle.save_state()
        circle.generate_target()
        self.project_mobject(circle.target)

        circle_points = self.get_sample_circle_points()
        dots = VGroup(*[Dot(p) for p in circle_points])
        dots.set_sheen(-0.2, DR)
        dots.set_stroke(DARK_GREY, 2, background=True)
        arrows = VGroup()
        for dot in dots:
            dot.scale(0.75)
            dot.generate_target()
            dot.target.move_to(
                self.project(dot.get_center())
            )
            arrow = Arrow(
                dot.get_center(),
                dot.target.get_center(),
            )
            arrows.add(arrow)
        neg_one_point = plane.number_to_point(-1)
        neg_one_dot = Dot(neg_one_point)
        neg_one_dot.set_fill(YELLOW)

        lines = self.get_lines()

        special_index = self.n_sample_points // 2 + 1
        line = lines[special_index]
        dot = dots[special_index]
        arrow = arrows[special_index]
        dot_target_outline = dot.target.copy()
        dot_target_outline.set_stroke(RED, 2)
        dot_target_outline.set_fill(opacity=0)
        dot_target_outline.scale(1.5)

        v_line = Line(UP, DOWN)
        v_line.set_height(FRAME_HEIGHT)
        v_line.set_stroke(RED, 5)

        self.play(LaggedStart(FadeInFromLarge, dots))
        self.play(FadeInFromLarge(neg_one_dot))
        self.add(lines, neg_one_dot, dots)
        self.play(LaggedStart(ShowCreation, lines))
        self.wait()
        self.play(
            lines.set_stroke, {"width": 0.5},
            line.set_stroke, {"width": 4},
        )
        self.play(ShowCreation(dot_target_outline))
        self.play(ShowCreationThenDestruction(v_line))
        self.play(MoveToTarget(dot))
        self.wait()
        self.play(
            lines.set_stroke, {"width": 1},
            FadeOut(dot_target_outline),
            MoveToTarget(circle),
            *map(MoveToTarget, dots),
            run_time=2,
        )
        self.wait()

        self.lines = lines
        self.dots = dots

    def describe_individual_points(self):
        plane = self.plane
        one_point, zero_point, i_point, neg_i_point, neg_one_point = [
            plane.number_to_point(n)
            for n in [1, 0, complex(0, 1), complex(0, -1), -1]
        ]
        i_pin = PushPin()
        i_pin.pin_to(i_point)
        neg_i_pin = PushPin()
        neg_i_pin.pin_to(neg_i_point)

        dot = Dot()
        dot.set_stroke(RED, 3)
        dot.set_fill(opacity=0)
        dot.scale(1.5)
        dot.move_to(one_point)

        arc1 = Arc(angle=TAU / 4, radius=2)
        arc2 = Arc(
            angle=85 * DEGREES, radius=2,
            start_angle=TAU / 4,
        )
        arc3 = Arc(
            angle=-85 * DEGREES, radius=2,
            start_angle=-TAU / 4,
        )
        VGroup(arc1, arc2, arc3).set_stroke(RED)

        frame = self.camera_frame
        frame_height_tracker = ValueTracker(frame.get_height())
        frame_height_growth = ContinualGrowValue(
            frame_height_tracker, rate=0.4
        )

        neg_one_tangent = VGroup(
            Line(ORIGIN, UP),
            Line(ORIGIN, DOWN),
        )
        neg_one_tangent.set_height(25)
        neg_one_tangent.set_stroke(YELLOW, 5)
        neg_one_tangent.move_to(neg_one_point)

        self.play(ShowCreation(dot))
        self.wait()
        self.play(dot.move_to, zero_point)
        self.wait()
        dot.move_to(i_point)
        self.play(ShowCreation(dot))
        self.play(FadeInFrom(i_pin, UL))
        self.wait()
        self.play(
            dot.move_to, neg_i_point,
            path_arc=-60 * DEGREES
        )
        self.play(FadeInFrom(neg_i_pin, UL))
        self.wait()
        self.play(
            dot.move_to, one_point,
            path_arc=-60 * DEGREES
        )
        frame.add_updater(
            lambda m: m.set_height(frame_height_tracker.get_value())
        )

        triplets = [
            (arc1, i_point, TAU / 4),
            (arc2, neg_one_point, TAU / 4),
            (arc3, neg_one_point, -TAU / 4),
        ]
        for arc, point, path_arc in triplets:
            self.play(
                ShowCreation(arc),
                dot.move_to, point, path_arc=path_arc,
                run_time=2
            )
            self.wait()
            self.play(
                ApplyFunction(self.project_mobject, arc, run_time=2)
            )
            self.wait()
            self.play(FadeOut(arc))
            self.wait()
            if arc is arc1:
                self.add(frame, frame_height_growth)
            elif arc is arc2:
                self.play(dot.move_to, neg_i_point)
        frame_height_growth.begin_wind_down()
        self.wait(2)
        self.play(*map(ShowCreation, neg_one_tangent))
        self.wait()
        self.play(FadeOut(neg_one_tangent))
        self.wait(2)
        frame.clear_updaters()
        self.play(
            frame.set_height, FRAME_HEIGHT,
            self.lines.set_stroke, {"width": 0.5},
            FadeOut(self.dots),
            FadeOut(dot),
            run_time=2,
        )

    def remind_that_most_points_are_not_projected(self):
        plane = self.plane
        circle = self.circle

        sample_values = [0, complex(1, 1), complex(-2, -1)]
        sample_points = [
            plane.number_to_point(value)
            for value in sample_values
        ]
        sample_dots = VGroup(*[Dot(p) for p in sample_points])
        sample_dots.set_fill(GREEN)

        self.play(
            FadeOut(self.lines),
            Restore(circle),
        )

        for value, dot in zip(sample_values, sample_dots):
            cross = Cross(dot)
            cross.scale(2)
            label = Integer(value)
            label.next_to(dot, UR, SMALL_BUFF)
            self.play(
                FadeInFromLarge(dot, 3),
                FadeInFromDown(label)
            )
            self.play(ShowCreation(cross))
            self.play(*map(FadeOut, [dot, cross, label]))
        self.wait()
        self.play(
            FadeIn(self.lines),
            MoveToTarget(circle, run_time=2),
        )
        self.wait()

    # Helpers
    def get_plane(self):
        plane = ComplexPlane(
            unit_size=2,
            color=GREY,
            secondary_color=DARK_GREY,
            x_radius=FRAME_WIDTH,
            y_radius=FRAME_HEIGHT,
            stroke_width=2,
        )
        plane.add_coordinates()
        return plane

    def get_circle(self):
        circle = CheckeredCircle(
            **self.circle_config
        )
        circle.set_stroke(width=7)
        return circle

    def get_circle_shadow(self):
        circle_shadow = CheckeredCircle(
            **self.circle_config
        )
        circle_shadow.set_stroke(opacity=0.65)
        return circle_shadow

    def get_sample_circle_points(self):
        plane = self.plane
        n = self.n_sample_points
        numbers = [
            np.exp(complex(0, TAU * x / n))
            for x in range(-(n // 2) + 1, n // 2)
        ]
        return [
            plane.number_to_point(number)
            for number in numbers
        ]

    def get_lines(self):
        plane = self.plane
        neg_one_point = plane.number_to_point(-1)
        circle_points = self.get_sample_circle_points()

        lines = VGroup(*[
            Line(neg_one_point, point)
            for point in circle_points
        ])
        for line in lines:
            line.scale(
                20 / line.get_length(),
                about_point=neg_one_point
            )
            line.set_stroke(YELLOW, 1)
        return lines

    def project(self, point):
        return stereo_project_point(point, axis=0, r=2)

    def project_mobject(self, mobject):
        return stereo_project(mobject, axis=0, r=2, outer_r=6)


class IntroduceStereographicProjectionLinusView(IntroduceStereographicProjection):
    def construct(self):
        self.describe_individual_points()
        self.point_at_infinity()
        self.show_90_degree_rotation()
        self.talk_through_90_degree_rotation()
        self.show_four_rotations()
        self.show_example_angles()

    def describe_individual_points(self):
        plane = self.plane = self.get_plane()
        circle = self.circle = self.get_circle()
        linus = self.linus = self.get_linus()

        angles = np.arange(-135, 180, 45) * DEGREES
        sample_numbers = [
            np.exp(complex(0, angle))
            for angle in angles
        ]
        sample_points = [
            plane.number_to_point(number)
            for number in sample_numbers
        ]
        projected_sample_points = [
            self.project(point)
            for point in sample_points
        ]
        dots = VGroup(*[Dot() for x in range(8)])
        dots.set_fill(WHITE)
        dots.set_stroke(BLACK, 1)

        def generate_dot_updater(circle_piece):
            return lambda d: d.move_to(circle_piece.points[0])

        for dot, piece in zip(dots, circle[::(len(circle) // 8)]):
            dot.add_updater(generate_dot_updater(piece))

        stot = "(\\sqrt{2} / 2)"
        labels_tex = [
            "-{}-{}i".format(stot, stot),
            "-i",
            "{}-{}i".format(stot, stot),
            "1",
            "{}+{}i".format(stot, stot),
            "i",
            "-{}+{}i".format(stot, stot),
        ]
        labels = VGroup(*[TexMobject(tex) for tex in labels_tex])
        vects = it.cycle([RIGHT, RIGHT])
        arrows = VGroup()
        for label, point, vect in zip(labels, projected_sample_points, vects):
            arrow = Arrow(vect, ORIGIN)
            arrow.next_to(point, vect, 2 * SMALL_BUFF)
            arrows.add(arrow)
            label.set_stroke(width=0, background=True)
            if stot in label.get_tex_string():
                label.set_height(0.5)
            else:
                label.set_height(0.5)
                label.set_stroke(WHITE, 2, background=True)
            label.next_to(arrow, vect, SMALL_BUFF)

        frame = self.camera_frame
        frame.set_height(12)

        self.add(linus)
        self.add(circle, *dots)
        self.play(
            ApplyFunction(self.project_mobject, circle),
            run_time=2
        )
        self.play(linus.change, "confused")
        self.wait()
        for i in [1, 0]:
            self.play(
                LaggedStart(GrowArrow, arrows[i::2]),
                LaggedStart(Write, labels[i::2])
            )
            self.play(Blink(linus))

        self.dots = dots

    def point_at_infinity(self):
        circle = self.circle
        linus = self.linus

        label = TextMobject(
            "$-1$ is \\\\ at $\\pm \\infty$"
        )
        label.scale(1.5)
        label.next_to(circle, LEFT, buff=1.25)
        arrows = VGroup(*[
            Vector(3 * v + 0.0 * RIGHT).next_to(label, v, buff=MED_LARGE_BUFF)
            for v in [UP, DOWN]
        ])
        arrows.set_color(YELLOW)

        self.play(
            Write(label),
            linus.change, "awe", label,
            *map(GrowArrow, arrows)
        )

        self.neg_one_label = VGroup(label, arrows)

    def show_90_degree_rotation(self):
        angle_tracker = ValueTracker(0)
        circle = self.circle
        linus = self.linus
        hand = Hand()
        hand.flip()
        one_dot = self.dots[0]
        hand.add_updater(
            lambda h: h.move_to(one_dot.get_center(), RIGHT)
        )

        def update_circle(circle):
            angle = angle_tracker.get_value()
            new_circle = self.get_circle()
            new_circle.rotate(angle)
            self.project_mobject(new_circle)
            circle.become(new_circle)

        circle.add_updater(update_circle)

        self.play(
            FadeIn(hand),
            one_dot.set_fill, RED,
        )
        for angle in 90 * DEGREES, 0:
            self.play(
                ApplyMethod(
                    angle_tracker.set_value, angle,
                    run_time=3,
                ),
                linus.change, "confused", hand
            )
            self.wait()
            self.play(Blink(linus))

        self.hand = hand
        self.angle_tracker = angle_tracker

    def talk_through_90_degree_rotation(self):
        linus = self.linus
        dots = self.dots
        one_dot = dots[0]
        i_dot = dots[2]
        neg_i_dot = dots[-2]

        kwargs1 = {
            "use_rectangular_stem": False,
            "path_arc": -90 * DEGREES,
            "buff": SMALL_BUFF,
        }
        kwargs2 = dict(kwargs1)
        kwargs2["path_arc"] = -40 * DEGREES
        arrows = VGroup(
            Arrow(one_dot, i_dot, **kwargs1),
            Arrow(i_dot, 6 * UP + LEFT, **kwargs2),
            Arrow(6 * DOWN + LEFT, neg_i_dot, **kwargs2),
            Arrow(neg_i_dot, one_dot, **kwargs1)
        )
        arrows.set_stroke(WHITE, 3)
        one_to_i, i_to_neg_1, neg_one_to_neg_i, neg_i_to_one = arrows

        for arrow in arrows:
            self.play(
                ShowCreation(arrow),
                linus.look_at, arrow
            )
            self.wait(2)

        self.arrows = arrows

    def show_four_rotations(self):
        angle_tracker = self.angle_tracker
        linus = self.linus
        hand = self.hand
        linus.add_updater(lambda l: l.look_at(hand))
        linus.add_updater(lambda l: l.eyes.next_to(l.body, UP, 0))

        for angle in np.arange(TAU / 4, 5 * TAU / 4, TAU / 4):
            self.play(
                ApplyMethod(
                    angle_tracker.set_value, angle,
                    run_time=3,
                ),
            )
            self.wait()
        self.play(FadeOut(self.arrows))

    def show_example_angles(self):
        angle_tracker = self.angle_tracker
        angle_tracker.set_value(0)

        for angle in self.example_angles:
            self.play(
                ApplyMethod(
                    angle_tracker.set_value, angle,
                    run_time=4,
                ),
            )
            self.wait()

