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Simple mvcr animations
This commit is contained in:
Grant Sanderson
@ -332,8 +332,8 @@ class ShowDerivativeAtExamplePoint(Scene):
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["{\\partial f / \\partial x}"],
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["{\\partial f / \\partial y}"],
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])
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partial_vect.get_mob_matrix()[1, 0][-1].set_color(BLUE)
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partial_vect.get_mob_matrix()[0, 0][-1].set_color(RED)
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partial_vect.get_mob_matrix()[0, 0][-1].set_color(BLUE)
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partial_vect.get_mob_matrix()[1, 0][-1].set_color(RED)
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result_vector = self.get_result_vector("x", "y")
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gradient = VGroup(
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713
active_projects/mvcr.py
Normal file
713
active_projects/mvcr.py
Normal file
@ -0,0 +1,713 @@
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from big_ol_pile_of_manim_imports import *
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class HoldUpMultivariableChainRule(TeacherStudentsScene):
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def construct(self):
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title = TextMobject("Multivariable chain rule")
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title.to_edge(UP, buff=MED_SMALL_BUFF)
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screen = ScreenRectangle()
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screen.next_to(title, DOWN)
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morty = self.teacher
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self.add(title)
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self.play(
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morty.change, "raise_right_hand",
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FadeInFromDown(screen)
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)
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self.change_all_student_modes(
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"confused", look_at_arg=screen
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)
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self.look_at(screen)
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self.wait(10)
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class ComputationalNetwork(MovingCameraScene):
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CONFIG = {
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"x_color": YELLOW,
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"f_color": BLUE,
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"g_color": GREEN,
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"h_color": RED,
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}
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def construct(self):
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self.draw_network()
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self.walk_through_parts()
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self.write_dh_dx_goal()
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self.feed_forward_input()
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self.compare_x_and_h_wiggling()
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self.expand_out_h_as_function_of_x()
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self.show_four_derivatives()
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self.show_chain_rule()
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self.talk_through_mvcr_parts()
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self.plug_in_expressions()
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self.plug_in_values()
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self.discuss_meaning_of_result()
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def draw_network(self):
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x = TexMobject("x")
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f_formula = TexMobject("f", "=", "x", "^2")
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g_formula = TexMobject("g", "=", "\\cos(\\pi", "x", ")")
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h_formula = TexMobject("h", "=", "f", "^2", "g")
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self.tex_to_color_map = {
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"x": self.x_color,
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"f": self.f_color,
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"g": self.g_color,
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"h": self.h_color,
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}
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formulas = VGroup(x, f_formula, g_formula, h_formula)
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formula_groups = VGroup()
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for formula in formulas:
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formula.box = SurroundingRectangle(formula)
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formula.box.set_color(WHITE)
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formula.group = VGroup(formula, formula.box)
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formula.set_color_by_tex_to_color_map(
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self.tex_to_color_map
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)
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formula_groups.add(formula.group)
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f_formula.box.match_width(
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g_formula.box, stretch=True, about_edge=LEFT
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)
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fg_group = VGroup(f_formula.group, g_formula.group)
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fg_group.arrange_submobjects(DOWN, buff=LARGE_BUFF)
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fg_group.to_edge(UP)
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x.group.next_to(fg_group, LEFT, buff=2)
