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gram_talk.py
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from manim import *
import numpy as np
import itertools as it
from colour import Color
def rainbowify(textmobj):
colors_of_the_rainbow = [RED, ORANGE, YELLOW, GREEN, BLUE, PURPLE]
colors = it.cycle(colors_of_the_rainbow)
for letter in textmobj:
letter.set_color(next(colors))
class HeresWhereTheTalkReallyBegins(Scene):
def construct(self):
begin_text = Tex(r"Here's where the talk", r"\textbf{really} begins!", color=YELLOW).scale(2.5).arrange(DOWN)
self.play(Write(begin_text))
self.wait()
class FirstThingToAnimate(Scene):
def construct(self):
first_text = Tex(r"Matrices", r"$\Downarrow$", r"\textit{Linear Transformations}", color=YELLOW).scale(2).arrange(DOWN).shift(UP)
self.play(Write(first_text))
self.wait()
class LinearMapToMatrix(LinearTransformationScene):
def __init__(self, **kwargs):
LinearTransformationScene.__init__(
self,
show_basis_vectors=False,
x_min=-10,
x_max=10,
y_min=-10,
y_max=10,
**kwargs
)
def construct(self):
matrix_1 = np.transpose([[1, 3], [-1, 4]])
matrix_2 = np.transpose([[-1, 1], [2, 4]])
matrix_3 = np.transpose([[-0.7, 0.7], [0, -0.9]])
matrix_4 = np.transpose([[2, 1], [-1, 1]])
self.wait()
linear_text = Tex(r"Grid lines remain parallel and evenly spaced", color=YELLOW).to_edge(UP).add_background_rectangle()
self.add(linear_text)
self.play(Write(linear_text[1]))
self.wait()
self.play(ApplyMatrix(matrix_1, self.plane))
self.wait()
self.play(ApplyMatrix(np.linalg.inv(matrix_1), self.plane))
self.wait()
self.play(ApplyMatrix(matrix_2, self.plane))
self.wait()
self.play(ApplyMatrix(np.linalg.inv(matrix_2), self.plane))
self.wait()
self.play(ApplyMatrix(matrix_3, self.plane))
self.wait()
self.play(ApplyMatrix(np.linalg.inv(matrix_3), self.plane))
self.wait()
self.play(Rotate(self.plane, angle = 90 * DEGREES))
self.wait()
self.play(Rotate(self.plane, angle = -90 * DEGREES))
self.wait()
self.play(linear_text.animate.set_opacity(0))
follow_text = Tex(r"Follow basis vectors").to_edge(UP).add_background_rectangle()
i_hat = self.add_vector([1, 0, 0], color=GREEN)
j_hat = self.add_vector([0, 1, 0], color=RED)
self.wait()
i_hat_label = self.get_vector_label(i_hat, MathTex(r"\hat{\imath}", color=GREEN))
j_hat_label = self.get_vector_label(j_hat, MathTex(r"\hat{\jmath}", color=RED))
self.play(Create(i_hat_label), Create(j_hat_label))
self.wait()
vector_v = self.add_vector([-2, -1, 0], color=YELLOW)
v_label = self.get_vector_label(vector_v, MathTex(r"\vec{\mathbf{v} }", color=YELLOW))
self.play(Create(v_label))
self.wait()
transform_i = self.get_vector([-2, 0, 0], color=GREEN)
t_i_label = self.get_vector_label(transform_i, MathTex(r"-2\hat{\imath}", color=GREEN)).next_to(transform_i, UP)
transform_j = self.get_vector([0, -1, 0], color=RED).shift([-2, 0, 0])
t_j_label = self.get_vector_label(transform_j, MathTex(r"-\hat{\jmath}", color=RED)).shift([-2, 0, 0])
self.play(TransformFromCopy(i_hat, transform_i), TransformFromCopy(j_hat, transform_j), TransformFromCopy(i_hat_label, t_i_label), TransformFromCopy(j_hat_label, t_j_label))
self.wait()
v_eq = MathTex(r"\vec{\mathbf{v} }", r"=", r"-2 \hat{\imath}", r"-\hat{\jmath}").scale(1.5).to_edge(UP).to_edge(LEFT)
v_eq[0].set_color(YELLOW)
v_eq[2].set_color(GREEN)
v_eq[3].set_color(RED)
v_eq.add_background_rectangle()
self.add_foreground_mobject(v_eq)
self.play(Write(v_eq[1:]))
self.wait()
v_eq_cpy = v_eq.copy()
fade_grp = VGroup(v_label, i_hat_label, j_hat_label, t_i_label, t_j_label, transform_i, transform_j, v_eq)
self.play(fade_grp.animate.set_opacity(0))
self.apply_matrix(matrix_4)
self.wait()
self.add_foreground_mobject(v_eq_cpy)
self.play(Write(v_eq_cpy[1:]))
self.wait()
i_hat_label_2 = self.get_vector_label(i_hat, MathTex(r"L(\hat{\imath})", color=GREEN))