    #
    def get_linus(self):
        linus = Linus()
        linus.move_to(3 * RIGHT)
        linus.to_edge(DOWN)
        linus.look_at(ORIGIN)
        return linus


class ShowRotationUnderStereographicProjection(IntroduceStereographicProjection):
    def construct(self):
        self.setup_plane()
        self.apply_projection()
        self.show_90_degree_rotation()
        self.talk_through_90_degree_rotation()
        self.show_four_rotations()
        self.show_example_angles()

    def apply_projection(self):
        plane = self.plane
        circle = self.circle
        neg_one_point = plane.number_to_point(-1)
        neg_one_dot = Dot(neg_one_point)
        neg_one_dot.set_fill(YELLOW)

        lines = self.get_lines()

        def generate_dot_updater(circle_piece):
            return lambda d: d.move_to(circle_piece.points[0])

        for circ, color in [(self.circle_shadow, RED), (self.circle, WHITE)]:
            for piece in circ[::(len(circ) // 8)]:
                dot = Dot(color=color)
                dot.set_fill(opacity=circ.get_stroke_opacity())
                dot.add_updater(generate_dot_updater(piece))
                self.add(dot)

        self.add(lines, neg_one_dot)
        self.play(*map(ShowCreation, lines))
        self.play(
            ApplyFunction(self.project_mobject, circle),
            lines.set_stroke, {"width": 0.5},
            run_time=2
        )
        self.play(
            self.camera_frame.set_height, 12,
            run_time=2
        )
        self.wait()

    def show_90_degree_rotation(self):
        circle = self.circle
        circle_shadow = self.circle_shadow

        def get_rotated_one_point():
            return circle_shadow[0].points[0]

        def get_angle():
            return angle_of_vector(get_rotated_one_point())

        self.get_angle = get_angle

        one_dot = Dot(color=RED)
        one_dot.add_updater(
            lambda m: m.move_to(get_rotated_one_point())
        )
        hand = Hand()
        hand.move_to(one_dot.get_center(), LEFT)

        def update_circle(circle):
            new_circle = self.get_circle()
            new_circle.rotate(get_angle())
            self.project_mobject(new_circle)
            circle.become(new_circle)

        circle.add_updater(update_circle)

        self.add(one_dot, hand)
        hand.add_updater(
            lambda h: h.move_to(one_dot.get_center(), LEFT)
        )
        self.play(
            FadeInFrom(hand, RIGHT),
            FadeInFromLarge(one_dot, 3),
        )
        for angle in 90 * DEGREES, -90 * DEGREES:
            self.play(
                Rotate(circle_shadow, angle, run_time=3),
            )
            self.wait(2)

    def talk_through_90_degree_rotation(self):
        plane = self.plane
        points = [
            plane.number_to_point(z)
            for z in [1, complex(0, 1), -1, complex(0, -1)]
        ]
        arrows = VGroup()
        for p1, p2 in adjacent_pairs(points):
            arrow = Arrow(
                p1, p2, path_arc=180 * DEGREES,
                use_rectangular_stem=False,
            )
            arrow.set_stroke(LIGHT_GREY, width=3)
            arrow.tip.set_fill(LIGHT_GREY)
            arrows.add(arrow)

        for arrow in arrows:
            self.play(ShowCreation(arrow))
            self.wait(2)

        self.arrows = arrows

    def show_four_rotations(self):
        circle_shadow = self.circle_shadow
        for x in range(4):
            self.play(
                Rotate(circle_shadow, TAU / 4, run_time=3)
            )
            self.wait()
        self.play(FadeOut(self.arrows))

    def show_example_angles(self):
        circle_shadow = self.circle_shadow
        angle_label = Integer(0, unit="^\\circ")
        angle_label.scale(1.5)
        angle_label.next_to(
            circle_shadow.get_top(), UR,
        )

        self.play(FadeInFromDown(angle_label))
        self.add(angle_label)
        for angle in self.example_angles:
            d_angle = angle - self.get_angle()
            self.play(
                Rotate(circle_shadow, d_angle),
                ChangingDecimal(
                    angle_label,
                    lambda a: (self.get_angle() % TAU) / DEGREES
                ),
                run_time=4
            )
            self.wait()


class IntroduceFelix(PiCreatureScene, SpecialThreeDScene):
    def setup(self):
        PiCreatureScene.setup(self)
        SpecialThreeDScene.setup(self)

    def construct(self):
        self.introduce_felix()
        self.add_plane()
        self.show_in_three_d()

    def introduce_felix(self):
        felix = self.felix = self.pi_creature

        arrow = Vector(DL, color=WHITE)
        arrow.next_to(felix, UR)

        label = TextMobject("Felix the Flatlander")
        label.next_to(arrow.get_start(), UP)

        self.add(felix)
        self.play(
            felix.change, "wave_1", label,
            Write(label),
            GrowArrow(arrow),
        )
        self.play(Blink(felix))
        self.play(felix.change, "thinking", label)

        self.to_fade = VGroup(label, arrow)

    def add_plane(self):
        plane = NumberPlane(y_radius=10)
        axes = self.get_axes()
        to_fade = self.to_fade
        felix = self.felix

        self.add(axes, plane, felix)
        self.play(
            ShowCreation(axes),
            ShowCreation(plane),
            FadeOut(to_fade),
        )
        self.wait()

        self.plane = plane
        self.axes = axes

    def show_in_three_d(self):
        felix = self.pi_creature
        plane = self.plane
        axes = self.axes

        # back_plane = Rectangle().replace(plane, stretch=True)
        # back_plane.shade_in_3d = True
        # back_plane.set_fill(LIGHT_GREY, opacity=0.5)
        # back_plane.set_sheen(1, UL)
        # back_plane.shift(SMALL_BUFF * IN)
        # back_plane.set_stroke(width=0)
        # back_plane = ParametricSurface(
        #     lambda u, v: u * RIGHT + v * UP
        # )
        # back_plane.replace(plane, stretch=True)
        # back_plane.set_stroke(width=0)
        # back_plane.set_fill(LIGHT_GREY, opacity=0.5)

        sphere = self.get_sphere()
        # sphere.set_fill(BLUE_E, 0.5)

        self.move_camera(
            phi=70 * DEGREES,
            theta=-110 * DEGREES,
            added_anims=[FadeOut(plane)],
            run_time=2
        )
        self.begin_ambient_camera_rotation()
        self.add(axes, sphere)
        self.play(
            Write(sphere),
            felix.change, "confused"
        )
        self.wait()

        axis_angle_pairs = [
            (RIGHT, 90 * DEGREES),
            (OUT, 45 * DEGREES),
            (UR + OUT, 120 * DEGREES),
            (RIGHT, 90 * DEGREES),
        ]
        for axis, angle in axis_angle_pairs:
            self.play(Rotate(
                sphere, angle=angle, axis=axis,
                run_time=2,
            ))
        self.wait(2)

    #
    def create_pi_creature(self):
        return Felix().move_to(4 * LEFT + 2 * DOWN)


class IntroduceThreeDNumbers(SpecialThreeDScene):
    CONFIG = {
        "camera_config": {
            "exponential_projection": False,
        }
    }

    def construct(self):
        self.add_third_axis()
        self.reorient_axes()
        self.show_example_number()

    def add_third_axis(self):
        plane = ComplexPlane(
            y_radius=FRAME_WIDTH / 4,
            unit_size=2,
            secondary_line_ratio=1,
        )
        plane.add_coordinates()
        title = TextMobject("Complex Plane")
        title.scale(1.8)
        title.add_background_rectangle()
        title.to_corner(UL, buff=MED_SMALL_BUFF)

        real_line = Line(LEFT, RIGHT).set_width(FRAME_WIDTH)
        imag_line = Line(DOWN, UP).set_height(FRAME_HEIGHT)
        real_line.set_color(YELLOW)
        imag_line.set_color(RED)

        for label in plane.coordinate_labels:
            label.remove(label.background_rectangle)
            label.shift(SMALL_BUFF * IN)
            self.add_fixed_orientation_mobjects(label)
        reals = plane.coordinate_labels[:7]
        imags = plane.coordinate_labels[7:]

        self.add(plane, title)
        for line, group in (real_line, reals), (imag_line, imags):
            line.set_stroke(width=5)
            self.play(
                ShowCreationThenDestruction(line),
                LaggedStart(
                    Indicate, group,
                    rate_func=there_and_back,
                    color=line.get_color(),
                ),
                run_time=2,
            )

        self.plane = plane
        self.title = title

    def reorient_axes(self):
        z_axis = NumberLine(unit_size=2)
        z_axis.rotate(90 * DEGREES, axis=DOWN)
        z_axis.rotate(90 * DEGREES, axis=OUT)
        z_axis.set_color(WHITE)
        z_axis_top = Line(
            z_axis.number_to_point(0),
            z_axis.main_line.get_end(),
        )
        z_axis_top.match_style(z_axis.main_line)

        z_unit_line = Line(
            z_axis.number_to_point(0),
            z_axis.number_to_point(1),
            color=RED,
            stroke_width=5
        )

        j_labels = VGroup(
            TexMobject("-2j"),
            TexMobject("-j"),
            TexMobject("j"),
            TexMobject("2j"),
        )
        for label, num in zip(j_labels, [-2, -1, 1, 2]):
            label.next_to(z_axis.number_to_point(num), RIGHT, MED_SMALL_BUFF)
            self.add_fixed_orientation_mobjects(label)

        plane = self.plane

        x_line = Line(LEFT, RIGHT).set_width(FRAME_WIDTH)
        y_line = Line(DOWN, UP).set_height(FRAME_WIDTH)
        z_line = Line(IN, OUT).set_depth(FRAME_WIDTH)
        x_line.set_stroke(GREEN, 5)
        y_line.set_stroke(RED, 5)
        z_line.set_stroke(YELLOW, 5)
        coord_lines = VGroup(x_line, y_line, z_line)

        self.add(z_axis, plane, z_axis_top)
        self.move_camera(
            phi=70 * DEGREES,
            theta=-80 * DEGREES,
            added_anims=[
                plane.set_stroke, {"opacity": 0.5},
            ],
            run_time=2,
        )
        self.begin_ambient_camera_rotation(rate=0.02)
        self.wait()
        self.play(FadeInFrom(j_labels, IN))
        z_axis.add(j_labels)
        self.play(
            ShowCreationThenDestruction(z_unit_line),
            run_time=2
        )
        self.wait(4)

        group = VGroup(*it.chain(plane.coordinate_labels, j_labels))
        for label in group:
            label.generate_target()
            axis = np.ones(3)
            label.target.rotate_about_origin(-120 * DEGREES, axis=axis)
            label.target.rotate(120 * DEGREES, axis=axis)
        for y, label in zip([-2, -1, 1, 2], j_labels):
            label.target.scale(0.65)
            label.target.next_to(
                2 * y * UP, RIGHT, 2 * SMALL_BUFF
            )
        self.play(
            LaggedStart(MoveToTarget, group, lag_ratio=0.8),
            FadeOut(self.title),
            run_time=3
        )
        self.wait(3)
        for line, wait in zip(coord_lines, [False, True, True]):
            self.play(
                ShowCreationThenDestruction(line),
                run_time=2
            )
            if wait:
                self.wait()

    def show_example_number(self):
        x, y, z = coords = 2 * np.array([1.5, -1, 1.25])
        dot = Sphere(radius=0.05)
        dot.set_fill(LIGHT_GREY)
        dot.move_to(coords)
        point_line = Line(ORIGIN, coords)
        point_line.set_stroke(WHITE, 1)

        z_line = Line(ORIGIN, z * OUT)
        x_line = Line(z_line.get_end(), z_line.get_end() + x * RIGHT)
        y_line = Line(x_line.get_end(), x_line.get_end() + y * UP)

        x_line.set_stroke(GREEN, 5)
        y_line.set_stroke(RED, 5)
        z_line.set_stroke(YELLOW, 5)
        lines = VGroup(z_line, x_line, y_line)

        number_label = TexMobject(
            str(z / 2), "+", str(x / 2), "i", "+", str(y / 2), "j",
            tex_to_color_map={
                str(z / 2): YELLOW,
                str(x / 2): GREEN,
                str(y / 2): RED,
            }
        )
        number_label.next_to(ORIGIN, RIGHT, LARGE_BUFF)
        number_label.to_edge(UP)

        self.add_fixed_in_frame_mobjects(number_label)
        self.play(
            ShowCreation(point_line),
            FadeInFrom(dot, -coords),
            FadeInFromDown(number_label)
        )
        self.wait()
        for num, line in zip([z, x, y], lines):
            tex = number_label.get_part_by_tex(str(num / 2))
            rect = SurroundingRectangle(tex)
            rect.set_color(WHITE)

            self.add_fixed_in_frame_mobjects(rect)
            self.play(
                ShowCreation(line),
                ShowCreationThenDestruction(rect),
                run_time=2
            )
            self.remove_fixed_in_frame_mobjects(rect)
            self.wait()
        self.wait(15)


class MentionImpossibilityOf3dNumbers(TeacherStudentsScene):
    def construct(self):
        equations = VGroup(
            TexMobject("ij = ?"),
            TexMobject("ji = ?"),
        )
        equations.arrange_submobjects(RIGHT, buff=LARGE_BUFF)
        equations.scale(1.5)
        equations.to_edge(UP)
        self.add(equations)

        why = TextMobject("Why not?")
        why.next_to(self.students[1], UP)

        self.teacher_says(
            "Such 3d number \\\\ have no good \\\\ multiplication rule",
            bubble_kwargs={"width": 4, "height": 3},
        )
        self.change_all_student_modes("confused")
        self.wait(2)
        self.play(
            self.students[1].change, "maybe",
            FadeInFromLarge(why),
        )
        self.wait(4)