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h_formula.group.next_to(fg_group, RIGHT, buff=2)
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xf_line = Line(x.box.get_right(), f_formula.box.get_left())
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xg_line = Line(x.box.get_right(), g_formula.box.get_left())
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fh_line = Line(f_formula.box.get_right(), h_formula.box.get_left())
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gh_line = Line(g_formula.box.get_right(), h_formula.box.get_left())
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graph_edges = VGroup(
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xf_line, xg_line, fh_line, gh_line
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)
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self.play(
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Write(x),
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FadeIn(x.box),
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)
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self.play(
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ShowCreation(xf_line),
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ShowCreation(xg_line),
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ReplacementTransform(x.box.copy(), f_formula.box),
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ReplacementTransform(x.box.copy(), g_formula.box),
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)
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self.play(
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Write(f_formula),
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Write(g_formula),
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)
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self.play(
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ShowCreation(fh_line),
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ShowCreation(gh_line),
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ReplacementTransform(f_formula.box.copy(), h_formula.box),
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ReplacementTransform(g_formula.box.copy(), h_formula.box),
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)
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self.play(Write(h_formula))
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self.wait()
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network = VGroup(formula_groups, graph_edges)
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self.set_variables_as_attrs(
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x, f_formula, g_formula, h_formula,
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xf_line, xg_line, fh_line, gh_line,
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formulas, formula_groups,
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graph_edges, network
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)
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def walk_through_parts(self):
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x = self.x
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f_formula = self.f_formula
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g_formula = self.g_formula
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h_formula = self.h_formula
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def indicate(mob):
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return ShowCreationThenDestructionAround(
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mob,
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surrounding_rectangle_config={
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"buff": 0.5 * SMALL_BUFF,
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"color": mob.get_color()
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}
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)
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for formula in f_formula, g_formula:
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self.play(indicate(formula[0]))
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self.play(ReplacementTransform(
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x.copy(),
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formula.get_parts_by_tex("x"),
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path_arc=PI / 3
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))
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self.wait()
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self.play(indicate(h_formula[0]))
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self.play(ReplacementTransform(
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f_formula[0].copy(),
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h_formula.get_part_by_tex("f"),
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path_arc=PI / 3
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))
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self.play(ReplacementTransform(
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g_formula[0].copy(),
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h_formula.get_part_by_tex("g"),
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path_arc=PI / 3
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))
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self.wait()
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def write_dh_dx_goal(self):
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deriv = TexMobject(
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"{dh", "\\over", "dx}", "(", "2", ")"
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)
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deriv.set_color_by_tex_to_color_map(
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self.tex_to_color_map
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)
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deriv.scale(1.5)
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deriv.move_to(DOWN)
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self.play(FadeInFromDown(deriv[:3]))
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self.play(ShowCreationThenDestructionAround(deriv[:3]))
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self.wait(2)
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self.play(Write(deriv[3:]))
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self.wait()