j_hat_label_2 = self.get_vector_label(j_hat, MathTex(r"L(\hat{\jmath})", color=RED))
v_label_2 = self.get_vector_label(vector_v, MathTex(r"L(\vec{\mathbf{v} })", color=YELLOW))
self.play(Create(i_hat_label_2), Create(j_hat_label_2), Create(v_label_2))
self.wait()
transform_i_2 = self.get_vector([-2, 0, 0], color=GREEN)
t_i_label_2 = self.get_vector_label(transform_i, MathTex(r"-2L(\hat{\imath})", color=GREEN)).next_to(transform_i, UP).shift(2*RIGHT + UP)
transform_j_2 = self.get_vector([0, -1, 0], color=RED).shift(-2 * np.array([2, 1, 0]))
t_j_label_2 = self.get_vector_label(transform_j, MathTex(r"-L(\hat{\jmath})", color=RED)).shift(-2 * np.array([2, 1, 0]))
self.play(TransformFromCopy(i_hat, transform_i_2), TransformFromCopy(j_hat, transform_j_2), TransformFromCopy(i_hat_label, t_i_label_2), TransformFromCopy(j_hat_label, t_j_label_2))
self.wait()
v_eq_2 = MathTex(r"L(\vec{\mathbf{v} })", r"=", r"-2 L(\hat{\imath})", r"-L(\hat{\jmath})").scale(1.5).next_to(v_eq_cpy, DOWN).to_edge(LEFT)
v_eq_2[0].set_color(YELLOW)
v_eq_2[2].set_color(GREEN)
v_eq_2[3].set_color(RED)
v_eq_2.add_background_rectangle()
self.add_foreground_mobject(v_eq_2)
self.play(Write(v_eq_2[1:]))
self.wait()
fade_grp_2 = VGroup(v_eq_cpy, v_eq_2, transform_i_2, t_i_label_2, transform_j_2, t_j_label_2, vector_v, v_label_2)
self.play(fade_grp_2.animate.set_opacity(0))
self.wait()
determine_text = Tex(r"Transformation determined by basis vectors").to_edge(UP).add_background_rectangle()
self.add(determine_text)
self.play(Write(determine_text[1]))
self.wait()
i_hat_matrix = Matrix([[2], [1]], include_background_rectangle=True)
i_hat_matrix[1].set_color(GREEN)
i_hat_matrix.next_to(i_hat, UP + RIGHT)
j_hat_matrix = Matrix([[-1], [1]], include_background_rectangle=True)
j_hat_matrix[1].set_color(RED)
j_hat_matrix.next_to(j_hat, UP + LEFT)
self.play(Create(i_hat_matrix), Create(j_hat_matrix))
self.wait()
target_matrix = Matrix([[2, -1], [1, 1]], include_background_rectangle=True)
target_matrix.set_column_colors(GREEN, RED)
for entry in target_matrix.get_entries():
entry.set_opacity(0)
target_matrix.next_to(determine_text, DOWN).to_edge(LEFT)
self.play(Create(target_matrix))
self.wait()
i_hat_matrix_cpy = i_hat_matrix.copy()
self.play(i_hat_matrix_cpy.animate.move_to(target_matrix[1][0].get_center()).align_to(target_matrix, UP))
self.wait()
self.play(FadeOut(i_hat_matrix_cpy), target_matrix[1][0].animate.set_opacity(1), target_matrix[1][2].animate.set_opacity(1))
self.wait()
j_hat_matrix_cpy = j_hat_matrix.copy()
self.play(j_hat_matrix_cpy.animate.move_to(target_matrix[1][1].get_center()).align_to(target_matrix, UP))
self.wait()
self.play(FadeOut(j_hat_matrix_cpy), target_matrix[1][1].animate.set_opacity(1), target_matrix[1][3].animate.set_opacity(1))
self.wait()
class SecondThingToAnimate(Scene):
def construct(self):
second_text = Tex(r"Determinant", r"$\Downarrow$", r"\textit{Volume scale factor}").scale(2).arrange(DOWN).shift(UP)
self.play(Write(second_text))
self.wait()
question_text = Tex(r"Why should it even exist?", color=YELLOW).scale(2).next_to(second_text, DOWN)
self.play(Write(question_text))
self.wait()
class WhyDeterminantShouldExist(LinearTransformationScene):
def __init__(self, **kwargs):
LinearTransformationScene.__init__(
self,
show_basis_vectors=False,
x_min=-10,
x_max=10,
y_min=-10,
y_max=10,
include_background_plane=False,
**kwargs
)
def add_square_custom(self, size, left_corner):
square = Rectangle(
color=YELLOW,
width=size,
height=size,
stroke_color=YELLOW,
stroke_width=3,
fill_color=YELLOW,
fill_opacity=0.3,
)
square.move_to(self.plane.coords_to_point(*left_corner), DL)
return square
def approximate_circ_with_num(self, num):
square_size = 2/num
squares = []
for left_edge in range(-1*num, 1*num):
for bottom_edge in range(-1*num, 1*num):
# Check if in circle or not
center_x = (left_edge + 0.5) * square_size
center_y = (bottom_edge + 0.5) * square_size
