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Deriminant puzzle solution animations
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Grant Sanderson
parent
b82a713d16
commit
bc229f4f29
1 changed files with 277 additions and 3 deletions
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@ -1,12 +1,286 @@
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from big_ol_pile_of_manim_imports import *
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class WorkOutNumerically(Scene):
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CONFIG = {
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"M1_COLOR": TEAL,
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"M2_COLOR": PINK,
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}
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def construct(self):
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pass
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self.add_question()
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self.add_example()
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self.compute_rhs()
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self.compute_lhs()
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def add_question(self):
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equation = self.original_equation = TexMobject(
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"\\det(", "M_1", "M_2", ")", "=",
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"\\det(", "M_1", ")",
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"\\det(", "M_2", ")",
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)
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equation.set_color_by_tex_to_color_map({
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"M_1": self.M1_COLOR,
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"M_2": self.M2_COLOR,
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})
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challenge = TextMobject("Explain in one sentence")
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challenge.set_color(YELLOW)
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group = VGroup(challenge, equation)
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group.arrange_submobjects(DOWN)
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group.to_edge(UP)
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self.add(equation)
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self.play(Write(challenge))
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self.wait()
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def add_example(self):
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M1 = self.M1 = Matrix([[2, -1], [1, 1]])
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M1.set_color(self.M1_COLOR)
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self.M1_copy = M1.copy()
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M2 = self.M2 = Matrix([[-1, 4], [1, 1]])
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M2.set_color(self.M2_COLOR)
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self.M2_copy = M2.copy()
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eq_parts = TexMobject(
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"\\det", "\\big(", "\\big)", "=",
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"\\det", "\\big(", "\\big)",
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"\\det", "\\big(", "\\big)",
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)
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for part in eq_parts.get_parts_by_tex("\\big"):
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part.scale(2)
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part.stretch(1.5, 1)
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i1, i2, i3 = [
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eq_parts.index_of_part(part) + 1
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for part in eq_parts.get_parts_by_tex("\\big(")
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]
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equation = self.equation_with_numbers = VGroup(*it.chain(
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eq_parts[:i1], [M1, M2],
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eq_parts[i1:i2], [self.M1_copy],
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eq_parts[i2:i3], [self.M2_copy],
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eq_parts[i3:],
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))
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equation.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
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eq_parts.get_part_by_tex("=").shift(0.2 * SMALL_BUFF * DOWN)
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equation.scale_to_fit_width(FRAME_WIDTH - 2 * LARGE_BUFF)
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equation.next_to(self.original_equation, DOWN, MED_LARGE_BUFF)
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self.play(LaggedStart(FadeIn, equation))
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def compute_rhs(self):
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M1, M2 = self.M1_copy, self.M2_copy
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line1 = VGroup(
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TexMobject(
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"\\big(", "2", "\\cdot", "2", "-",
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"(-1)", "\\cdot", "1", "\\big)"
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),
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TexMobject(
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"\\big(", "-1", "\\cdot", "1", "-",
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"4", "\\cdot", "1", "\\big)"
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),
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)
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line1.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
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line1[0].set_color(self.M1_COLOR)
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line1[1].set_color(self.M2_COLOR)
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indices = [1, 3, 5, 7]
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line2 = TexMobject("(3)", "(-5)")
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line2.match_style(line1)
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line3 = TexMobject("-15")
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arrows = [TexMobject("\\downarrow") for x in range(2)]
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lines = VGroup(line1, arrows[0], line2, arrows[1], line3)
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lines.arrange_submobjects(DOWN, buff=MED_SMALL_BUFF)
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lines.next_to(self.equation_with_numbers, DOWN, buff=MED_LARGE_BUFF)
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lines.to_edge(RIGHT)
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for matrix, det in zip([M1, M2], line1):
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numbers = VGroup(*[det[i] for i in indices])
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numbers_iter = iter(numbers)
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non_numbers = VGroup(*filter(
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lambda m: m not in numbers,
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det
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))
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matrix_numbers = VGroup(*[
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matrix.mob_matrix[i][j].copy()
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for i, j in (0, 0), (1, 1), (0, 1), (1, 0)
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])
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self.play(
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LaggedStart(FadeIn, non_numbers, run_time=1),
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LaggedStart(
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ReplacementTransform,
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matrix_numbers,
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lambda m: (m, numbers_iter.next()),
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path_arc=TAU / 6
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),
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)
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self.play(LaggedStart(FadeIn, lines[1:], run_time=3))
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def compute_lhs(self):
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matrix = Matrix([[-3, 7], [0, 5]])
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matrix.set_color(BLUE)
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matrix.scale(0.8)
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empty_det_tex = TexMobject("\\det", "\\big(", "\\big)")
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empty_det_tex[1:].scale(1.5)
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empty_det_tex[1:].match_height(matrix, stretch=True)
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det_tex = VGroup(empty_det_tex[:2], matrix, *empty_det_tex[2:])
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det_tex.arrange_submobjects(RIGHT, buff=SMALL_BUFF)
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group = VGroup(
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det_tex,
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TexMobject("\\downarrow"),
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TexMobject("(-3)(5) - (7)(0)").scale(0.8),
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TexMobject("\\downarrow"),
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TexMobject("-15"),
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)
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group.arrange_submobjects(DOWN, buff=2 * SMALL_BUFF)
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# group.scale_to_fit_height(0.4*FRAME_HEIGHT)
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group.next_to(self.equation_with_numbers, DOWN)
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group.shift(FRAME_WIDTH * LEFT / 4)
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self.play(FadeIn(empty_det_tex))
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self.play(*[
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ReplacementTransform(M.copy(), matrix)
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for M in self.M1, self.M2
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])
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self.play(LaggedStart(FadeIn, group[1:]))
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self.wait()
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class LetsGoInOneSentence(TeacherStudentsScene):
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def construct(self):
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self.teacher_says(
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"Here we go, \\\\", "one sentence!"