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self.dh_dx_at_two = deriv
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def feed_forward_input(self):
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formula_groups = self.formula_groups
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x, f_formula, g_formula, h_formula = self.formulas
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dh_dx_at_two = self.dh_dx_at_two
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values = [2, 4, 1, 16]
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value_labels = VGroup()
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for formula_group, value in zip(formula_groups, values):
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label = TexMobject("=", str(value))
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eq, value_mob = label
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eq.rotate(90 * DEGREES)
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eq.next_to(value_mob, UP, SMALL_BUFF)
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var = formula_group[0][0]
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label[1].match_color(var)
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# label.next_to(formula_group, DOWN, SMALL_BUFF)
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label.next_to(var, DOWN, SMALL_BUFF)
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eq.add_background_rectangle(buff=SMALL_BUFF, opacity=1)
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value_labels.add(label)
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x_val_label, f_val_label, g_val_label, h_val_label = value_labels
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two, four, one, sixteen = [vl[1] for vl in value_labels]
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self.play(
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ReplacementTransform(
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dh_dx_at_two.get_part_by_tex("2").copy(),
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two,
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),
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Write(x_val_label[0])
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)
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self.wait()
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two_copy1 = two.copy()
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two_copy2 = two.copy()
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four_copy = four.copy()
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one_copy = one.copy()
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x_in_f = f_formula.get_part_by_tex("x")
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x_in_g = g_formula.get_part_by_tex("x")
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f_in_h = h_formula.get_part_by_tex("f")
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g_in_h = h_formula.get_part_by_tex("g")
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self.play(
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two_copy1.move_to, x_in_f, DOWN,
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x_in_f.set_fill, {"opacity": 0},
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)
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self.wait()
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self.play(Write(f_val_label))
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self.wait()
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self.play(
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two_copy2.move_to, x_in_g, DOWN,
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x_in_g.set_fill, {"opacity": 0},
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)
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self.wait()
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self.play(Write(g_val_label))
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self.wait()
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self.play(
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four_copy.move_to, f_in_h, DOWN,
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f_in_h.set_fill, {"opacity": 0},
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)
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self.play(
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one_copy.move_to, g_in_h, DOWN,
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g_in_h.set_fill, {"opacity": 0},
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)
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self.wait()
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self.play(Write(h_val_label))
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self.wait()
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self.value_labels = value_labels
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self.revert_to_formula_animations = [
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ApplyMethod(term.set_fill, {"opacity": 1})
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for term in x_in_f, x_in_g, f_in_h, g_in_h
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] + [
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FadeOut(term)
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for term in two_copy1, two_copy2, four_copy, one_copy
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]
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def compare_x_and_h_wiggling(self):
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x_val = self.value_labels[0][1]
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h_val = self.value_labels[3][1]
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x_line = NumberLine(
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x_min=0,
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x_max=4,
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include_numbers=True,
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numbers_to_show=[0, 2, 4],
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unit_size=0.75,
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)
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x_line.next_to(
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x_val, DOWN, LARGE_BUFF,
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aligned_edge=RIGHT
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)