class SphereExamplePointsDecimal(Scene):
    CONFIG = {
        "point_rotation_angle_axis_pairs": [
            (45 * DEGREES, DOWN),
            (120 * DEGREES, OUT),
            (35 * DEGREES, rotate_vector(RIGHT, 30 * DEGREES)),
            (90 * DEGREES, IN),
        ]
    }

    def construct(self):
        decimals = VGroup(*[
            DecimalNumber(
                0,
                num_decimal_places=3,
                color=color,
                include_sign=True,
                edge_to_fix=RIGHT,
            )
            for color in [YELLOW, GREEN, RED]
        ])
        number_label = VGroup(
            decimals[0], TexMobject("+"),
            decimals[1], TexMobject("i"), TexMobject("+"),
            decimals[2], TexMobject("j"),
        )
        number_label.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
        number_label.to_corner(UL)

        point = VectorizedPoint(OUT)

        def generate_decimal_updater(decimal, index):
            shifted_i = (index - 1) % 3
            return ContinualChangingDecimal(
                decimal,
                lambda a: point.get_location()[shifted_i]
            )

        for i, decimal in enumerate(decimals):
            self.add(generate_decimal_updater(decimal, i))

        decimal_braces = VGroup()
        for decimal, char in zip(decimals, "wxy"):
            brace = Brace(decimal, DOWN, buff=SMALL_BUFF)
            label = brace.get_tex(char, buff=SMALL_BUFF)
            label.match_color(decimal)
            brace.add(label)
            decimal_braces.add(brace)

        equation = TexMobject(
            "w^2 + x^2 + y^2 = 1",
            tex_to_color_map={
                "w": YELLOW,
                "x": GREEN,
                "y": RED,
            }
        )
        equation.next_to(decimal_braces, DOWN, MED_LARGE_BUFF)

        self.add(number_label)
        self.add(decimal_braces)
        self.add(equation)

        pairs = self.point_rotation_angle_axis_pairs
        for angle, axis in pairs:
            self.play(
                Rotate(point, angle, axis=axis, about_point=ORIGIN),
                run_time=2
            )
            self.wait()


class TwoDStereographicProjection(IntroduceFelix):
    CONFIG = {
        "camera_config": {
            "exponential_projection": False,
        },
        "sphere_sample_point_u_range": np.arange(
            0, PI, PI / 16,
        ),
        "sphere_sample_point_v_range": np.arange(
            0, TAU, TAU / 16,
        ),
        "n_sample_rotation_cycles": 2,
    }

    def construct(self):
        self.add_parts()
        self.talk_through_sphere()
        self.draw_projection_lines()
        self.show_point_at_infinity()
        self.show_a_few_rotations()

    def add_parts(self, run_time=1):
        felix = self.felix = self.pi_creature
        felix.shift(1.5 * DL)
        axes = self.axes = self.get_axes()
        sphere = self.sphere = self.get_sphere()

        c2p = axes.coords_to_point
        labels = VGroup(
            TexMobject("i").next_to(c2p(1, 0, 0), DR, SMALL_BUFF),
            TexMobject("-i").next_to(c2p(-1, 0, 0), DL, SMALL_BUFF),
            TexMobject("j").next_to(c2p(0, 1, 0), UL, SMALL_BUFF),
            TexMobject("-j").next_to(c2p(0, -1, 0), DL, SMALL_BUFF),
            TexMobject("1").rotate(
                90 * DEGREES, RIGHT,
            ).next_to(c2p(0, 0, 1), RIGHT + OUT, SMALL_BUFF),
            TexMobject("-1").rotate(
                90 * DEGREES, RIGHT,
            ).next_to(c2p(0, 0, -1), RIGHT + IN, SMALL_BUFF),
        )
        for sm in labels[:4].family_members_with_points():
            sm.add(VectorizedPoint(
                0.25 * DOWN + 0.25 * OUT
            ))
        labels.set_stroke(width=0, background=True)
        for submob in labels.get_family():
            submob.shade_in_3d = True

        self.add(felix, axes, sphere, labels)
        self.move_camera(
            **self.get_default_camera_position(),
            run_time=run_time
        )
        self.begin_ambient_camera_rotation(rate=0.01)
        self.play(
            felix.change, "pondering", sphere,
            run_time=run_time,
        )

    def talk_through_sphere(self):
        point = VectorizedPoint(OUT)
        arrow = Vector(IN, shade_in_3d=True)
        arrow.set_fill(PINK)
        arrow.set_stroke(BLACK, 1)

        def get_dot():
            dot = Sphere(radius=0.05, u_max=PI / 2)
            dot.set_fill(PINK)
            dot.set_stroke(width=0)
            dot.move_to(2.05 * OUT)
            dot.apply_matrix(
                z_to_vector(normalize(point.get_location())),
                about_point=ORIGIN
            )
            return dot

        dot = get_dot()
        dot.add_updater(
            lambda d: d.become(get_dot())
        )

        def update_arrow(arrow):
            target_point = 2.1 * point.get_location()
            rot_matrix = np.dot(
                z_to_vector(normalize(target_point)),
                np.linalg.inv(
                    z_to_vector(normalize(-arrow.get_vector()))
                )
            )
            arrow.apply_matrix(rot_matrix)
            arrow.shift(target_point - arrow.get_end())
            return arrow
        arrow.add_updater(update_arrow)

        self.add(self.sphere, dot, arrow)
        pairs = SphereExamplePointsDecimal.CONFIG.get(
            "point_rotation_angle_axis_pairs"
        )
        for angle, axis in pairs:
            self.play(
                Rotate(point, angle, axis=axis, about_point=ORIGIN),
                run_time=2
            )
            self.wait()
        self.play(FadeOut(dot), FadeOut(arrow))

    def draw_projection_lines(self):
        sphere = self.sphere
        axes = self.axes
        radius = sphere.get_width() / 2

        neg_one_point = axes.coords_to_point(0, 0, -1)
        neg_one_dot = Dot(
            neg_one_point,
            color=YELLOW,
            shade_in_3d=True
        )

        xy_plane = StereoProjectedSphere(
            u_max=15 * PI / 16,
            **self.sphere_config
        )
        xy_plane.set_fill(WHITE, 0.25)
        xy_plane.set_stroke(width=0)

        point_mob = VectorizedPoint(2 * OUT)
        point_mob.add_updater(
            lambda m: m.move_to(radius * normalize(m.get_center()))
        )
        point_mob.move_to([1, -1, 1])
        point_mob.update(0)

        def get_projection_line(sphere_point):
            to_sphere = Line(neg_one_point, sphere_point)
            to_plane = Line(
                sphere_point,
                self.project_point(sphere_point)
            )
            line = VGroup(to_sphere, to_plane)
            line.set_stroke(YELLOW, 3)
            for submob in line:
                submob.shade_in_3d = True
            return line

        def get_sphere_dot(sphere_point):
            dot = Dot(shade_in_3d=True)
            dot.set_fill(PINK)
            dot.insert_n_anchor_points(12)  # Helps with flashing?
            dot.apply_matrix(
                z_to_vector(sphere_point),
                about_point=ORIGIN,
            )
            dot.move_to(1.01 * sphere_point)
            dot.add(VectorizedPoint(5 * sphere_point))
            return dot

        def get_projection_dot(sphere_point):
            projection = self.project_point(sphere_point)
            dot = Dot(projection, shade_in_3d=True)
            dot.add(VectorizedPoint(dot.get_center() + 0.1 * OUT))
            dot.set_fill(WHITE)
            return dot

        point = point_mob.get_location()
        dot = get_sphere_dot(point)
        line = get_projection_line(point)
        projection_dot = get_projection_dot(point)

        sample_points = [
            radius * sphere.func(u, v)
            for u in self.sphere_sample_point_u_range
            for v in self.sphere_sample_point_v_range
        ]

        lines = VGroup(*[get_projection_line(p) for p in sample_points])
        lines.set_stroke(width=1)
        north_lines = lines[:len(lines) // 2]
        south_lines = lines[len(lines) // 2:]

        self.add(xy_plane, sphere)
        self.play(Write(xy_plane))
        self.wait(2)
        self.play(sphere.set_fill, BLUE_E, 0.5)
        self.play(FadeInFromLarge(dot))
        self.play(
            FadeIn(neg_one_dot),
            ShowCreation(line),
        )
        self.wait(2)
        self.play(ReplacementTransform(
            dot.copy(), projection_dot
        ))

        def get_point():
            return 2 * normalize(point_mob.get_location())

        dot.add_updater(
            lambda d: d.become(get_sphere_dot(get_point()))
        )
        line.add_updater(
            lambda l: l.become(get_projection_line(get_point()))
        )
        projection_dot.add_updater(
            lambda d: d.become(get_projection_dot(get_point()))
        )

        self.play(
            point_mob.move_to,
            radius * normalize(np.array([1, -1, -1])),
            run_time=3
        )
        self.move_camera(
            theta=-150 * DEGREES,
            run_time=3
        )
        self.add(axes, sphere, xy_plane, dot, line)
        for point in np.array([-2, 1, -0.5]), np.array([-0.01, -0.01, 1]):
            self.play(
                point_mob.move_to,
                radius * normalize(point),
                run_time=3
            )
        self.wait(2)

        # Project norther hemisphere
        north_hemisphere = self.get_sphere()
        n = len(north_hemisphere)
        north_hemisphere.remove(*north_hemisphere[n // 2:])
        north_hemisphere.generate_target()
        self.project_mobject(north_hemisphere.target)
        north_hemisphere.set_fill(opacity=0.8)

        self.play(
            LaggedStart(ShowCreation, north_lines),
            FadeIn(north_hemisphere)
        )
        self.play(
            MoveToTarget(north_hemisphere),
            run_time=3,
            rate_func=lambda t: smooth(0.99 * t)
        )
        self.play(FadeOut(north_lines))
        self.wait(2)

        # Unit circle
        circle = Sphere(
            radius=2.01,
            u_min=PI / 2 - 0.01,
            u_max=PI / 2 + 0.01,
            resolution=(1, 24),
        )
        for submob in circle:
            submob.add(VectorizedPoint(1.5 * submob.get_center()))
        circle.set_fill(YELLOW)
        circle_path = Circle(radius=2)
        circle_path.rotate(-90 * DEGREES)

        self.play(FadeInFromLarge(circle))
        self.play(point_mob.move_to, circle_path.points[0])
        self.play(MoveAlongPath(point_mob, circle_path, run_time=6))
        self.move_camera(
            phi=0,
            theta=-90 * DEGREES,
            rate_func=there_and_back_with_pause,
            run_time=6,
        )
        self.play(point_mob.move_to, OUT)
        self.wait()

        # Southern hemisphere
        south_hemisphere = self.get_sphere()
        n = len(south_hemisphere)
        south_hemisphere.remove(*south_hemisphere[:n // 2])
        south_hemisphere.remove(
            *south_hemisphere[-sphere.resolution[1]:]
        )
        south_hemisphere.generate_target()
        self.project_mobject(south_hemisphere.target)
        south_hemisphere.set_fill(opacity=0.8)
        south_hemisphere.target[-sphere.resolution[1] // 2:].set_fill(
            opacity=0
        )

        self.play(
            LaggedStart(ShowCreation, south_lines),
            FadeIn(south_hemisphere)
        )
        self.play(
            MoveToTarget(south_hemisphere),
            FadeOut(south_lines),
            FadeOut(xy_plane),
            run_time=3,
            rate_func=lambda t: smooth(0.99 * t)
        )
        self.wait(3)

        self.projected_sphere = VGroup(
            north_hemisphere,
            south_hemisphere,
        )
        self.equator = circle
        self.point_mob = point_mob

    def show_point_at_infinity(self):
        points = list(compass_directions(
            12, start_vect=rotate_vector(RIGHT, 3.25 * DEGREES)
        ))
        points.pop(7)
        points.pop(2)
        arrows = VGroup(*[
            Arrow(6 * p, 11 * p)
            for p in points
        ])
        arrows.set_fill(RED)
        arrows.set_stroke(RED, 10)
        neg_ones = VGroup(*[
            TexMobject("-1").next_to(arrow.get_start(), -p)
            for p, arrow in zip(points, arrows)
        ])
        neg_ones.set_stroke(width=0, background=True)

        sphere_arcs = VGroup()
        for angle in np.arange(0, TAU, TAU / 12):
            arc = Arc(PI, radius=2)
            arc.set_stroke(RED)
            arc.rotate(PI / 2, axis=DOWN, about_point=ORIGIN)
            arc.rotate(angle, axis=OUT, about_point=ORIGIN)
            sphere_arcs.add(arc)
        sphere_arcs.set_stroke(RED)

        self.play(
            LaggedStart(GrowArrow, arrows),
            LaggedStart(Write, neg_ones)
        )
        self.wait(3)
        self.play(
            FadeOut(self.projected_sphere),
            FadeOut(arrows),
            FadeOut(neg_ones),
        )
        for x in range(2):
            self.play(
                ShowCreationThenDestruction(
                    sphere_arcs,
                    submobject_mode="all_at_once",
                    run_time=3,
                )
            )

    def show_a_few_rotations(self):
        sphere = self.sphere
        felix = self.felix
        point_mob = self.point_mob
        point_mob.add_updater(
            lambda m: m.move_to(sphere.get_all_points()[0])
        )
        coord_point_mobs = VGroup(
            VectorizedPoint(RIGHT),
            VectorizedPoint(UP),
            VectorizedPoint(OUT),
        )
        for pm in coord_point_mobs:
            pm.shade_in_3d = True

        def get_rot_matrix():
            return np.array([
                pm.get_location()
                for pm in coord_point_mobs
            ]).T

