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https://github.com/3b1b/manim.git
synced 2025-08-02 11:03:03 +08:00
More animations on Cramer
This commit is contained in:
Grant Sanderson
@ -1,11 +1,10 @@
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from big_ol_pile_of_manim_imports import *
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X_COLOR = GREEN
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Y_COLOR = RED
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Z_COLOR = BLUE
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OUTPUT_COLOR = MAROON_B
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INPUT_COLOR = YELLOW
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OUTPUT_COLOR = YELLOW
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INPUT_COLOR = MAROON_B
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def get_cramer_matrix(matrix, output_vect, index=0):
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@ -248,7 +247,9 @@ class NotTheMostComputationallyEfficient(Scene):
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class SetupSimpleSystemOfEquations(LinearTransformationScene):
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CONFIG = {
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"matrix": [[3, 2], [-1, 2]],
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"output_vect": [-5, -1],
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"output_vect": [-4, -2],
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"quit_before_final_transformation": False,
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"array_scale_factor": 0.75
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}
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def construct(self):
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@ -376,7 +377,7 @@ class SetupSimpleSystemOfEquations(LinearTransformationScene):
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np.linalg.inv(first_half_matrix),
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)
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scale_factor = 0.75
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scale_factor = self.array_scale_factor
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column_mobs = VGroup()
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for i in 0, 1:
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@ -438,11 +439,18 @@ class SetupSimpleSystemOfEquations(LinearTransformationScene):
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])
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self.play(ShowPassingFlash(q_marks_rect))
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self.wait(2)
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self.apply_matrix(self.matrix, added_anims=[
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FadeOut(q_marks),
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ReplacementTransform(input_vect_mob, output_vect_mob),
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FadeIn(column_mobs),
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])
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if not self.quit_before_final_transformation:
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self.apply_matrix(self.matrix, added_anims=[
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FadeOut(q_marks),
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ReplacementTransform(input_vect_mob, output_vect_mob),
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FadeIn(column_mobs),
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])
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self.wait()
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self.q_marks = q_marks
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self.input_vect_mob = input_vect_mob
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self.output_vect_mob = output_vect_mob
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self.column_mobs = column_mobs
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# Helpers
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@ -490,3 +498,320 @@ class SetupSimpleSystemOfEquations(LinearTransformationScene):
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index_of_submobject_to_align=-2
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)
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return system
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class ShowZeroDeterminantCase(LinearTransformationScene):
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CONFIG = {
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"show_basis_vectors": True,
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"matrix": [[3, -2.0], [1, -2.0 / 3]],
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"tex_scale_factor": 1.25,
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"det_eq_symbol": "=",
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}
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def construct(self):
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self.add_equation()
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self.show_det_zero()
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def add_equation(self):
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equation = self.equation = TexMobject(
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"A", "\\vec{\\textbf{x}}", "=", "\\vec{\\textbf{v}}"
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)
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equation.scale(self.tex_scale_factor)
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equation.set_color_by_tex("{x}", INPUT_COLOR)
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equation.set_color_by_tex("{v}", OUTPUT_COLOR)
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equation.add_background_rectangle()
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equation.to_corner(UL)
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self.add(equation)
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self.add_foreground_mobject(equation)
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def show_det_zero(self):
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matrix = self.matrix
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# vect_in_span = [2, 2.0 / 3]
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vect_in_span = [2, 2.0 / 3]
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vect_off_span = [1, 2]
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vect_in_span_mob = self.get_vector(vect_in_span, color=OUTPUT_COLOR)
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vect_off_span_mob = self.get_vector(vect_off_span, color=OUTPUT_COLOR)
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for vect_mob in vect_in_span_mob, vect_off_span_mob:
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circle = Circle(color=WHITE, radius=0.15, stroke_width=2)
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circle.move_to(vect_mob.get_end())
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vect_mob.circle = circle
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vect_off_span_words = TextMobject("No input lands here")
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vect_off_span_words.next_to(vect_off_span_mob.circle, UP)
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vect_off_span_words.add_background_rectangle()
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vect_in_span_words = TextMobject("Many inputs lands here")
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vect_in_span_words.next_to(vect_in_span_mob.circle, DR)
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vect_in_span_words.shift_onto_screen()
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vect_in_span_words.add_background_rectangle()
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moving_group = VGroup(self.plane, self.basis_vectors)
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moving_group.save_state()
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solution = np.dot(np.linalg.pinv(matrix), vect_in_span)
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import sympy
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null_space_basis = np.array(sympy.Matrix(matrix).nullspace())
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null_space_basis = null_space_basis.flatten().astype(float)
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solution_vectors = VGroup(*[
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self.get_vector(
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solution + x * null_space_basis,
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rectangular_stem_width=0.025,
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tip_length=0.2,
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)
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for x in np.linspace(-4, 4, 20)
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])
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solution_vectors.set_color_by_gradient(YELLOW, MAROON_B)
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self.apply_matrix(matrix, path_arc=0)
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self.wait()
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self.show_det_equation()