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h_line = NumberLine(
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x_min=0,
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x_max=32,
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include_numbers=True,
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numbers_with_elongated_ticks=[0, 16, 32],
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numbers_to_show=[0, 16, 32],
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tick_frequency=1,
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tick_size=0.05,
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unit_size=1.0 / 12,
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)
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h_line.next_to(
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h_val, DOWN, LARGE_BUFF,
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aligned_edge=LEFT
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)
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x_dot = Dot(color=self.x_color)
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x_dot.move_to(x_line.number_to_point(2))
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x_arrow = Arrow(self.x.get_bottom(), x_dot.get_top())
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x_arrow.match_color(x_dot)
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h_dot = Dot(color=self.h_color)
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h_dot.move_to(h_line.number_to_point(16))
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h_arrow = Arrow(self.h_formula[0].get_bottom(), h_dot.get_top())
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h_arrow.match_color(h_dot)
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self.play(
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ShowCreation(x_line),
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ShowCreation(h_line),
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)
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self.play(
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GrowArrow(x_arrow),
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GrowArrow(h_arrow),
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ReplacementTransform(x_val.copy(), x_dot),
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ReplacementTransform(h_val.copy(), h_dot),
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)
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self.wait()
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for x in range(2):
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self.play(
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x_dot.shift, 0.25 * RIGHT,
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h_dot.shift, 0.35 * RIGHT,
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rate_func=wiggle,
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run_time=1,
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)
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self.set_variables_as_attrs(
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x_line, h_line,
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x_dot, h_dot,
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x_arrow, h_arrow,
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)
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def expand_out_h_as_function_of_x(self):
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self.play(*self.revert_to_formula_animations)
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deriv = self.dh_dx_at_two
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expanded_formula = TexMobject(
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"h = x^4 \\cos(\\pi x)",
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tex_to_color_map=self.tex_to_color_map
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)
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expanded_formula.move_to(deriv)
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cross = Cross(expanded_formula)
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self.play(
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FadeInFromDown(expanded_formula),
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deriv.scale, 1.0 / 1.5,
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deriv.shift, DOWN,
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)
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self.wait()
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self.play(ShowCreation(cross))
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self.wait()
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self.play(
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FadeOut(VGroup(expanded_formula, cross)),
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deriv.shift, UP,
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)
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for edge in self.graph_edges:
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self.play(ShowCreationThenDestruction(
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edge.copy().set_stroke(YELLOW, 6)
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))
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def show_four_derivatives(self):
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lines = self.graph_edges
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xf_line, xg_line, fh_line, gh_line = lines
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df_dx = TexMobject("df", "\\over", "dx")
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dg_dx = TexMobject("dg", "\\over", "dx")
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dh_df = TexMobject("\\partial h", "\\over", "\\partial f")
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dh_dg = TexMobject("\\partial h", "\\over", "\\partial g")
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derivatives = VGroup(df_dx, dg_dx, dh_df, dh_dg)
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df_dx.next_to(xf_line.get_center(), UP, SMALL_BUFF)
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dg_dx.next_to(xg_line.get_center(), DOWN, SMALL_BUFF)
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dh_df.next_to(fh_line.get_center(), UP, SMALL_BUFF)
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dh_dg.next_to(gh_line.get_center(), DOWN, SMALL_BUFF)
