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Halfway through ShowEqualAngleSlices
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Grant Sanderson
@ -9,7 +9,7 @@ COBALT = "#0047AB"
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class Orbiting(ContinualAnimation):
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CONFIG = {
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"rate": 0.3,
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"rate": 7.5,
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}
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def __init__(self, planet, star, ellipse, **kwargs):
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@ -24,18 +24,28 @@ class Orbiting(ContinualAnimation):
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def update_mobject(self, dt):
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# time = self.internal_time
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rate = self.rate
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planet = self.planet
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star = self.star
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ellipse = self.ellipse
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rate *= 1 / np.linalg.norm(
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planet.get_center() - star.get_center()
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rate = self.rate
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radius_vector = planet.get_center() - star.get_center()
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rate *= 1.0 / np.linalg.norm(radius_vector)
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prop = self.proportion
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d_prop = 0.001
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ds = np.linalg.norm(op.add(
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ellipse.point_from_proportion((prop + d_prop) % 1),
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-ellipse.point_from_proportion(prop),
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))
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delta_prop = (d_prop / ds) * rate * dt
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self.proportion = (self.proportion + delta_prop) % 1
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planet.move_to(
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ellipse.point_from_proportion(self.proportion)
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)
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self.proportion += rate * dt
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self.proportion = self.proportion % 1
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planet.move_to(ellipse.point_from_proportion(self.proportion))
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class SunAnimation(ContinualAnimation):
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@ -812,7 +822,7 @@ class AskAboutEllipses(TheMotionOfPlanets):
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)
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self.focus_points = [
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self.sun.get_center(),
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self.sun.get_center() + c * LEFT,
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self.sun.get_center() + 2 * c * LEFT,
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]
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return ellipse
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@ -2515,7 +2525,7 @@ class FeynmanRecountingNewton(Scene):
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class IntroduceShapeOfVelocities(AskAboutEllipses, MovingCameraScene):
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CONFIG = {
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"animate_sun": False,
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"animate_sun": True,
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"sun_center": 2 * RIGHT,
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"a": 4.0,
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"b": 3.5,
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@ -2535,14 +2545,15 @@ class IntroduceShapeOfVelocities(AskAboutEllipses, MovingCameraScene):
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self.add_foreground_mobjects(self.comet)
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def warp_orbit(self):
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def func(z, c=6):
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def func(z, c=3.5):
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return 1 * (np.exp((1.0 / c) * (z) + 1) - np.exp(1))
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ellipse = self.ellipse
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ellipse.save_state()
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ellipse.generate_target()
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ellipse.target.stretch(0.7, 1)
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ellipse.target.apply_complex_function(func)
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ellipse.target.replace(ellipse, dim_to_match=0)
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ellipse.target.replace(ellipse, dim_to_match=1)
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self.wait(5)
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self.play(MoveToTarget(ellipse, run_time=2))
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@ -2646,6 +2657,7 @@ class IntroduceShapeOfVelocities(AskAboutEllipses, MovingCameraScene):
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circle = Circle(color=YELLOW)
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circle.replace(vector_targets)
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circle.scale(1.04)
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velocity_space = TextMobject("Velocity space")
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velocity_space.next_to(circle, UP)
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@ -2712,3 +2724,284 @@ class IntroduceShapeOfVelocities(AskAboutEllipses, MovingCameraScene):
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vector.copy().set_stroke(BLACK, 5)
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)
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return vector
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class AskWhy(TeacherStudentsScene):
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def construct(self):
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self.student_says(
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"Um...why?",
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target_mode="confused",
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student_index=2,
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bubble_kwargs={"direction": LEFT},
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)
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self.play(
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self.teacher.change, "happy",
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self.get_student_changes(
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"raise_left_hand", "sassy", "confused"
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)
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)
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self.wait(5)
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class FeynmanConfusedByNewton(Scene):
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def construct(self):
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pass
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class ShowEqualAngleSlices(IntroduceShapeOfVelocities):
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CONFIG = {
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"animate_sun": False,
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"theta": 30 * DEGREES,
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}
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def construct(self):
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self.setup_orbit()
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self.show_equal_angle_slices()