        def get_projected_sphere():
            result = StereoProjectedSphere(
                get_rot_matrix(),
                max_r=10,
                **self.sphere_config,
            )
            result.set_fill(opacity=0.2)
            result.fade_far_out_submobjects(32)
            for submob in result:
                if submob.get_center()[1] < -11:
                    submob.fade(1)
            return result

        projected_sphere = get_projected_sphere()
        projected_sphere.add_updater(
            lambda m: m.become(get_projected_sphere())
        )

        def get_projected_equator():
            equator = CheckeredCircle(
                n_pieces=24,
                radius=2,
            )
            for submob in equator.get_family():
                submob.shade_in_3d = True
            equator.set_stroke(YELLOW, 5)
            equator.apply_matrix(get_rot_matrix())
            self.project_mobject(equator)
            return equator

        projected_equator = get_projected_equator()
        projected_equator.add_updater(
            lambda m: m.become(get_projected_equator())
        )

        self.add(sphere, projected_sphere)
        self.move_camera(phi=60 * DEGREES)
        self.play(
            sphere.set_fill_by_checkerboard,
            BLUE_E, BLUE_D, {"opacity": 0.8},
            FadeIn(projected_sphere)
        )
        sphere.add(coord_point_mobs)
        sphere.add(self.equator)
        self.add(projected_equator)
        pairs = self.get_sample_rotation_angle_axis_pairs()
        for x in range(self.n_sample_rotation_cycles):
            for angle, axis in pairs:
                self.play(
                    Rotate(
                        sphere, angle=angle, axis=axis,
                        about_point=ORIGIN,
                        run_time=3,
                    ),
                    felix.change, "confused",
                )
                self.wait()

        self.projected_sphere = projected_sphere

    #
    def project_mobject(self, mobject):
        return stereo_project(mobject, axis=2, r=2, outer_r=20)

    def project_point(self, point):
        return stereo_project_point(point, axis=2, r=2)

    def get_sample_rotation_angle_axis_pairs(self):
        return SphereExamplePointsDecimal.CONFIG.get(
            "point_rotation_angle_axis_pairs"
        )


class FelixViewOfProjection(TwoDStereographicProjection):
    CONFIG = {}

    def construct(self):
        self.add_axes()
        self.show_a_few_rotations()

    def add_axes(self):
        axes = Axes(
            number_line_config={
                "unit_size": 2,
                "color": WHITE,
            }
        )
        labels = VGroup(
            TexMobject("i"),
            TexMobject("-i"),
            TexMobject("j"),
            TexMobject("-j"),
        )
        coords = [(1, 0), (-1, 0), (0, 1), (0, -1)]
        vects = [DOWN, DOWN, RIGHT, RIGHT]
        for label, coords, vect in zip(labels, coords, vects):
            point = axes.coords_to_point(*coords)
            label.next_to(point, vect, buff=MED_SMALL_BUFF)

        self.add(axes, labels)
        self.pi_creature.change("confused")

    def show_a_few_rotations(self):
        felix = self.pi_creature
        coord_point_mobs = VGroup([
            VectorizedPoint(point)
            for point in [RIGHT, UP, OUT]
        ])

        def get_rot_matrix():
            return np.array([
                pm.get_location()
                for pm in coord_point_mobs
            ]).T

        def get_projected_sphere():
            return StereoProjectedSphere(
                get_rot_matrix(),
                **self.sphere_config,
            )

        def get_projected_equator():
            equator = Circle(radius=2, num_anchors=24)
            equator.set_stroke(YELLOW, 5)
            equator.apply_matrix(get_rot_matrix())
            self.project_mobject(equator)
            return equator

        projected_sphere = get_projected_sphere()
        projected_sphere.add_updater(
            lambda m: m.become(get_projected_sphere())
        )

        equator = get_projected_equator()
        equator.add_updater(
            lambda m: m.become(get_projected_equator())
        )

        dot = Dot(color=PINK)
        dot.add_updater(
            lambda d: d.move_to(
                self.project_point(
                    np.dot(2 * OUT, get_rot_matrix().T)
                )
            )
        )
        hand = Hand()
        hand.add_updater(
            lambda h: h.move_to(dot.get_center(), LEFT)
        )
        felix.add_updater(lambda f: f.look_at(dot))

        self.add(projected_sphere)
        self.add(equator)
        self.add(dot)
        self.add(hand)

        pairs = self.get_sample_rotation_angle_axis_pairs()
        for x in range(self.n_sample_rotation_cycles):
            for angle, axis in pairs:
                self.play(
                    Rotate(
                        coord_point_mobs, angle=angle, axis=axis,
                        about_point=ORIGIN,
                        run_time=3,
                    ),
                )
                self.wait()


class ShowRotationsJustWithReferenceCircles(TwoDStereographicProjection):
    def construct(self):
        self.add_parts(run_time=1)
        self.begin_ambient_camera_rotation(rate=0.03)
        self.edit_parts()
        self.show_1i_circle()
        self.show_1j_circle()
        self.show_random_circle()
        self.show_rotations()

    def edit_parts(self):
        sphere = self.sphere
        axes = self.axes
        axes.set_stroke(width=1)
        xy_plane = StereoProjectedSphere(u_max=15 * PI / 16)
        xy_plane.set_fill(WHITE, 0.2)
        xy_plane.set_stroke(width=0, opacity=0)

        self.add(xy_plane, sphere)
        self.play(
            FadeIn(xy_plane),
            sphere.set_fill, BLUE_E, {"opacity": 0.2},
            sphere.set_stroke, {"width": 0.1, "opacity": 0.5}
        )

    def show_1i_circle(self):
        axes = self.axes

        circle = self.get_circle(GREEN_E, GREEN)
        circle.rotate(TAU / 4, RIGHT)
        circle.rotate(TAU / 4, DOWN)

        projected = self.get_projected_circle(circle)

        labels = VGroup(*map(TexMobject, ["0", "2i", "3i"]))
        labels.set_shade_in_3d(True)
        for label, x in zip(labels, [0, 2, 3]):
            label.next_to(
                axes.coords_to_point(x, 0, 0), DR, SMALL_BUFF
            )

        self.play(ShowCreation(circle, run_time=3))
        self.wait()
        self.play(ReplacementTransform(
            circle.copy(), projected,
            run_time=3
        ))
        # self.axes.x_axis.pieces.set_stroke(width=0)
        self.wait(7)
        self.move_camera(
            phi=60 * DEGREES,
        )
        self.play(
            LaggedStart(
                FadeInFrom, labels,
                lambda m: (m, UP)
            )
        )
        self.wait(2)
        self.play(FadeOut(labels))

        self.one_i_circle = circle
        self.projected_one_i_circle = projected

    def show_1j_circle(self):
        circle = self.get_circle(RED_E, RED)
        circle.rotate(TAU / 4, DOWN)

        projected = self.get_projected_circle(circle)

        self.move_camera(theta=-170 * DEGREES)
        self.play(ShowCreation(circle, run_time=3))
        self.wait()
        self.play(ReplacementTransform(
            circle.copy(), projected, run_time=3
        ))
        # self.axes.y_axis.pieces.set_stroke(width=0)
        self.wait(3)

        self.one_j_circle = circle
        self.projected_one_j_circle = projected

    def show_random_circle(self):
        sphere = self.sphere

        circle = self.get_circle(BLUE_E, BLUE)
        circle.set_width(2 * sphere.radius * np.sin(30 * DEGREES))
        circle.shift(sphere.radius * np.cos(30 * DEGREES) * OUT)
        circle.rotate(150 * DEGREES, UP, about_point=ORIGIN)

        projected = self.get_projected_circle(circle)

        self.play(ShowCreation(circle, run_time=2))
        self.wait()
        self.play(ReplacementTransform(
            circle.copy(), projected,
            run_time=2
        ))
        self.wait(3)
        self.play(
            FadeOut(circle),
            FadeOut(projected),
        )

    def show_rotations(self):
        sphere = self.sphere
        c1i = self.one_i_circle
        pc1i = self.projected_one_i_circle
        c1j = self.one_j_circle
        pc1j = self.projected_one_j_circle
        cij = self.get_circle(YELLOW_E, YELLOW)
        pcij = self.get_projected_circle(cij)

        circles = VGroup(c1i, c1j, cij)
        x_axis = self.axes.x_axis
        y_axis = self.axes.y_axis

        arrow = Arrow(
            2 * RIGHT, 2 * UP,
            buff=SMALL_BUFF,
            path_arc=PI,
            use_rectangular_stem=False,
        )
        arrow.set_stroke(LIGHT_GREY, 3)
        arrow.tip.set_fill(LIGHT_GREY)
        arrows = VGroup(arrow, *[
            arrow.copy().rotate(angle, about_point=ORIGIN)
            for angle in np.arange(TAU / 4, TAU, TAU / 4)
        ])
        arrows.rotate(TAU / 4, RIGHT, about_point=ORIGIN)
        arrows.rotate(TAU / 2, OUT, about_point=ORIGIN)
        arrows.rotate(TAU / 4, UP, about_point=ORIGIN)
        arrows.space_out_submobjects(1.2)

        self.play(FadeInFromLarge(cij))
        sphere.add(circles)

        pc1i.add_updater(
            lambda c: c.become(self.get_projected_circle(c1i))
        )
        pc1j.add_updater(
            lambda c: c.become(self.get_projected_circle(c1j))
        )
        pcij.add_updater(
            lambda c: c.become(self.get_projected_circle(cij))
        )
        self.add(pcij)

        # About j-axis
        self.play(ShowCreation(arrows, run_time=3, rate_func=None))
        self.wait(3)
        for x in range(2):
            y_axis.pieces.set_stroke(width=1)
            self.play(
                Rotate(sphere, 90 * DEGREES, axis=UP),
                run_time=4,
            )
            y_axis.pieces.set_stroke(width=0)
            self.wait(2)

        # About i axis
        self.move_camera(theta=-45 * DEGREES)
        self.play(Rotate(arrows, TAU / 4, axis=OUT))
        self.wait(2)
        for x in range(2):
            x_axis.pieces.set_stroke(width=1)
            self.play(
                Rotate(sphere, -90 * DEGREES, axis=RIGHT),
                run_time=4,
            )
            x_axis.pieces.set_stroke(width=0)
            self.wait(2)
        self.wait(2)

        # About real axis
        self.move_camera(
            theta=-135 * DEGREES,
            added_anims=[FadeOut(arrows)]
        )
        self.ambient_camera_rotation.rate = 0.01
        for x in range(2):
            x_axis.pieces.set_stroke(width=1)
            y_axis.pieces.set_stroke(width=1)
            self.play(
                Rotate(sphere, 90 * DEGREES, axis=OUT),
                run_time=4,
            )
            # x_axis.pieces.set_stroke(width=0)
            # y_axis.pieces.set_stroke(width=0)
            self.wait(2)

    #
    def get_circle(self, *colors):
        sphere = self.sphere
        circle = CheckeredCircle(colors=colors, n_pieces=48)
        circle.set_shade_in_3d(True)
        circle.match_width(sphere)

        return circle

    def get_projected_circle(self, circle):
        result = circle.deepcopy()
        self.project_mobject(result)
        result[::2].fade(1)
        for sm in result:
            if sm.get_width() > FRAME_WIDTH:
                sm.fade(1)
            if sm.get_height() > FRAME_HEIGHT:
                sm.fade(1)
        return result


class IntroduceQuaternions(Scene):
    def construct(self):
        self.compare_three_number_systems()
        self.mention_four_perpendicular_axes()
        self.bring_back_complex()
        self.show_components_of_quaternion()

    def compare_three_number_systems(self):
        numbers = self.get_example_numbers()
        labels = VGroup(
            TextMobject("Complex number"),
            TextMobject("Not-actually-a-number-system 3d number"),
            TextMobject("Quaternion"),
        )

        for number, label in zip(numbers, labels):
            label.next_to(number, UP, aligned_edge=LEFT)

            self.play(
                FadeInFromDown(number),
                Write(label),
            )
            self.play(CircleThenFadeAround(
                number[2:],
                surrounding_rectangle_config={"color": BLUE}
            ))
            self.wait()

        shift_size = FRAME_HEIGHT / 2 - labels[2].get_top()[1] - MED_LARGE_BUFF
        self.play(
            numbers.shift, shift_size * UP,
            labels.shift, shift_size * UP,
        )

        self.numbers = numbers
        self.labels = labels

    def mention_four_perpendicular_axes(self):
        number = self.numbers[2]
        three_axes = VGroup(*[
            self.get_simple_axes(label, color)
            for label, color in zip(
                ["i", "j", "k"],
                [GREEN, RED, BLUE],
            )
        ])
        three_axes.arrange_submobjects(RIGHT, buff=LARGE_BUFF)
        three_axes.next_to(number, DOWN, LARGE_BUFF)

        self.play(LaggedStart(FadeInFromLarge, three_axes))
        self.wait(2)

        self.three_axes = three_axes

    def bring_back_complex(self):
        numbers = self.numbers
        labels = self.labels
        numbers[0].move_to(numbers[1], LEFT)
        labels[0].move_to(labels[1], LEFT)
        numbers.remove(numbers[1])
        labels.remove(labels[1])

        group = VGroup(numbers, labels)
        self.play(
            group.to_edge, UP,
            FadeOutAndShift(self.three_axes, DOWN)
        )
        self.wait()

    def show_components_of_quaternion(self):
        quat = self.numbers[-1]
        real_part = quat[0]
        imag_part = quat[2:]
        real_brace = Brace(real_part, DOWN)
        imag_brace = Brace(imag_part, DOWN)
        real_word = TextMobject("Real \\\\ part")
        imag_word = TextMobject("Imaginary \\\\ part")
        scalar_word = TextMobject("Scalar \\\\ part")
        vector_word = TextMobject("``Vector'' \\\\ part")
        for word in real_word, scalar_word:
            word.next_to(real_brace, DOWN, SMALL_BUFF)
        for word in imag_word, vector_word:
            word.next_to(imag_brace, DOWN, SMALL_BUFF)
        braces = VGroup(real_brace, imag_brace)
        VGroup(scalar_word, vector_word).set_color(YELLOW)

        self.play(
            LaggedStart(GrowFromCenter, braces),
            LaggedStart(
                FadeInFrom, VGroup(real_word, imag_word),
                lambda m: (m, UP)
            )
        )
        self.wait()
        self.play(
            FadeOutAndShift(real_word, DOWN),
            FadeInFrom(scalar_word, DOWN),
        )
        self.wait(2)
        self.play(ChangeDecimalToValue(real_part, 0))
        self.wait()
        self.play(
            FadeOutAndShift(imag_word, DOWN),
            FadeInFrom(vector_word, DOWN)
        )
        self.wait(2)