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)
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self.change_all_student_modes("hooray")
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self.teacher_says(
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"Or three...", "",
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target_mode="guilty"
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)
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self.change_all_student_modes("sassy")
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self.wait(4)
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class SuccessiveLinearTransformations(LinearTransformationScene):
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CONFIG = {
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"matrix_2": [[3, -1], [0, 1]],
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"matrix_1": [[2, 3], [-1. / 3, 2]],
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}
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def construct(self):
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self.apply_transposed_matrix([[0, 1], [1, 0]])
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self.apply_transposed_matrix([[1, 1], [0, 1]])
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self.create_product_and_inverse()
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self.scale_area_successively()
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self.apply_transformations_successively()
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self.scale_area_successively(
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"\\det(M_2)", "\\det(M_1)", "\\det(M_1 M_2)",
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reset=False
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)
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# self.show_det_as_scaling_factor()
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def create_product_and_inverse(self):
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self.matrix_product = np.dot(self.matrix_1, self.matrix_2)
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self.matrix_product_inverse = np.linalg.inv(self.matrix_product)
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def scale_area_successively(self, tex2="3", tex1="5", tex_prod="15", reset=True):
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self.add_unit_square()
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t1 = "$%s \\, \\cdot $" % tex1
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t2 = "$%s \\, \\cdot $" % tex2
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t3 = "Area"
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areas = VGroup(
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TextMobject("", "", t3),
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TextMobject("", t2, t3),
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TextMobject(t1, t2, t3),
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)
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areas.scale(0.8)
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areas.move_to(self.square)
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area = areas[0]
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self.add_moving_mobject(area, areas[1])
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self.play(
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FadeIn(self.square),
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Write(area),
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Animation(self.basis_vectors)
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)
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self.apply_matrix(self.matrix_2)
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self.wait()
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self.add_moving_mobject(area, areas[2])
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self.apply_matrix(self.matrix_1)
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self.wait()
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product = VGroup(area[:2])
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brace = Brace(product, DOWN, buff=SMALL_BUFF)
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brace_tex = brace.get_tex(tex_prod, buff=SMALL_BUFF)
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brace_tex.scale(0.8, about_edge=UP)
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self.play(
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GrowFromCenter(brace),
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Write(brace_tex)
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)
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self.wait()
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if reset:
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self.play(
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FadeOut(VGroup(self.square, area, brace, brace_tex)),
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Animation(self.plane),
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Animation(self.basis_vectors)
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)
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self.transformable_mobjects.remove(self.square)
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self.moving_mobjects = []
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self.reset_plane()
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self.wait()
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def apply_transformations_successively(self):
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M1, M2, all_space = expression = TexMobject(
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"M_1", "M_2", "\\text{(All 2d space)}"
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)
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expression.set_color_by_tex_to_color_map({
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"M_1": TEAL,
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"M_2": PINK,
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})
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expression.shift(FRAME_WIDTH * LEFT / 4)
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expression.to_edge(UP)
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for part in expression:
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part.add_background_rectangle()
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part.background_rectangle.stretch(1.05, 0)
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M1.save_state()
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M1.move_to(ORIGIN)
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M1.fade(1)
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# Apply one after the other
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self.play(
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FocusOn(M2, run_time=1),
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FadeIn(VGroup(M2, all_space))
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)
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self.add_foreground_mobjects(M2, all_space)
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self.apply_matrix(self.matrix_2)
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self.wait()
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self.play(M1.restore)
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self.add_foreground_mobjects(M1)
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self.apply_matrix(self.matrix_1)
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self.wait()
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# Show full composition
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rp, lp = parens = TexMobject("()")
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matrices = VGroup(M1, M2)
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matrices.generate_target()
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parens.match_height(matrices)
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lp.move_to(matrices, RIGHT)
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matrices.target.next_to(lp, LEFT, SMALL_BUFF)
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rp.next_to(matrices.target, LEFT, SMALL_BUFF)
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self.reset_plane()
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self.play(
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MoveToTarget(matrices),
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*map(GrowFromCenter, parens)
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)
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self.apply_matrix(self.matrix_product)
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self.wait()
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self.reset_plane()
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def show_det_as_scaling_factor(self):
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pass
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###
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def reset_plane(self):
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plane_and_bases = VGroup(self.plane, self.basis_vectors)
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self.play(FadeOut(plane_and_bases))
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self.apply_matrix(self.matrix_product_inverse, run_time=0)
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self.play(FadeIn(plane_and_bases))
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