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self.label_points_P_theta()
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self.ask_about_time_per_slice()
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self.areas_are_proportional_to_radius_squared()
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self.show_inverse_square_law()
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self.show_change_in_velocity()
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self.directly_compare_velocity_vectors()
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self.show_equal_angle_changes()
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def setup_orbit(self):
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IntroduceShapeOfVelocities.setup_orbit(self)
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self.remove(self.orbit)
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self.add(self.comet)
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def show_equal_angle_slices(self):
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sun_center = self.sun.get_center()
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ellipse = self.ellipse
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def get_cos_angle_diff(v1, v2):
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return np.dot(
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v1 / np.linalg.norm(v1),
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v2 / np.linalg.norm(v2),
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)
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lines = VGroup()
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angle_arcs = VGroup()
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thetas = VGroup()
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angles = np.arange(0, TAU, self.theta)
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for angle in angles:
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prop = angle / TAU
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vect = rotate_vector(RIGHT, angle)
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end_point = ellipse.point_from_proportion(prop)
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curr_cos = get_cos_angle_diff(
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end_point - sun_center, vect
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)
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coss_diff = (1 - curr_cos)
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while abs(coss_diff) > 0.00001:
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d_prop = 0.001
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alt_end = ellipse.point_from_proportion(
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(prop + d_prop) % 1
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)
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alt_cos = get_cos_angle_diff(alt_end - sun_center, vect)
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d_cos = (alt_cos - curr_cos)
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delta_prop = (coss_diff / d_cos) * d_prop
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prop += delta_prop
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end_point = ellipse.point_from_proportion(prop)
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curr_cos = get_cos_angle_diff(end_point - sun_center, vect)
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coss_diff = 1 - curr_cos
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line = Line(sun_center, end_point)
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line.prop = prop
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lines.add(line)
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angle_arc = AnnularSector(
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angle=self.theta,
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inner_radius=1,
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outer_radius=1.05,
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)
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angle_arc.rotate(angle, about_point=ORIGIN)
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angle_arc.scale(0.5, about_point=ORIGIN)
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angle_arc.shift(sun_center)
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angle_arc.mid_angle = angle + self.theta / 2
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angle_arcs.add(angle_arc)
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theta = TexMobject("\\theta")
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theta.scale(0.6)
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vect = rotate_vector(RIGHT, angle_arc.mid_angle)
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theta.move_to(
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angle_arc.get_center() + 0.2 * vect
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)
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thetas.add(theta)
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arcs = VGroup()
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wedges = VGroup()
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for l1, l2 in adjacent_pairs(lines):
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arc = VMobject()
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arc.pointwise_become_partial(
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ellipse, l1.prop, (l2.prop or 1.0)
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)
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arcs.add(arc)
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wedge = VMobject()
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wedge.append_vectorized_mobject(
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Line(sun_center, arc.points[0])
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)
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wedge.append_vectorized_mobject(arc)
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wedge.append_vectorized_mobject(
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Line(arc.points[-1], sun_center)
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)
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wedges.add(wedge)
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lines.set_stroke(LIGHT_GREY, 2)
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angle_arcs.set_color_by_gradient(
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YELLOW, BLUE, RED, PINK, YELLOW
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)
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arcs.set_color_by_gradient(BLUE, YELLOW)
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wedges.set_stroke(width=0)
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wedges.set_fill(opacity=1)
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wedges.set_color_by_gradient(BLUE, COBALT, BLUE_E, BLUE)
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kwargs = {
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"run_time": 6,
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"lag_ratio": 0.2,
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"rate_func": there_and_back,
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}
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faders = VGroup(wedges, angle_arcs, thetas)
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faders.set_fill(opacity=0.4)
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thetas.set_fill(opacity=0)
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self.play(
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FadeIn(faders),
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*map(ShowCreation, lines)
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)
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self.play(*[
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LaggedStart(
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ApplyMethod, fader,
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lambda m: (m.set_fill, {"opacity": 1}),
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**kwargs
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)
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for fader in faders