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partial_terms = VGroup(
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dh_df[0][0],
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dh_df[2][0],
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dh_dg[0][0],
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dh_dg[2][0],
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)
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partial_term_rects = VGroup(*[
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SurroundingRectangle(pt, buff=0.05)
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for pt in partial_terms
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])
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partial_term_rects.set_stroke(width=0)
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partial_term_rects.set_fill(TEAL, 0.5)
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self.play(FadeOut(self.value_labels))
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for derivative in derivatives:
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derivative.set_color_by_tex_to_color_map(self.tex_to_color_map)
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derivative.add_to_back(derivative.copy().set_stroke(BLACK, 5))
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self.play(FadeInFromDown(derivative))
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self.wait()
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self.play(
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LaggedStart(FadeIn, partial_term_rects),
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Animation(derivatives)
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)
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self.wait()
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self.play(
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LaggedStart(FadeOut, partial_term_rects),
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Animation(derivatives)
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)
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self.derivatives = derivatives
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def show_chain_rule(self):
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dh_dx_at_two = self.dh_dx_at_two
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dh_dx = dh_dx_at_two[:3]
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at_two = dh_dx_at_two[3:]
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derivatives = self.derivatives.copy()
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df_dx, dg_dx, dh_df, dh_dg = derivatives
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frame = self.camera_frame
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self.play(
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frame.shift, 3 * UP,
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dh_dx.to_edge, UP,
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dh_dx.shift, 3 * LEFT + 3 * UP,
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at_two.set_fill, {"opacity": 0},
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)
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for deriv in derivatives:
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deriv.generate_target()
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rhs = VGroup(
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TexMobject("="),
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df_dx.target, dh_df.target,
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TexMobject("+"),
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dg_dx.target, dh_dg.target
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)
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rhs.arrange_submobjects(
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RIGHT,
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buff=2 * SMALL_BUFF,
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)
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rhs.next_to(dh_dx, RIGHT)
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for deriv in derivatives:
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y = rhs[0].get_center()[1]
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alt_y = deriv.target[2].get_center()[1]
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deriv.target.shift((y - alt_y) * UP)
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self.play(
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Write(rhs[::3]),
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LaggedStart(MoveToTarget, derivatives)
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)
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self.wait()
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self.chain_rule_derivatives = derivatives
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self.chain_rule_rhs = rhs
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def talk_through_mvcr_parts(self):
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derivatives = self.derivatives
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cr_derivatives = self.chain_rule_derivatives
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df_dx, dg_dx, dh_df, dh_dg = cr_derivatives
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df, dx1 = df_dx[1::2]
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dg, dx2 = dg_dx[1::2]
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del_h1, del_f = dh_df[1::2]
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del_h2, del_g = dh_dg[1::2]
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terms = VGroup(df, dx1, dg, dx2, del_h1, del_f, del_h2, del_g)
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for term in terms:
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||||
term.rect = SurroundingRectangle(
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||||
term,
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||||
buff=0.5 * SMALL_BUFF,
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||||
stroke_width=0,
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||||
fill_color=TEAL,
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||||
fill_opacity=0.5
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||||
)
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||||
for derivative in derivatives:
|
||||
derivative.rect = SurroundingRectangle(
|