    #
    def get_example_numbers(self):
        number_2d = VGroup(
            DecimalNumber(3.14),
            TexMobject("+"),
            DecimalNumber(1.59),
            TexMobject("i")
        )
        number_3d = VGroup(
            DecimalNumber(2.65),
            TexMobject("+"),
            DecimalNumber(3.58),
            TexMobject("i"),
            TexMobject("+"),
            DecimalNumber(9.79),
            TexMobject("j"),
        )
        number_4d = VGroup(
            DecimalNumber(3.23),
            TexMobject("+"),
            DecimalNumber(8.46),
            TexMobject("i"),
            TexMobject("+"),
            DecimalNumber(2.64),
            TexMobject("j"),
            TexMobject("+"),
            DecimalNumber(3.38),
            TexMobject("k"),
        )
        numbers = VGroup(number_2d, number_3d, number_4d)
        for number in numbers:
            number.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
            for part in number:
                if isinstance(part, TexMobject):
                    # part.set_color_by_tex_to_color_map({
                    #     "i": GREEN,
                    #     "j": RED,
                    #     "k": BLUE,
                    # })
                    if part.get_tex_string() == "j":
                        part.shift(0.5 * SMALL_BUFF * DL)
            number[2].set_color(GREEN)
            if len(number) > 5:
                number[5].set_color(RED)
            if len(number) > 8:
                number[8].set_color(BLUE)
        numbers.arrange_submobjects(
            DOWN, buff=2, aligned_edge=LEFT
        )
        numbers.center()
        numbers.shift(LEFT)
        return numbers

    def get_simple_axes(self, label, color):
        axes = Axes(
            x_min=-2.5,
            x_max=2.5,
            y_min=-2.5,
            y_max=2.5,
        )
        axes.set_height(2.5)
        label_mob = TexMobject(label)
        label_mob.set_color(color)
        label_mob.next_to(axes.coords_to_point(0, 1.5), RIGHT, SMALL_BUFF)
        reals_label_mob = TextMobject("Reals")
        reals_label_mob.next_to(
            axes.coords_to_point(1, 0), DR, SMALL_BUFF
        )
        axes.add(label_mob, reals_label_mob)
        return axes


class SimpleImaginaryQuaternionAxes(SpecialThreeDScene):
    def construct(self):
        self.three_d_axes_config.update({
            "number_line_config": {"unit_size": 2},
            "x_min": -2,
            "x_max": 2,
            "y_min": -2,
            "y_max": 2,
            "z_min": -1.25,
            "z_max": 1.25,
        })
        axes = self.get_axes()
        labels = VGroup(*[
            TexMobject(tex).set_color(color)
            for tex, color in zip(
                ["i", "j", "k"],
                [GREEN, RED, BLUE]
            )
        ])
        labels[0].next_to(axes.coords_to_point(1, 0, 0), DOWN + IN, SMALL_BUFF)
        labels[1].next_to(axes.coords_to_point(0, 1, 0), RIGHT, SMALL_BUFF)
        labels[2].next_to(axes.coords_to_point(0, 0, 1), RIGHT, SMALL_BUFF)

        self.add(axes)
        self.add(labels)
        for label in labels:
            self.add_fixed_orientation_mobjects(label)

        self.move_camera(**self.get_default_camera_position())
        self.begin_ambient_camera_rotation(rate=0.05)
        self.wait(15)


class ShowDotProductCrossProductFromOfQMult(Scene):
    def construct(self):
        v_tex = "\\vec{\\textbf{v}}"
        product = TexMobject(
            "(", "w_1", "+",
            "x_1", "i", "+", "y_1", "j", "+", "z_1", "k", ")"
            "(", "w_2", "+",
            "x_2", "i", "+", "y_2", "j", "+", "z_2", "k", ")",
            "=",
            "(w_1", ",", v_tex + "_1", ")",
            "(w_2", ",", v_tex + "_2", ")",
            "="
        )
        product.set_width(FRAME_WIDTH - 1)

        i1 = product.index_of_part_by_tex("x_1")
        i2 = product.index_of_part_by_tex(")")
        i3 = product.index_of_part_by_tex("x_2")
        i4 = product.index_of_part_by_tex("z_2") + 2
        vector_parts = [product[i1:i2], product[i3:i4]]

        vector_defs = VGroup()
        braces = VGroup()
        for i, vp in zip(it.count(1), vector_parts):
            brace = Brace(vp, UP)
            vector = Matrix([
                ["x_" + str(i)],
                ["y_" + str(i)],
                ["z_" + str(i)],
            ])
            colors = [GREEN, RED, BLUE]
            for mob, color in zip(vector.get_entries(), colors):
                mob.set_color(color)
            group = VGroup(
                TexMobject("{}_{} = ".format(v_tex, i)),
                vector,
            )
            group.arrange_submobjects(RIGHT, SMALL_BUFF)
            group.next_to(brace, UP)

            braces.add(brace)
            vector_defs.add(group)

        result = TexMobject(
            "\\left(", "w_1", "w_2",
            "-", v_tex + "_1", "\\cdot", v_tex, "_2", ",\\,",
            "w_1", v_tex + "_2", "+", "w_2", v_tex + "_1",
            "+", "{}_1 \\times {}_2".format(v_tex, v_tex),
            "\\right)"
        )
        result.match_width(product)
        result.next_to(product, DOWN, LARGE_BUFF)
        for mob in product, result:
            mob.set_color_by_tex_to_color_map({
                "w": YELLOW,
                "x": GREEN,
                "y": RED,
                "z": BLUE,
            })
            mob.set_color_by_tex(v_tex, WHITE)

        self.add(product)
        self.add(braces)
        self.add(vector_defs)
        self.play(LaggedStart(FadeInFromLarge, result))
        self.wait()


class ShowComplexMagnitude(ShowComplexMultiplicationExamples):
    def construct(self):
        self.add_planes()
        plane = self.plane
        tex_to_color_map = {
            "a": YELLOW,
            "b": GREEN,
        }

        z = complex(3, 2)
        z_point = plane.number_to_point(z)
        z_dot = Dot(z_point)
        z_dot.set_color(PINK)
        z_line = Line(plane.number_to_point(0), z_point)
        z_line.set_stroke(WHITE, 2)
        z_label = TexMobject(
            "z", "=", "a", "+", "b", "i",
            tex_to_color_map=tex_to_color_map
        )
        z_label.add_background_rectangle()
        z_label.next_to(z_dot, UR, buff=SMALL_BUFF)
        z_norm_label = TexMobject("||z||")
        z_norm_label.add_background_rectangle()
        z_norm_label.next_to(ORIGIN, UP, SMALL_BUFF)
        z_norm_label.rotate(z_line.get_angle(), about_point=ORIGIN)
        z_norm_label.shift(z_line.get_center())

        h_line = Line(
            plane.number_to_point(0),
            plane.number_to_point(z.real),
            stroke_color=YELLOW,
            stroke_width=5,
        )
        v_line = Line(
            plane.number_to_point(z.real),
            plane.number_to_point(z),
            stroke_color=GREEN,
            stroke_width=5,
        )

        z_norm_equation = TexMobject(
            "||z||", "=", "\\sqrt", "{a^2", "+", "b^2", "}",
            tex_to_color_map=tex_to_color_map
        )
        z_norm_equation.set_background_stroke(width=0)
        z_norm_equation.add_background_rectangle()
        z_norm_equation.next_to(z_label, UP)

        self.add(z_line, h_line, v_line, z_dot, z_label)
        self.play(ShowCreation(z_line))
        self.play(FadeInFromDown(z_norm_label))
        self.wait()
        self.play(
            FadeIn(z_norm_equation[0]),
            FadeIn(z_norm_equation[2:]),
            TransformFromCopy(
                z_norm_label[1:],
                VGroup(z_norm_equation[1]),
            ),
        )
        self.wait()


class BreakUpQuaternionMultiplicationInParts(Scene):
    def construct(self):
        q1_color = MAROON_B
        q2_color = YELLOW

        product = TexMobject(
            "q_1", "\\cdot", "q_2", "=",
            "\\left(", "{q_1", "\\over", "||", "q_1", "||}", "\\right)",
            "||", "q_1", "||", "\\cdot", "q_2",
        )
        product.set_color_by_tex("q_1", q1_color)
        product.set_color_by_tex("q_2", q2_color)
        lhs = product[:3]
        scale_part = product[-5:]
        rotate_part = product[4:-5]
        lhs_rect = SurroundingRectangle(lhs)
        lhs_rect.set_color(YELLOW)
        lhs_words = TextMobject("Quaternion \\\\ multiplication")
        lhs_words.next_to(lhs_rect, UP, LARGE_BUFF)
        scale_brace = Brace(scale_part, UP)
        rotate_brace = Brace(rotate_part, DOWN)
        scale_words = TextMobject("Scale", "$q_2$")
        scale_words.set_color_by_tex("q_2", q2_color)
        scale_words.next_to(scale_brace, UP)
        rotate_words = TextMobject("Apply special \\\\ 4d rotation")
        rotate_words.next_to(rotate_brace, DOWN)

        norm_equation = TexMobject(
            "||", "q_1", "||", "=",
            "||", "w_1", "+",
            "x_1", "i", "+",
            "y_1", "j", "+",
            "z_1", "k", "||", "=",
            "\\sqrt",
            "{w_1^2", "+",
            "x_1^2", "+",
            "y_1^2", "+",
            "z_1^2", "}",
        )
        # norm_equation.set_color_by_tex_to_color_map({
        #     "w": YELLOW,
        #     "x": GREEN,
        #     "y": RED,
        #     "z": BLUE,
        # })
        norm_equation.set_color_by_tex("q_1", q1_color)
        norm_equation.to_edge(UP)
        norm_equation.set_background_stroke(width=0)

        line1 = Line(ORIGIN, 0.5 * LEFT + 3 * UP)
        line2 = Line(ORIGIN, UR)
        zero_dot = Dot()
        zero_label = TexMobject("0")
        zero_label.next_to(zero_dot, DOWN, SMALL_BUFF)
        q1_dot = Dot(line1.get_end())
        q2_dot = Dot(line2.get_end())
        q1_label = TexMobject("q_1").next_to(q1_dot, UP, SMALL_BUFF)
        q2_label = TexMobject("q_2").next_to(q2_dot, UR, SMALL_BUFF)
        VGroup(q1_dot, q1_label).set_color(q1_color)
        VGroup(q2_dot, q2_label).set_color(q2_color)
        dot_group = VGroup(
            line1, line2, q1_dot, q2_dot, q1_label, q2_label,
            zero_dot, zero_label,
        )
        dot_group.set_height(3)
        dot_group.center()
        dot_group.to_edge(LEFT)

        q1_dot.add_updater(lambda d: d.move_to(line1.get_end()))
        q1_label.add_updater(lambda l: l.next_to(q1_dot, UP, SMALL_BUFF))
        q2_dot.add_updater(lambda d: d.move_to(line2.get_end()))
        q2_label.add_updater(lambda l: l.next_to(q2_dot, UR, SMALL_BUFF))

        self.add(norm_equation)
        self.wait()
        self.play(
            FadeInFromDown(lhs),
            Write(dot_group),
        )
        self.add(*dot_group)
        self.add(
            VGroup(line2, q2_dot, q2_label).copy().fade(0.5)
        )
        self.play(
            ShowCreation(lhs_rect),
            FadeIn(lhs_words)
        )
        self.play(FadeOut(lhs_rect))
        self.wait()
        self.play(
            TransformFromCopy(lhs, product[3:]),
            # FadeOut(lhs_words)
        )
        self.play(
            GrowFromCenter(scale_brace),
            Write(scale_words),
        )
        self.play(
            line2.scale, 2, {"about_point": line2.get_start()}
        )
        self.wait()
        self.play(
            GrowFromCenter(rotate_brace),
            FadeInFrom(rotate_words, UP),
        )
        self.play(
            Rotate(
                line2, -line1.get_angle(),
                about_point=line2.get_start(),
                run_time=3
            )
        )
        self.wait()