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self.wait()
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# Mention zero determinants
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self.play(GrowArrow(vect_off_span_mob))
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self.play(
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ShowCreation(vect_off_span_mob.circle),
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Write(vect_off_span_words),
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)
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self.wait()
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self.play(
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FadeOut(vect_off_span_mob.circle),
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ReplacementTransform(vect_off_span_mob, vect_in_span_mob),
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ReplacementTransform(vect_off_span_words, vect_in_span_words),
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ReplacementTransform(vect_off_span_mob.circle, vect_in_span_mob.circle),
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)
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self.wait(2)
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self.play(
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FadeOut(vect_in_span_words),
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FadeOut(vect_in_span_mob.circle),
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ApplyMethod(moving_group.restore, run_time=2),
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ReplacementTransform(
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VGroup(vect_in_span_mob),
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solution_vectors,
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run_time=2,
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),
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)
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self.wait()
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# Helpers
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def show_det_equation(self):
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equation = self.equation
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det_equation = TexMobject(
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"\\det(", "A", ")", self.det_eq_symbol, "0"
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)
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det_equation.scale(self.tex_scale_factor)
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det_equation.next_to(equation, DOWN, buff=MED_LARGE_BUFF, aligned_edge=LEFT)
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det_rect = BackgroundRectangle(det_equation)
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self.play(
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FadeIn(det_rect),
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Write(det_equation[0]),
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ReplacementTransform(
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equation.get_part_by_tex("A").copy(),
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det_equation.get_part_by_tex("A").copy(),
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),
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Write(det_equation[2:]),
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)
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self.add_foreground_mobject(det_rect, det_equation)
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class NonZeroDeterminantCase(ShowZeroDeterminantCase, SetupSimpleSystemOfEquations):
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CONFIG = {
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"det_eq_symbol": "\\neq"
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}
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def construct(self):
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self.add_equation()
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self.show_det_equation()
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output_vect = self.output_vect
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matrix = self.matrix
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input_vect = np.dot(np.linalg.inv(matrix), output_vect)
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input_vect_mob = self.get_vector(input_vect, color=INPUT_COLOR)
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output_vect_mob = self.get_vector(output_vect, color=OUTPUT_COLOR)
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input_vect_label = TextMobject("Input")
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input_vect_label.next_to(input_vect_mob.get_end(), DOWN, SMALL_BUFF)
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input_vect_label.match_color(input_vect_mob)
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output_vect_label = TextMobject("Output")
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output_vect_label.next_to(output_vect_mob.get_end(), DOWN, SMALL_BUFF)
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output_vect_label.match_color(output_vect_mob)
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for label in input_vect_label, output_vect_label:
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label.scale(1.25, about_edge=UP)
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label.add_background_rectangle()
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self.apply_matrix(matrix)
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self.wait()
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self.apply_inverse(matrix)
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self.wait()
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self.play(
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GrowArrow(input_vect_mob),
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Write(input_vect_label),
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run_time=1
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)
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self.wait()
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self.remove(input_vect_mob, input_vect_label)
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self.apply_matrix(matrix, added_anims=[
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ReplacementTransform(input_vect_mob.copy(), output_vect_mob),
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ReplacementTransform(input_vect_label.copy(), output_vect_label),
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], run_time=2)
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self.wait()
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self.remove(output_vect_mob, output_vect_label)
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self.apply_inverse(matrix, added_anims=[
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ReplacementTransform(output_vect_mob.copy(), input_vect_mob),
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ReplacementTransform(output_vect_label.copy(), input_vect_label),
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], run_time=2)
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self.wait()
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class ThinkOfPuzzleAsLinearCombination(SetupSimpleSystemOfEquations):
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CONFIG = {
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"output_vect": [-4, -2],
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}
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def construct(self):
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self.force_skipping()
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super(ThinkOfPuzzleAsLinearCombination, self).construct()
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self.revert_to_original_skipping_status()
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self.rearrange_equation_as_linear_combination()
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self.show_linear_combination_of_vectors()
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def rearrange_equation_as_linear_combination(self):
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system = self.matrix_system
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corner_rect = self.corner_rect
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matrix, input_vect, equals, output_vect = system
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columns = VGroup(*[
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VGroup(*matrix.mob_matrix[:, i].flatten())
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for i in 0, 1
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])
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column_arrays = VGroup(*[
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MobjectMatrix(matrix.deepcopy().mob_matrix[:, i])
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for i in 0, 1
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])
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for column_array in column_arrays:
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column_array.match_height(output_vect)