||||
derivative,
|
||||
color=TEAL
|
||||
)
|
||||
|
||||
del_h_sub_f = TexMobject("f")
|
||||
del_h_sub_f.scale(0.5)
|
||||
del_h_sub_f.next_to(del_h1.get_corner(DR), RIGHT, buff=0)
|
||||
del_h_sub_f.set_color(self.f_color)
|
||||
|
||||
lines = self.graph_edges
|
||||
top_lines = lines[::2].copy()
|
||||
bottom_lines = lines[1::2].copy()
|
||||
for group in top_lines, bottom_lines:
|
||||
group.set_stroke(YELLOW, 6)
|
||||
|
||||
self.add_foreground_mobjects(cr_derivatives)
|
||||
rect = dx1.rect.copy()
|
||||
rect.save_state()
|
||||
rect.scale(3)
|
||||
rect.set_fill(opacity=0)
|
||||
|
||||
self.play(
|
||||
rect.restore,
|
||||
FadeIn(derivatives[0].rect)
|
||||
)
|
||||
self.wait()
|
||||
self.play(Transform(rect, df.rect))
|
||||
self.wait()
|
||||
self.play(
|
||||
rect.replace, df_dx, {"stretch": True},
|
||||
rect.scale, 1.1,
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
Transform(rect, del_f.rect),
|
||||
FadeOut(derivatives[0].rect),
|
||||
FadeIn(derivatives[2].rect),
|
||||
)
|
||||
self.wait()
|
||||
self.play(Transform(rect, del_h1.rect))
|
||||
self.wait()
|
||||
self.play(ReplacementTransform(
|
||||
del_f[1].copy(), del_h_sub_f,
|
||||
path_arc=PI,
|
||||
))
|
||||
self.wait()
|
||||
self.play(
|
||||
del_h_sub_f.shift, UR,
|
||||
del_h_sub_f.fade, 1,
|
||||
rate_func=running_start,
|
||||
remover=True
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
Transform(rect, del_f.rect),
|
||||
ReplacementTransform(rect.copy(), df.rect),
|
||||
)
|
||||
self.wait()
|
||||
for x in range(3):
|
||||
self.play(ShowCreationThenDestruction(
|
||||
top_lines,
|
||||
submobject_mode="one_at_a_time"
|
||||
))
|
||||
self.wait()
|
||||
self.play(
|
||||
rect.replace, cr_derivatives[1::2], {"stretch": True},
|
||||
rect.scale, 1.1,
|
||||
FadeOut(df.rect),
|
||||
FadeOut(derivatives[2].rect),
|
||||
FadeIn(derivatives[1].rect),
|
||||
FadeIn(derivatives[3].rect),
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
Transform(rect, dg.rect),
|
||||
FadeOut(derivatives[3].rect)
|
||||
)
|
||||
self.wait()
|
||||
self.play(Transform(rect, dx2.rect))
|
||||
self.wait()
|
||||
self.play(
|
||||
Transform(rect, del_h2.rect),
|
||||
FadeOut(derivatives[1].rect),
|
||||
FadeIn(derivatives[3].rect),
|
||||
)
|
||||
self.wait()
|
||||
self.play(Transform(rect, del_g.rect))
|
||||
self.wait()
|
||||
self.play(
|
||||
rect.replace, cr_derivatives, {"stretch": True},
|
||||
rect.scale, 1.1,
|
||||
FadeOut(derivatives[3].rect)
|
||||
)
|
||||
for x in range(3):
|
||||
self.play(*[
|
||||
ShowCreationThenDestruction(
|
||||
group,
|
||||
submobject_mode="one_at_a_time"
|
||||
)
|
||||
for group in top_lines, bottom_lines
|
||||
])
|
||||
self.wait()
|
||||
self.play(FadeOut(rect))
|
||||
self.remove_foreground_mobject(cr_derivatives)
|
||||
|
||||
def plug_in_expressions(self):
|
||||
lhs = VGroup(
|
||||
self.dh_dx_at_two[:3],
|
||||
self.chain_rule_rhs[::3],
|
||||
self.chain_rule_derivatives,
|
||||
)
|
||||
lhs.generate_target()
|
||||
lhs.target.to_edge(LEFT)
|
||||
df_dx, dg_dx, dh_df, dh_dg = self.chain_rule_derivatives
|
||||
|
||||
formulas = self.formulas
|
||||
x, f_formula, g_formula, h_formula = formulas
|
||||
|
||||
full_derivative = TexMobject(
|
||||
"=",
|
||||
"(", "2", "x", ")",
|
||||
"(", "2", "f", "g", ")",
|
||||
"+",
|
||||
"(", "-\\sin(", "\\pi", "x", ")", "\\pi", ")",
|
||||
"(", "f", "^2", ")"
|
||||
)
|
||||
full_derivative.next_to(lhs.target, RIGHT)
|
||||
full_derivative.set_color_by_tex_to_color_map(
|
||||
self.tex_to_color_map
|
||||
)
|
||||
|
||||
self.play(MoveToTarget(lhs))
|
||||
self.play(Write(full_derivative[0]))
|
||||
|
||||
# df/dx
|
||||
self.play(
|
||||
f_formula.shift, UP,
|
||||
df_dx.shift, 0.5 * DOWN
|
||||
)
|
||||
self.play(
|
||||
ReplacementTransform(
|
||||
f_formula[2:].copy(),
|
||||
full_derivative[2:4],
|
||||
),
|
||||
FadeIn(full_derivative[1:5:3])
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
f_formula.shift, DOWN,
|
||||
df_dx.shift, 0.5 * UP
|
||||
)
|
||||
self.wait()
|
||||
|
||||
# dg/dx
|
||||
self.play(
|
||||
g_formula.shift, 0.75 * UP,
|
||||
dg_dx.shift, 0.5 * DOWN
|
||||
)
|
||||
self.play(
|
||||
ReplacementTransform(
|
||||
g_formula[2:].copy(),
|
||||
full_derivative[12:17],
|
||||
),
|
||||
FadeIn(full_derivative[11:18:6]),
|
||||
Write(full_derivative[10]),
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
g_formula.shift, 0.75 * DOWN,
|
||||
dg_dx.shift, 0.5 * UP
|
||||
)
|
||||
self.wait()
|
||||
|
||||
# dh/df
|
||||
self.play(
|
||||
h_formula.shift, UP,
|
||||
dh_df.shift, 0.5 * DOWN
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
ReplacementTransform(
|
||||
h_formula[2:].copy(),
|
||||
full_derivative[6:9],
|
||||
),
|
||||
FadeIn(full_derivative[5:10:4])
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
dh_df.shift, 0.5 * UP
|
||||
)
|
||||
|
||||
# dh/dg
|
||||
self.play(
|
||||
dh_dg.shift, 0.5 * DOWN,
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
ReplacementTransform(
|
||||
h_formula[2:].copy(),
|
||||
full_derivative[19:21],
|
||||
),
|
||||
FadeIn(full_derivative[18:22:3])
|
||||
)
|
||||
self.wait()
|
||||
self.play(
|
||||
h_formula.shift, DOWN,
|
||||
dh_dg.shift, 0.5 * UP
|
||||
)
|
||||
self.wait()
|
||||
|
||||
self.full_derivative = full_derivative
|
||||
|
||||
def plug_in_values(self):
|
||||
full_derivative = self.full_derivative
|
||||
value_labels = self.value_labels
|
||||
|
||||
rhs = TexMobject(
|
||||
"""
|
||||
=
|
||||
(2 \\cdot 2)
|
||||
(2 \\cdot 4 \\cdot 1) +
|
||||
(-\\sin(\\pi 2)\\pi)(4^2)
|
||||
""",
|
||||
tex_to_color_map={
|
||||
"2": self.x_color,
|
||||
"4": self.f_color,
|
||||
"1": self.g_color,
|
||||
"^2": WHITE,
|
||||
}
|
||||
)
|
||||
rhs.next_to(full_derivative, DOWN, aligned_edge=LEFT)
|
||||
|
||||
result = TexMobject("=", "32", "+", "0")
|
||||
result.next_to(rhs, DOWN, aligned_edge=LEFT)
|
||||
|
||||
self.play(LaggedStart(Write, value_labels))
|
||||
self.wait()
|
||||
self.play(ReplacementTransform(
|
||||
full_derivative.copy(), rhs,
|
||||
submobject_mode="lagged_start",
|
||||
run_time=2
|
||||
))
|
||||
self.wait()
|
||||
self.play(Write(result))
|
||||
self.wait()
|
||||
|
||||
def discuss_meaning_of_result(self):
|
||||
x_dot = self.x_dot
|
||||
h_dot = self.h_dot
|
||||
|
||||
for x in range(3):
|
||||
self.play(
|
||||
x_dot.shift, 0.25 * RIGHT,
|
||||
h_dot.shift, RIGHT,
|
||||
run_time=2,
|
||||
rate_func=lambda t: wiggle(t, 4)
|
||||
)
|
||||
self.wait()
|
||||
Reference in New Issue
Block a user