        # Ask
        randy = Randolph(height=2)
        randy.flip()
        randy.next_to(rotate_words, RIGHT)
        randy.to_edge(DOWN)
        q_marks = TexMobject("???")
        random.shuffle(q_marks.submobjects)
        q_marks.next_to(randy, UP)
        self.play(
            FadeIn(randy)
        )
        self.play(
            randy.change, "confused", rotate_words,
            CircleThenFadeAround(rotate_words),
        )
        self.play(LaggedStart(
            FadeInFrom, q_marks,
            lambda m: (m, LEFT),
            lag_ratio=0.8,
        ))
        self.play(Blink(randy))
        self.wait(2)


class SphereProjectionsWrapper(Scene):
    def construct(self):
        rect_rows = VGroup(*[
            VGroup(*[
                ScreenRectangle(height=3)
                for x in range(3)
            ]).arrange_submobjects(RIGHT, buff=LARGE_BUFF)
            for y in range(2)
        ]).arrange_submobjects(DOWN, buff=2 * LARGE_BUFF)
        rect_rows.set_width(FRAME_WIDTH - 1)

        sphere_labels = VGroup(
            TextMobject("Circle in 2d"),
            TextMobject("Sphere in 3d"),
            TextMobject("Hypersphere in 4d"),
        )
        for label, rect in zip(sphere_labels, rect_rows[0]):
            label.next_to(rect, UP)

        projected_labels = VGroup(
            TextMobject("Sterographically projected \\\\ circle in 1d"),
            TextMobject("Sterographically projected \\\\ sphere in 2d"),
            TextMobject("Sterographically projected \\\\ hypersphere in 3d"),
        )
        for label, rect in zip(projected_labels, rect_rows[1]):
            label.match_width(rect)
            label.next_to(rect, UP)

        q_marks = TexMobject("???")
        q_marks.scale(2)
        q_marks.move_to(rect_rows[0][2])

        self.add(rect_rows)
        for l1, l2 in zip(sphere_labels, projected_labels):
            added_anims = []
            if l1 is sphere_labels[2]:
                added_anims.append(FadeIn(q_marks))
            self.play(FadeIn(l1), *added_anims)
            self.play(FadeIn(l2))
            self.wait()


class HypersphereStereographicProjection(SpecialThreeDScene):
    CONFIG = {
        # "fancy_dot": False,
        "fancy_dot": True,
        "initial_quaternion_sample_values": [
            [0, 1, 0, 0],
            [-1, 1, 0, 0],
            [0, 0, 1, 1],
            [0, 1, -1, 1],
        ]
    }

    def construct(self):
        self.setup_axes()
        self.introduce_quaternion_label()
        self.show_one()
        self.show_unit_sphere()
        self.show_quaternions_with_nonzero_real_part()
        self.emphasize_only_units()
        self.show_reference_spheres()

    def setup_axes(self):
        axes = self.axes = self.get_axes()
        axes.set_stroke(width=1)
        self.add(axes)
        self.move_camera(
            **self.get_default_camera_position(),
            run_time=0
        )
        self.begin_ambient_camera_rotation(rate=0.01)

    def introduce_quaternion_label(self):
        q_tracker = QuaternionTracker()
        coords = [
            DecimalNumber(0, color=color, include_sign=sign, edge_to_fix=RIGHT)
            for color, sign in zip(
                [YELLOW, GREEN, RED, BLUE],
                [False, True, True, True],
            )
        ]
        label = VGroup(
            coords[0], VectorizedPoint(),
            coords[1], TexMobject("i"),
            coords[2], TexMobject("j"),
            coords[3], TexMobject("k"),
        )
        label.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
        label.to_corner(UR)

        def update_label(label):
            self.remove_fixed_in_frame_mobjects(label)
            quat = q_tracker.get_value()
            for value, coord in zip(quat, label[::2]):
                coord.set_value(value)
            self.add_fixed_in_frame_mobjects(label)
            return label

        label.add_updater(update_label)

        def get_pq_point():
            point = self.project_quaternion(q_tracker.get_value())
            if get_norm(point) > 100:
                return point * 100 / get_norm(point)
            return point

        pq_dot = self.get_dot()
        pq_dot.add_updater(lambda d: d.move_to(get_pq_point()))
        dot_radius = pq_dot.get_width() / 2

        def get_pq_line():
            point = get_pq_point()
            norm = get_norm(point)
            if norm > dot_radius:
                point *= (norm - dot_radius) / norm
            result = Line(ORIGIN, point)
            result.set_stroke(width=1)
            return result

        pq_line = get_pq_line()
        pq_line.add_updater(lambda cl: cl.become(get_pq_line()))

        self.add(q_tracker, label, pq_line, pq_dot)

        self.q_tracker = q_tracker
        self.q_label = label
        self.pq_line = pq_line
        self.pq_dot = pq_dot

        rect = SurroundingRectangle(label, color=WHITE)
        self.add_fixed_in_frame_mobjects(rect)
        self.play(ShowCreation(rect))
        self.play(FadeOut(rect))
        self.remove_fixed_orientation_mobjects(rect)

        for value in self.initial_quaternion_sample_values:
            self.set_quat(value)
            self.wait()

    def show_one(self):
        q_tracker = self.q_tracker

        one_label = TexMobject("1")
        one_label.rotate(TAU / 4, RIGHT)
        one_label.next_to(ORIGIN, IN + RIGHT, SMALL_BUFF)
        one_label.set_shade_in_3d(True)
        one_label.set_background_stroke(width=0)

        self.play(
            ApplyMethod(
                q_tracker.set_value, [1, 0, 0, 0],
                run_time=2
            ),
            FadeInFromDown(one_label)
        )
        self.wait(4)

    def show_unit_sphere(self):
        sphere = self.sphere = self.get_projected_sphere(
            quaternion=[1, 0, 0, 0], null_axis=0,
            solid=False,
            stroke_width=0.5
        )
        self.specially_color_sphere(sphere)
        labels = self.get_unit_labels()
        labels.remove(labels[3])

        real_part = self.q_label[0]
        brace = Brace(real_part, DOWN)
        words = TextMobject("Real part zero")
        words.next_to(brace, DOWN, SMALL_BUFF, LEFT)

        self.play(Write(sphere))
        self.play(LaggedStart(
            FadeInFrom, labels,
            lambda m: (m, IN)
        ))
        self.add_fixed_in_frame_mobjects(brace, words)
        self.set_quat(
            [0, 1, 0, 0],
            added_anims=[
                GrowFromCenter(brace),
                Write(words),
            ]
        )
        self.wait()
        self.set_quat([0, 1, -1, 1])
        self.wait(2)
        self.set_quat([0, -1, -1, 1])
        self.wait(2)
        self.set_quat([0, 0, 0, 1])
        self.wait(2)
        self.set_quat([0, 0, -1, 0])
        self.wait(2)
        self.set_quat([0, 1, 0, 0])
        self.wait(2)
        self.play(FadeOut(words))
        self.remove_fixed_in_frame_mobjects(words)

        self.real_part_brace = brace

    def show_quaternions_with_nonzero_real_part(self):
        # Positive real part
        self.set_quat([1, 1, 2, 0])
        self.wait(2)
        self.set_quat([4, 0, -1, -1])
        self.wait(2)
        # Negative real part
        self.set_quat([-1, 1, 2, 0])
        self.wait(2)
        self.set_quat([-2, 0, -1, 1])
        self.wait(2)
        self.set_quat([-1, 1, 0, 0])
        self.move_camera(theta=-160 * DEGREES, run_time=3)
        self.set_quat([-1, 0.001, 0, 0])
        self.wait(2)

    def emphasize_only_units(self):
        q_label = self.q_label
        brace = self.real_part_brace

        brace.target = Brace(q_label, DOWN, buff=SMALL_BUFF)
        words = TextMobject(
            "Only those where \\\\",
            "$w^2 + x^2 + y^2 + z^2 = 1$"
        )
        words.next_to(brace.target, DOWN, SMALL_BUFF)

        self.add_fixed_in_frame_mobjects(words)
        self.play(
            MoveToTarget(brace),
            Write(words)
        )
        self.set_quat([1, 1, 1, 1])
        self.wait(2)
        self.set_quat([1, 1, -1, 1])
        self.wait(2)
        self.set_quat([-1, 1, -1, 1])
        self.wait(8)
        self.play(FadeOut(brace), FadeOut(words))
        self.remove_fixed_in_frame_mobjects(brace, words)

    # TODO
    def show_reference_spheres(self):
        sphere = self.sphere
        self.move_camera(
            phi=60 * DEGREES,
            theta=-150 * DEGREES,
            added_anims=[
                self.q_tracker.set_value, [1, 0, 0, 0]
            ]
        )
        sphere_ijk = self.get_projected_sphere(null_axis=0)
        sphere_1jk = self.get_projected_sphere(null_axis=1)
        sphere_1ik = self.get_projected_sphere(null_axis=2)
        sphere_1ij = self.get_projected_sphere(null_axis=3)
        circle = StereoProjectedCircleFromHypersphere(axes=[0, 1])

        circle_words = TextMobject(
            "Circle through\\\\", "$1, i, -1, -i$"
        )
        sphere_1ij_words = TextMobject(
            "Sphere through\\\\", "$1, i, j, -1, -i, -j$"
        )
        sphere_1jk_words = TextMobject(
            "Sphere through\\\\", "$1, j, k, -1, -j, -k$"
        )
        sphere_1ik_words = TextMobject(
            "Sphere through\\\\", "$1, i, k, -1, -i, -k$"
        )
        for words in [circle_words, sphere_1ij_words, sphere_1jk_words, sphere_1ik_words]:
            words.to_corner(UL)
            self.add_fixed_in_frame_mobjects(words)

        self.play(
            ShowCreation(circle),
            Write(circle_words),
        )
        self.set_quat([0, 1, 0, 0])
        self.set_quat([1, 0, 0, 0])
        self.remove(sphere)
        sphere_ijk.match_style(sphere)
        self.add(sphere_ijk)

        # Show xy plane
        self.play(
            FadeOutAndShift(circle_words, DOWN),
            FadeInFromDown(sphere_1ij_words),
            FadeOut(circle),
            sphere_ijk.set_stroke, {"width": 0.0}
        )
        self.play(Write(sphere_1ij))
        self.wait(10)
        return

        # Show yz plane
        self.play(
            FadeOutAndShift(sphere_1ij_words, DOWN),
            FadeInFromDown(sphere_1jk_words),
            sphere_1ij.set_fill, BLUE_E, 0.25,
            sphere_1ij.set_stroke, {"width": 0.0},
            Write(sphere_1jk)
        )
        self.wait(5)

        # Show xz plane
        self.play(
            FadeOutAndShift(sphere_1jk_words, DOWN),
            FadeInFromDown(sphere_1ik_words),
            sphere_1jk.set_fill, GREEN_E, 0.25,
            sphere_1jk.set_stroke, {"width": 0.0},
            Write(sphere_1ik)
        )
        self.wait(5)
        self.play(
            sphere_1ik.set_fill, RED_E, 0.25,
            sphere_1ik.set_stroke, {"width": 0.0},
            FadeOut(sphere_1ik_words)
        )

        # Start applying quaternion multiplication
        kwargs = {"solid": False, "stroke_width": 0}
        sphere_ijk.add_updater(
            lambda s: s.become(self.get_projected_sphere(0, **kwargs))
        )
        sphere_1jk.add_updater(
            lambda s: s.become(self.get_projected_sphere(1, **kwargs))
        )
        sphere_1ik.add_updater(
            lambda s: s.become(self.get_projected_sphere(2, **kwargs))
        )
        sphere_1ij.add_updater(
            lambda s: s.become(self.get_projected_sphere(3, **kwargs))
        )

        self.set_quat([0, 1, 1, 1])

    #
    def project_quaternion(self, quat):
        return self.axes.coords_to_point(
            *stereo_project_point(quat, axis=0, r=1)[1:]
        )

    def get_dot(self):
        if self.fancy_dot:
            sphere = self.get_sphere()
            sphere.set_width(0.2)
            sphere.set_stroke(width=0)
            sphere.set_fill(PINK)
            return sphere
        else:
            return VGroup(
                Dot(color=PINK),
                Dot(color=PINK).rotate(TAU / 4, RIGHT),
            )

    def get_unit_labels(self):
        c2p = self.axes.coords_to_point
        tex_coords_vects = [
            ("i", [1, 0, 0], IN + RIGHT),
            ("-i", [-1, 0, 0], IN + LEFT),
            ("j", [0, 1, 0], UP + OUT + RIGHT),
            ("-j", [0, -1, 0], RIGHT + DOWN),
            ("k", [0, 0, 1], OUT + RIGHT),
            ("-k", [0, 0, -1], IN + RIGHT),
        ]
        labels = VGroup()
        for tex, coords, vect in tex_coords_vects:
            label = TexMobject(tex)
            label.rotate(90 * DEGREES, RIGHT)
            label.next_to(c2p(*coords), vect, SMALL_BUFF)
            labels.add(label)
        labels.set_shade_in_3d(True)
        labels.set_background_stroke(width=0)
        return labels

    def set_quat(self, value, run_time=3, added_anims=None):
        if added_anims is None:
            added_anims = []
        self.play(
            self.q_tracker.set_value, value,
            *added_anims,
            run_time=run_time
        )

    def get_projected_sphere(self, null_axis, quaternion=None, solid=True, **kwargs):
        if quaternion is None:
            quaternion = self.get_multiplier()
        axes_to_color = {
            0: interpolate_color(YELLOW, BLACK, 0.5),
            1: GREEN_E,
            2: RED_D,
            3: BLUE_E,
        }
        color = axes_to_color[null_axis]
        config = dict(self.sphere_config)
        config.update({
            "stroke_color": WHITE,
            "stroke_width": 0.5,
            "stroke_opacity": 0.5,
            "max_r": 24,
        })
        if solid:
            config.update({
                "checkerboard_colors": [
                    color, interpolate_color(color, BLACK, 0.5)
                ],
                "fill_opacity": 1,
            })
        else:
            config.update({
                "checkerboard_colors": [],
                "fill_color": color,
                "fill_opacity": 0.25,
            })
        config.update(kwargs)
        sphere = StereoProjectedSphereFromHypersphere(
            quaternion=quaternion,
            null_axis=null_axis,
            **config
        )
        sphere.set_shade_in_3d(True)
        return sphere

    def get_projected_circle(self, quaternion=None, **kwargs):
        if quaternion is None:
            quaternion = self.get_multiplier()
        return StereoProjectedCircleFromHypersphere(quaternion, **kwargs)

    def get_multiplier(self):
        return self.q_tracker.get_value()

    def specially_color_sphere(self, sphere):
        for submob in sphere:
            u, v = submob.u1, submob.v1
            x = np.cos(v) * np.sin(u)
            y = np.sin(v) * np.sin(u)
            z = np.cos(u)
            rgb = sum([
                (x**2) * hex_to_rgb(GREEN),
                (y**2) * hex_to_rgb(RED),
                (z**2) * hex_to_rgb(BLUE),
            ])
            clip_in_place(rgb, 0, 1)
            submob.set_fill(rgb_to_hex(rgb))
        return sphere


class RuleOfQuaternionMultiplicationOverlay(Scene):
    def construct(self):
        q_mob, times_mob, p_mob = q_times_p = TexMobject(
            "q", "\\cdot", "p"
        )
        q_times_p.scale(2)
        q_mob.set_color(MAROON_B)
        p_mob.set_color(YELLOW)
        q_arrow = Vector(DOWN, color=WHITE)
        q_arrow.next_to(q_mob, UP)
        p_arrow = Vector(UP, color=WHITE)
        p_arrow.next_to(p_mob, DOWN)

        q_words = TextMobject("Think of as\\\\ an action")
        q_words.next_to(q_arrow, UP)
        p_words = TextMobject("Think of as\\\\ a point")
        p_words.next_to(p_arrow, DOWN)

        i_mob = TexMobject("i")[0]
        i_mob.scale(2)
        i_mob.move_to(q_mob, RIGHT)
        i_mob.set_color(GREEN)

        self.add(q_times_p)
        self.play(
            FadeInFrom(q_words, UP),
            GrowArrow(q_arrow),
        )
        self.play(
            FadeInFrom(p_words, DOWN),
            GrowArrow(p_arrow),
        )
        self.wait()
        self.play(*map(FadeOut, [
            q_words, q_arrow,
            p_words, p_arrow,
        ]))
        self.play(
            FadeInFromDown(i_mob),
            FadeOutAndShift(q_mob, UP)
        )
        product = VGroup(i_mob, times_mob, p_mob)
        self.play(product.to_edge, UP)