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x, y = input_vect.elements
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movers = VGroup(x, y, equals, output_vect)
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for mover in movers:
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mover.generate_target()
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plus = TexMobject("+")
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new_system = VGroup(
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x.target, column_arrays[0], plus,
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y.target, column_arrays[1],
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equals.target,
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output_vect.target
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)
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new_system.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
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new_system.move_to(matrix, LEFT)
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corner_rect.generate_target()
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corner_rect.target.stretch_to_fit_width(
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new_system.get_width() + MED_LARGE_BUFF,
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about_edge=LEFT
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)
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self.remove_foreground_mobjects(corner_rect, system)
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self.play(
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MoveToTarget(corner_rect),
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FadeOut(input_vect.brackets),
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ReplacementTransform(matrix.brackets, column_arrays[0].brackets),
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ReplacementTransform(matrix.brackets.copy(), column_arrays[1].brackets),
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ReplacementTransform(columns[0], column_arrays[0].elements),
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ReplacementTransform(columns[1], column_arrays[1].elements),
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Write(plus, rate_func=squish_rate_func(smooth, 0.5, 1)),
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*[
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MoveToTarget(mover, replace_mobject_with_target_in_scene=True)
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for mover in movers
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],
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path_arc=90 * DEGREES,
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run_time=2
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)
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self.add_foreground_mobject(corner_rect, new_system)
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self.wait()
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def show_linear_combination_of_vectors(self):
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basis_vectors = self.basis_vectors
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input_vect = np.dot(np.linalg.inv(self.matrix), self.output_vect)
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origin = self.plane.coords_to_point(0, 0)
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for basis, scalar in zip(basis_vectors, input_vect):
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basis.ghost = basis.copy()
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basis.ghost.set_color(average_color(basis.get_color(), BLACK))
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self.add_foreground_mobjects(basis.ghost, basis)
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basis.generate_target()
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basis_coords = np.array(self.plane.point_to_coords(basis.get_end()))
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new_coords = scalar * basis_coords
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basis.target.put_start_and_end_on(
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origin, self.plane.coords_to_point(*new_coords),
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)
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dashed_lines = VGroup(*[DashedLine(LEFT, RIGHT) for x in range(2)])
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def update_dashed_lines(lines):
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for i in 0, 1:
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lines[i].put_start_and_end_on(
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basis_vectors[i].get_start(),
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basis_vectors[i].get_end(),
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)
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lines[i].shift(basis_vectors[1 - i].get_end() - origin)
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return lines
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update_dashed_lines(dashed_lines)
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self.play(LaggedStart(ShowCreation, dashed_lines, lag_ratio=0.7))
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for basis in basis_vectors:
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self.play(
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MoveToTarget(basis, run_time=2),
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UpdateFromFunc(dashed_lines, update_dashed_lines)
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)
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self.wait()
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class WrongButHelpful(TeacherStudentsScene):
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def construct(self):
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self.teacher_says("What's next is wrong, \\\\ but helpful")
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self.change_student_modes("sassy", "sad", "angry")
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self.wait(3)
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class LookAtDotProducts(SetupSimpleSystemOfEquations):
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CONFIG = {
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"quit_before_final_transformation": True,
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}
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def construct(self):
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self.force_skipping()
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super(LookAtDotProducts, self).construct()
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self.revert_to_original_skipping_status()
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self.remove_corner_system()
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self.show_dot_products()
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self.show_transformation()
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self.show_transformed_dot_products()
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def remove_corner_system(self):
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to_remove = [self.corner_rect, self.matrix_system]
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self.remove_foreground_mobjects(*to_remove)
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self.remove(*to_remove)
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q_marks = self.q_marks
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input_vect_mob = self.input_vect_mob
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xy_vect_mob = Matrix(["x", "y"], add_background_rectangle=True)
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xy_vect_mob.scale(self.array_scale_factor)
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xy_vect_mob.next_to(input_vect_mob.get_end(), DOWN, SMALL_BUFF)
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q_marks.next_to(xy_vect_mob, RIGHT)
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self.add(xy_vect_mob)
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def show_dot_products(self):
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pass
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def show_transformation(self):
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pass
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def show_transformed_dot_products(self):
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pass
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@ -325,6 +325,7 @@ class LinearTransformationScene(VectorScene):
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)
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self.moving_vectors += list(self.basis_vectors)
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self.i_hat, self.j_hat = self.basis_vectors
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self.add(self.basis_vectors)
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def add_special_mobjects(self, mob_list, *mobs_to_add):
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for mobject in mobs_to_add:
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Reference in New Issue
Block a user