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] + [Animation(lines)])
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self.wait()
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self.lines = lines
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self.wedges = wedges
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self.arcs = arcs
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self.angle_arcs = angle_arcs
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self.thetas = thetas
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def label_points_P_theta(self):
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sun_center = self.sun.get_center()
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lines = self.lines
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thick_lines = lines.copy().set_stroke(YELLOW, 4)
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frame_scale_factor = 1.2
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frame = self.camera_frame
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P_labels = VGroup()
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dots = VGroup()
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for line in lines:
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dot = Dot(line.get_end())
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dot.scale(0.5)
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dot.set_color(YELLOW)
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dots.add(dot)
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angle = angle_of_vector(line.get_end() - sun_center)
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angle = angle % TAU
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angle_degrees = angle * (360 / TAU)
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P_label = TexMobject(
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"P_{%d^\\circ}" % int(np.round(angle_degrees, -1))
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)
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P_label.next_to(
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dot, line.get_unit_vector(),
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buff=SMALL_BUFF,
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submobject_to_align=P_label[:2],
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)
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P_labels.add(P_label)
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self.play(
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LaggedStart(GrowFromCenter, dots),
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LaggedStart(Write, P_labels),
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Animation(self.comet),
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frame.scale, frame_scale_factor
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)
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self.wait(2)
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self.play(*map(FadeOut, filter(
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lambda pl: pl is not P_labels[1],
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P_labels
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)))
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self.play(ShowCreationThenDestruction(thick_lines[1]))
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self.wait()
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for i in 1, 2, 3:
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self.play(
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FadeOut(P_labels[i]),
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FadeIn(P_labels[i + 1]),
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)
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self.play(ShowCreationThenDestruction(thick_lines[i + 1]))
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self.wait()
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self.play(
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FadeOut(P_labels[4]),
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# frame.scale, 1.0 / frame_scale_factor
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)
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self.dots = dots
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self.P_labels = P_labels
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def ask_about_time_per_slice(self):
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wedge1 = self.wedges[0]
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wedge2 = self.wedges[6]
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arc1 = self.arcs[0]
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arc2 = self.arcs[6]
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comet = self.comet
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frame = self.camera_frame
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words1 = TextMobject(
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"Time spent \\\\ traversing \\\\ this slice?"
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)
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words2 = TextMobject("How about \\\\ this one?")
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words1.to_corner(UR)
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words2.next_to(wedge2, LEFT, MED_LARGE_BUFF)
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arrow1 = Arrow(
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words1.get_bottom(),
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wedge1.get_center() + 0.5 * DOWN,
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color=WHITE,
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)
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arrow2 = Arrow(
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words2.get_right(),
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wedge2.get_center() + 0.5 * UL,
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color=WHITE
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)
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foreground = VGroup(
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self.ellipse, self.angle_arcs, self.lines,
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self.dots, comet,
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)
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self.play(
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Write(words1),
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wedge1.set_fill, {"opacity": 1},
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GrowArrow(arrow1),
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Animation(foreground),
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)
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self.play(MoveAlongPath(comet, arc1, rate_func=None))
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self.play(
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Write(words2),
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wedge2.set_fill, {"opacity": 1},
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Write(arrow2),
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Animation(foreground),
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)
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self.play(MoveAlongPath(comet, arc2, rate_func=None, run_time=3))
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self.wait()
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def areas_are_proportional_to_radius_squared(self):
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pass
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def show_inverse_square_law(self):
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pass
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def show_change_in_velocity(self):
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pass
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def directly_compare_velocity_vectors(self):
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pass
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def show_equal_angle_changes(self):
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pass
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