        # Show i products
        underline = Line(LEFT, RIGHT)
        underline.set_width(product.get_width() + MED_SMALL_BUFF)
        underline.next_to(product, DOWN)

        kwargs = {
            "tex_to_color_map": {
                "i": GREEN,
                "j": RED,
                "k": BLUE
            }
        }
        i_products = VGroup(
            TexMobject("i", "\\cdot", "1", "=", "1", **kwargs),
            TexMobject("i", "\\cdot", "i", "=", "-1", **kwargs),
            TexMobject("i", "\\cdot", "j", "=", "k", **kwargs),
            TexMobject("i", "\\cdot", "k", "=", "-j", **kwargs),
        )
        i_products.scale(2)
        i_products.arrange_submobjects(
            DOWN, buff=MED_LARGE_BUFF,
            aligned_edge=LEFT,
        )
        i_products.next_to(underline, DOWN, LARGE_BUFF)
        i_products.align_to(i_mob, LEFT)

        self.play(ShowCreation(underline))
        self.wait()
        for i_product in i_products:
            self.play(TransformFromCopy(
                product, i_product[:3]
            ))
            self.wait()
            self.play(TransformFromCopy(
                i_product[:3], i_product[3:],
            ))
            self.wait()

        rect = SurroundingRectangle(
            VGroup(product, i_products),
            buff=0.4
        )
        rect.set_stroke(WHITE, width=5)
        self.play(ShowCreation(rect))
        self.play(FadeOut(rect))


class RuleOfQuaternionMultiplication(HypersphereStereographicProjection):
    CONFIG = {
        "fancy_dot": True,
        "initial_quaternion_sample_values": [],
    }

    def construct(self):
        self.setup_all_trackers()
        self.show_multiplication_by_i_on_circle_1i()
        self.show_multiplication_by_i_on_circle_jk()
        self.show_multiplication_by_i_on_ijk_sphere()

    def setup_all_trackers(self):
        self.setup_multiplier_tracker()
        self.force_skipping()
        self.setup_axes()
        self.introduce_quaternion_label()
        self.add_unit_labels()
        self.revert_to_original_skipping_status()

    def setup_multiplier_tracker(self):
        self.multiplier_tracker = QuaternionTracker([1, 0, 0, 0])
        self.multiplier_tracker.add_updater(
            lambda m: m.set_value(normalize(
                m.get_value(),
                fall_back=[1, 0, 0, 0]
            ))
        )
        self.add(self.multiplier_tracker)

    def add_unit_labels(self):
        labels = self.unit_labels = self.get_unit_labels()
        one_label = TexMobject("1")
        one_label.set_shade_in_3d(True)
        one_label.rotate(90 * DEGREES, RIGHT)
        one_label.next_to(ORIGIN, IN + RIGHT, SMALL_BUFF)
        self.add(labels, one_label)

    def show_multiplication_by_i_on_circle_1i(self):
        m_tracker = self.multiplier_tracker

        def get_circle_1i():
            return self.get_projected_circle(
                basis_vectors=[
                    [1, 0, 0, 0],
                    [1, 1, 0, 0],
                ],
                colors=[GREEN, YELLOW],
                quaternion=m_tracker.get_value(),
            )
        circle = get_circle_1i()
        arrows = self.get_i_circle_arrows()

        def set_to_q_value(mt):
            mt.set_value(self.q_tracker.get_value())

        self.play(ShowCreation(circle, run_time=2))
        self.play(LaggedStart(ShowCreation, arrows, lag_ratio=0.25))
        self.wait()
        circle.add_updater(lambda c: c.become(get_circle_1i()))
        m_tracker.add_updater(set_to_q_value)
        self.add(m_tracker)
        self.set_quat([0, 1, 0, 0])
        self.wait()
        self.set_quat([-1, 0.001, 0, 0])
        self.wait()
        self.q_tracker.set_value([-1, -0.001, 0, 0])
        self.set_quat([0, -1, 0, 0])
        self.wait()
        self.set_quat([1, 0, 0, 0])
        self.wait(3)
        self.play(FadeOut(arrows))

        m_tracker.remove_updater(set_to_q_value)
        self.circle_1i = circle

    def show_multiplication_by_i_on_circle_jk(self):
        def get_circle_jk():
            return self.get_projected_circle(
                basis_vectors=[
                    [0, 0, 1, 0],
                    [0, 0, 0, 1],
                ],
                colors=[RED, BLUE_E]
            )
        circle = get_circle_jk()
        arrows = self.get_jk_circle_arrows()
        m_tracker = self.multiplier_tracker
        q_tracker = self.q_tracker

        def set_q_to_mj(qt):
            qt.set_value(q_mult(
                m_tracker.get_value(), [0, 0, 1, 0]
            ))

        self.move_camera(theta=-50 * DEGREES)
        self.play(ShowCreation(circle, run_time=2))
        circle.add_updater(lambda c: c.become(get_circle_jk()))
        self.wait(10)
        self.stop_ambient_camera_rotation()
        self.begin_ambient_camera_rotation(rate=-0.01)
        self.play(*map(ShowCreation, arrows))
        self.wait()
        self.set_quat([0, 0, 1, 0], run_time=1)
        q_tracker.add_updater(set_q_to_mj, index=0)
        self.add(self.circle_1i)
        self.play(
            m_tracker.set_value, [0, 1, 0, 0],
            run_time=3
        )
        self.wait()
        self.play(
            m_tracker.set_value, [-1, 0.001, 0, 0],
            run_time=3
        )
        self.wait()
        m_tracker.set_value([-1, 0.001, 0, 0])
        self.play(
            m_tracker.set_value, [0, -1, 0, 0],
            run_time=3
        )
        self.wait()
        self.play(
            m_tracker.set_value, [1, 0, 0, 0],
            run_time=3
        )
        self.wait()
        q_tracker.remove_updater(set_q_to_mj)
        self.play(
            FadeOut(arrows),
            q_tracker.set_value, [1, 0, 0, 0],
        )
        self.wait(10)

        self.circle_jk = circle

    def show_multiplication_by_i_on_ijk_sphere(self):
        m_tracker = self.multiplier_tracker
        q_tracker = self.q_tracker
        m_tracker.add_updater(lambda m: m.set_value(q_tracker.get_value()))

        def get_sphere():
            result = self.get_projected_sphere(null_axis=0, solid=False)
            self.specially_color_sphere(result)
            return result

        sphere = get_sphere()

        self.play(Write(sphere))
        sphere.add_updater(lambda s: s.become(get_sphere()))

        self.set_quat([0, 1, 0, 0])
        self.wait()
        self.set_quat([-1, 0.001, 0, 0])
        self.wait()
        self.q_tracker.set_value([-1, -0.001, 0, 0])
        self.set_quat([0, -1, 0, 0])
        self.wait()
        self.set_quat([1, 0, 0, 0])
        self.wait(3)

    #
    def get_multiplier(self):
        return self.multiplier_tracker.get_value()

    def get_i_circle_arrows(self):
        c2p = self.axes.coords_to_point
        i_arrow = Arrow(
            ORIGIN, 2 * RIGHT, path_arc=-120 * DEGREES,
            use_rectangular_stem=False,
            buff=SMALL_BUFF,
        )
        neg_one_arrow = Arrow(
            ORIGIN, 5.5 * RIGHT + UP,
            path_arc=-30 * DEGREES,
            use_rectangular_stem=False,
            buff=SMALL_BUFF,
        )
        neg_i_arrow = Arrow(
            4.5 * LEFT + 1.5 * UP, ORIGIN,
            path_arc=-30 * DEGREES,
            use_rectangular_stem=False,
            buff=SMALL_BUFF,
        )
        one_arrow = i_arrow.copy()
        result = VGroup(i_arrow, neg_one_arrow, neg_i_arrow, one_arrow)
        for arrow in result:
            arrow.set_color(LIGHT_GREY)
            arrow.set_stroke(width=3)
            arrow.rotate(90 * DEGREES, RIGHT)
        i_arrow.next_to(c2p(0, 0, 0), OUT + RIGHT, SMALL_BUFF)
        neg_one_arrow.next_to(c2p(1, 0, 0), OUT + RIGHT, SMALL_BUFF)
        neg_i_arrow.next_to(c2p(-1, 0, 0), OUT + LEFT, SMALL_BUFF)
        one_arrow.next_to(c2p(0, 0, 0), OUT + LEFT, SMALL_BUFF)
        return result

    def get_jk_circle_arrows(self):
        arrow = Arrow(
            1.5 * RIGHT, 1.5 * UP,
            path_arc=90 * DEGREES,
            buff=SMALL_BUFF,
            use_rectangular_stem=False
        )
        arrow.set_color(LIGHT_GREY)
        arrow.set_stroke(width=3)
        arrows = VGroup(*[
            arrow.copy().rotate(angle, about_point=ORIGIN)
            for angle in np.arange(0, TAU, TAU / 4)
        ])
        arrows.rotate(90 * DEGREES, RIGHT)
        arrows.rotate(90 * DEGREES, OUT)
        return arrows


class ShowDistributionOfI(TeacherStudentsScene):
    def construct(self):
        tex_to_color_map = {
            "q": PINK,
            "w": YELLOW,
            "x": GREEN,
            "y": RED,
            "z": BLUE,
        }
        top_product = TexMobject(
            "q", "\\cdot", "\\left(",
            "w", "+", "x", "i", "+", "y", "j", "+", "z", "k",
            "\\right)"
        )
        top_product.to_edge(UP)
        self.add(top_product)
        bottom_product = TexMobject(
            "q", "w",
            "+", "x", "q", "\\cdot", "i",
            "+", "y", "q", "\\cdot", "j",
            "+", "z", "q", "\\cdot", "k",
        )
        bottom_product.next_to(top_product, DOWN, MED_LARGE_BUFF)

        for product in [top_product, bottom_product]:
            for tex, color in tex_to_color_map.items():
                product.set_color_by_tex(tex, color, substring=False)

        self.student_says(
            "What does it do \\\\ to other quaternions?",
            target_mode="raise_left_hand"
        )
        self.change_student_modes(
            "pondering", "raise_left_hand", "erm",
            look_at_arg=top_product,
        )
        self.wait(2)
        self.play(
            self.teacher.change, "raise_right_hand",
            RemovePiCreatureBubble(self.students[1], target_mode="pondering"),
            *[
                TransformFromCopy(
                    top_product.get_parts_by_tex(tex, substring=False),
                    bottom_product.get_parts_by_tex(tex, substring=False),
                    run_time=2
                )
                for tex in ["w", "x", "i", "y", "j", "z", "k", "+"]
            ]
        )
        self.play(*[
            TransformFromCopy(
                top_product.get_parts_by_tex(tex, substring=False),
                bottom_product.get_parts_by_tex(tex, substring=False),
                run_time=2
            )
            for tex in ["q", "\\cdot"]
        ])
        self.change_all_student_modes("thinking")
        self.wait(3)


class ShowMultiplicationBy135Example(RuleOfQuaternionMultiplication):
    CONFIG = {
        "fancy_dot": True,
    }

    def construct(self):
        self.setup_all_trackers()
        self.add_circles()
        self.add_ijk_sphere()
        self.show_multiplication()

    def add_circles(self):
        self.circle_1i = self.add_auto_updating_circle(
            basis_vectors=[
                [1, 0, 0, 0],
                [0, 1, 0, 0],
            ],
            colors=[YELLOW, GREEN_E]
        )
        self.circle_jk = self.add_auto_updating_circle(
            basis_vectors=[
                [0, 0, 1, 0],
                [0, 0, 0, 1],
            ],
            colors=[RED, BLUE_E]

        )

    def add_auto_updating_circle(self, **circle_config):
        circle = self.get_projected_circle(**circle_config)
        circle.add_updater(
            lambda c: c.become(self.get_projected_circle(**circle_config))
        )
        self.add(circle)
        return circle

    def add_ijk_sphere(self):
        def get_sphere():
            result = self.get_projected_sphere(
                null_axis=0,
                solid=False,
                stroke_width=0.5,
                stroke_opacity=0.2,
                fill_opacity=0.1,
            )
            self.specially_color_sphere(result)
            return result
        sphere = get_sphere()
        sphere.add_updater(lambda s: s.become(get_sphere()))
        self.add(sphere)
        self.sphere = sphere

    def show_multiplication(self):
        m_tracker = self.multiplier_tracker

        quat = normalize(np.array([-1, 1, 0, 0]))
        point = self.project_quaternion(quat)
        arrow = Vector(DR)
        arrow.next_to(point, UL, MED_SMALL_BUFF)
        arrow.set_color(PINK)
        label = TexMobject(
            "-{\\sqrt{2} \\over 2}", "+",
            "{\\sqrt{2} \\over 2}", "i",
        )
        label.next_to(arrow.get_start(), UP)
        label.set_background_stroke(width=0)

        def get_one_point():
            return self.circle_1i[0].points[0]

        def get_j_point():
            return self.circle_jk[0].points[0]

        one_point = VectorizedPoint()
        one_point.add_updater(lambda v: v.set_location(get_one_point()))
        self.add(one_point)

        hand = Hand()
        hand.rotate(45 * DEGREES, RIGHT)
        hand.add_updater(
            lambda h: h.move_to(get_one_point(), LEFT)
        )

        j_line = Line(ORIGIN, get_j_point())
        moving_j_line = j_line.deepcopy()
        moving_j_line.add_updater(
            lambda m: m.put_start_and_end_on(ORIGIN, get_j_point())
        )

        self.add(j_line, moving_j_line)
        self.set_camera_orientation(
            phi=60 * DEGREES, theta=-70 * DEGREES
        )
        self.play(
            FadeInFromLarge(label, 3),
            GrowArrow(arrow)
        )
        self.set_quat(quat)
        self.wait(5)
        self.play(FadeInFromLarge(hand))
        self.add(m_tracker)
        for q in [quat, [1, 0, 0, 0], quat]:
            self.play(
                m_tracker.set_value, q,
                UpdateFromFunc(
                    m_tracker,
                    lambda m: m.set_value(normalize(m.get_value()))
                ),
                run_time=5
            )
            self.wait()


class JMultiplicationChart(Scene):
    def construct(self):
        # Largely copy-pasted....what are you gonna do about it?
        product = TexMobject("j", "\\cdot", "p")
        product[0].set_color(RED)
        product.scale(2)
        product.to_edge(UP)

        underline = Line(LEFT, RIGHT)
        underline.set_width(product.get_width() + MED_SMALL_BUFF)
        underline.next_to(product, DOWN)

        kwargs = {
            "tex_to_color_map": {
                "i": GREEN,
                "j": RED,
                "k": BLUE
            }
        }
        j_products = VGroup(
            TexMobject("j", "\\cdot", "1", "=", "1", **kwargs),
            TexMobject("j", "\\cdot", "j", "=", "-1", **kwargs),
            TexMobject("j", "\\cdot", "i", "=", "-k", **kwargs),
            TexMobject("j", "\\cdot", "k", "=", "i", **kwargs),
        )
        j_products.scale(2)
        j_products.arrange_submobjects(
            DOWN, buff=MED_LARGE_BUFF,
            aligned_edge=LEFT,
        )
        j_products.next_to(underline, DOWN, LARGE_BUFF)
        j_products.align_to(product, LEFT)

        self.play(FadeInFromDown(product))
        self.play(ShowCreation(underline))
        self.wait()
        for j_product in j_products:
            self.play(TransformFromCopy(
                product, j_product[:3]
            ))
            self.wait()
            self.play(TransformFromCopy(
                j_product[:3], j_product[3:],
            ))
            self.wait()

        rect = SurroundingRectangle(
            VGroup(product, j_products),
            buff=MED_SMALL_BUFF
        )
        rect.set_stroke(WHITE, width=5)
        self.play(ShowCreation(rect))
        self.play(FadeOut(rect))


class ShowJMultiplication(ShowMultiplicationBy135Example):
    CONFIG = {
        "fancy_dot": True,
        "run_time_per_rotation": 4,
    }

    def construct(self):
        self.setup_all_trackers()
        self.add_circles()
        self.add_ijk_sphere()
        self.show_multiplication()

    def add_circles(self):
        self.circle_1j = self.add_auto_updating_circle(
            basis_vectors=[
                [1, 0, 0, 0],
                [0, 0, 1, 0],
            ],
            colors=[YELLOW, RED]
        )
        self.circle_ik = self.add_auto_updating_circle(
            basis_vectors=[
                [0, 1, 0, 0],
                [0, 0, 0, 1],
            ],
            colors=[GREEN, BLUE_E]
        )

    def show_multiplication(self):
        self.set_camera_orientation(theta=-30 * DEGREES)

        q_tracker = self.q_tracker
        m_tracker = self.multiplier_tracker

        def normalize_tracker(t):
            t.set_value(normalize(t.get_value()))

        updates = [
            UpdateFromFunc(tracker, normalize_tracker)
            for tracker in (q_tracker, m_tracker)
        ]

        run_time = self.run_time_per_rotation
        self.play(
            m_tracker.set_value, [0, 0, 1, 0],
            q_tracker.set_value, [0, 0, 1, 0],
            *updates,
            run_time=run_time,
        )
        self.wait(2)
        self.play(
            m_tracker.set_value, [-1, 0, 1e-3, 0],
            q_tracker.set_value, [-1, 0, 1e-3, 0],
            *updates,
            run_time=run_time,
        )
        self.wait(2)

        # Show ik circle
        self.move_camera(theta=-110 * DEGREES)
        m_tracker.set_value([1, 0, 0, 0])
        q_tracker.set_value([0, 1, 0, 0])
        self.wait()
        self.play(
            m_tracker.set_value, [0, 0, 1, 0],
            q_tracker.set_value, [0, 0, 0, -1],
            *updates,
            run_time=run_time,
        )
        self.wait(2)
        self.play(
            m_tracker.set_value, [-1, 0, 1e-3, 0],
            q_tracker.set_value, [0, -1, 0, 0],
            *updates,
            run_time=run_time,
        )
        self.wait(2)


class ShowArbitraryMultiplication(ShowMultiplicationBy135Example):
    CONFIG = {
        "fancy_dot": True,
        "run_time_per_rotation": 4,
        "special_quaternion": [-0.5, 0.5, 0.5, 0.5],
    }

    def construct(self):
        self.setup_all_trackers()
        self.add_circles()
        self.add_ijk_sphere()
        self.show_multiplication()

    def add_circles(self):
        self.circle1 = self.add_auto_updating_circle(
            basis_vectors=[
                [1, 0, 0, 0],
                [0, 1, 1, 1],
            ],
            colors=[YELLOW_E, YELLOW]
        )
        bv1 = normalize([0, -1, -1, 2])
        bv2 = [0] + list(normalize(np.cross([1, 1, 1], bv1[1:])))
        self.circle2 = self.add_auto_updating_circle(
            basis_vectors=[bv1, bv2],
            colors=[WHITE, GREY]
        )

    def show_multiplication(self):
        q_tracker = self.q_tracker
        m_tracker = self.multiplier_tracker
        run_time = self.run_time_per_rotation

        def normalize_tracker(t):
            t.set_value(normalize(t.get_value()))

        # for tracker in q_tracker, m_tracker:
        #     self.add(ContinualUpdate(tracker, normalize_tracker))
        updates = [
            UpdateFromFunc(tracker, normalize_tracker)
            for tracker in (q_tracker, m_tracker)
        ]

        special_q = self.special_quaternion
        pq_point = self.project_quaternion(special_q)
        label = TextMobject("Some unit quaternion")
        label.set_color(PINK)
        label.rotate(90 * DEGREES, RIGHT)
        label.next_to(pq_point, IN + RIGHT, SMALL_BUFF)

        circle1, circle2 = self.circle1, self.circle2
        for circle in [circle1, circle2]:
            circle.tucked_away_updaters = circle1.updaters
            circle.clear_updaters()
            self.remove(circle)

        hand = Hand()
        hand.rotate(90 * DEGREES, RIGHT)
        hand.move_to(ORIGIN, LEFT)
        hand.set_shade_in_3d(True)
        one_dot = self.get_dot()
        one_dot.set_color(YELLOW_E)
        one_dot.move_to(ORIGIN)
        one_dot.add_updater(
            lambda m: m.move_to(circle1[0].points[0])
        )
        self.add(one_dot)

        self.stop_ambient_camera_rotation()
        self.begin_ambient_camera_rotation(rate=0.02)
        self.set_quat(special_q)
        self.play(FadeInFrom(label, IN))
        self.wait(3)
        for circle in [circle1, circle2]:
            self.play(ShowCreation(circle, run_time=3))
            circle.updaters = circle.tucked_away_updaters
            self.wait(2)
        self.play(
            FadeInFrom(hand, 2 * IN + 2 * RIGHT),
            run_time=2
        )
        hand.add_updater(
            lambda h: h.move_to(circle1[0].points[0], LEFT)
        )

        for quat in [special_q, [1, 0, 0, 0], special_q]:
            self.play(
                m_tracker.set_value, special_q,
                *updates,
                run_time=run_time,
            )
            self.wait()


class MentionCommutativity(TeacherStudentsScene):
    def construct(self):
        kwargs = {
            "tex_to_color_map": {
                "q": MAROON_B,
                "p": YELLOW,
                "i": GREEN,
                "j": RED,
                "k": BLUE,
            }
        }
        general_eq = TexMobject("q \\cdot p \\ne p \\cdot q", **kwargs)
        general_eq.get_part_by_tex("\\ne").submobjects.reverse()
        ij_eq = TexMobject("i \\cdot j = k", **kwargs)
        ji_eq = TexMobject("j \\cdot i = -k", **kwargs)

        for mob in [general_eq, ij_eq, ji_eq]:
            mob.move_to(self.hold_up_spot, DOWN)

        words = TextMobject("Multiplication doesn't \\\\ commute")
        words.next_to(general_eq, UP, MED_LARGE_BUFF)
        words.shift_onto_screen()

        joke = TextMobject("Quaternions work from home")
        joke.scale(0.75)
        joke.to_corner(UL, MED_SMALL_BUFF)

        self.play(
            FadeInFromDown(general_eq),
            self.teacher.change, "raise_right_hand",
            self.get_student_changes("erm", "confused", "sassy")
        )
        self.play(FadeInFrom(words, RIGHT))
        self.wait(2)
        self.play(
            ReplacementTransform(words, joke),
            general_eq.shift, UP,
            FadeInFromDown(ij_eq),
            self.get_student_changes(*["pondering"] * 3)
        )
        self.look_at(self.screen)
        self.wait(3)
        self.play(
            FadeInFrom(ji_eq),
            LaggedStart(
                ApplyMethod, VGroup(ij_eq, general_eq),
                lambda m: (m.shift, UP),
                lag_ratio=0.8,
            )
        )
        self.look_at(self.screen)
        self.wait(5)


class RubuiksCubeOperations(SpecialThreeDScene):
    def construct(self):
        self.set_camera_orientation(**self.get_default_camera_position())
        self.begin_ambient_camera_rotation()
        cube = RubiksCube()
        cube.shift(2.5 * RIGHT)
        # for square in cube:
        #     square.set_sheen(0.2, DOWN + LEFT + IN)
        cube2 = cube.copy()

        self.add(cube)
        self.play(
            Rotate(cube.get_face(RIGHT), 90 * DEGREES, RIGHT),
            run_time=2
        )
        self.play(
            Rotate(cube.get_face(DOWN), 90 * DEGREES, UP),
            run_time=2
        )
        self.wait()
        self.play(
            cube.shift, 5 * LEFT,
            FadeIn(cube2)
        )
        self.play(
            Rotate(cube2.get_face(DOWN), 90 * DEGREES, UP),
            run_time=2
        )
        self.play(
            Rotate(cube2.get_face(RIGHT), 90 * DEGREES, RIGHT),
            run_time=2
        )
        self.wait(6)


class RotationsOfCube(SpecialThreeDScene):
    def construct(self):
        self.set_camera_orientation(**self.get_default_camera_position())
        self.begin_ambient_camera_rotation(0.0001)
        cube = RubiksCube()
        cube2 = cube.copy()
        axes = self.get_axes()
        axes.scale(0.75)

        label1 = TextMobject(
            "z-axis\\\\",
            "then x-axis"
        )
        label2 = TextMobject(
            "x-axis\\\\",
            "then z-axis"
        )
        for label in [label1, label2]:
            for part in label:
                part.add_background_rectangle()
            label.rotate(90 * DEGREES, RIGHT)
            label.move_to(3 * OUT + 0.5 * IN)

        self.add(axes, cube)
        self.play(
            Rotate(cube, 90 * DEGREES, OUT, run_time=2),
            FadeInFrom(label1[0], IN),
        )
        self.play(
            Rotate(cube, 90 * DEGREES, RIGHT, run_time=2),
            FadeInFrom(label1[1], IN),
        )
        self.wait()
        self.play(
            cube.shift, 5 * RIGHT,
            label1.shift, 5 * RIGHT,
            Write(cube2, run_time=1)
        )
        self.play(
            Rotate(cube2, 90 * DEGREES, RIGHT, run_time=2),
            FadeInFrom(label2[0], IN),
        )
        self.play(
            Rotate(cube2, 90 * DEGREES, OUT, run_time=2),
            FadeInFrom(label2[1], IN),
        )
        self.wait(5)