if (center_x**2 + center_y**2) <= 1:
squares.append(self.add_square_custom(square_size, [left_edge * square_size, bottom_edge * square_size]))
return VGroup(*squares)
def construct(self):
matrix = np.transpose([[2, 1], [-1, 1]])
self.add_unit_square()
square_1 = self.square
self.apply_matrix(matrix)
self.wait()
scale_text = Tex(r"Scaled by $k$").to_edge(UP).to_edge(RIGHT).add_background_rectangle()
scale_arrow = Arrow(scale_text.get_corner(DL), square_1.get_center())
self.add(scale_text)
self.play(Write(scale_text[1:]), Create(scale_arrow))
self.wait()
down_implies = MathTex(r"\Downarrow").next_to(scale_text, DOWN)
all_text = Tex(r"All scaled by $k$", color=YELLOW).next_to(down_implies, DOWN).to_edge(RIGHT).add_background_rectangle()
self.add(all_text)
self.play(Create(down_implies), Write(all_text[1:]))
self.wait()
fade_grp = VGroup(all_text, down_implies, scale_text, scale_arrow)
self.play(fade_grp.animate.set_opacity(0))
self.apply_matrix(np.linalg.inv(matrix))
self.wait()
square_2 = self.add_square_custom(1, [-2, 1])
square_3 = self.add_square_custom(0.5, [2, 0])
square_4 = self.add_square_custom(2, [0, -3])
square_5 = self.add_square_custom(0.2, [-1, 0])
self.play(DrawBorderThenFill(square_2), DrawBorderThenFill(square_3), DrawBorderThenFill(square_4), DrawBorderThenFill(square_5))
self.wait()
self.moving_mobjects = []
self.add_transformable_mobject(square_2)
self.add_transformable_mobject(square_3)
self.add_transformable_mobject(square_4)
self.add_transformable_mobject(square_5)
self.apply_matrix(matrix)
square_text = Tex(r"All grid squares scaled by $k$").to_edge(UP).add_background_rectangle()
self.add(square_text)
self.play(Write(square_text[1:]))
self.wait()
self.play(square_text.animate.set_opacity(0))
self.wait()
self.moving_mobjects = []
self.apply_matrix(np.linalg.inv(matrix))
fade_grp_2 = VGroup(self.square, square_2, square_3, square_4, square_5)
self.play(fade_grp_2.animate.set_opacity(0))
test_circ = Circle(fill_color=RED,
fill_opacity=0.3,)
self.play(Create(test_circ))
self.add_transformable_mobject(test_circ)
squares_1 = self.approximate_circ_with_num(5)
self.play(Create(squares_1))
self.play(FadeOut(squares_1))
squares_2 = self.approximate_circ_with_num(10)
self.play(Create(squares_2))
self.play(FadeOut(squares_2))
squares_3 = self.approximate_circ_with_num(20)
self.play(Create(squares_3))
self.add_transformable_mobject(squares_3)
self.wait()
self.moving_mobjects = []
self.apply_matrix(matrix)
self.wait()
arb_text = Tex(r"Arbitrary region scaled by $k$", color=YELLOW).to_edge(UP).add_background_rectangle()
self.add(arb_text)
self.play(Write(arb_text[1:]))
self.wait()
class DeterminantToMeasureArea(Scene):
def construct(self):
title_text = Tex(r"What we want").scale(2).to_edge(UP)
self.play(Write(title_text))
self.wait()
interested_text_0 = Tex(r"Interested in determining area of parallelogram,")
interested_text_1 = Tex(r"or 3D volume of parallelepiped,")
interested_text_2 = Tex(r"or general volume of parallelotope")
text_grp = VGroup(interested_text_0, interested_text_1, interested_text_2)
text_grp.to_edge(UP).arrange(DOWN)
self.play(Write(interested_text_0))
self.wait()
self.play(Write(interested_text_1))
self.wait()
self.play(Write(interested_text_2))
self.wait()
what_text = Tex(r"What do we already know?", color=YELLOW).next_to(interested_text_2, DOWN)
self.play(Write(what_text))
self.wait()
class WhatWeCanDoSoFar0(Scene):
def construct(self):
title_text = Tex("What we know so far").scale(2)
title_text.to_edge(UP)
self.play(Write(title_text))
self.wait(2)
determinant_text = Tex(r"Given $n$ vectors in $\mathbb{R}^n$ $\longrightarrow$ ", "take the determinant")
determinant_text[1].set_color(GREEN)
self.play(Write(determinant_text))
self.wait(2)
self.play(determinant_text.animate.shift(UP))
cross_text = Tex(r"Given 2 vectors in $\mathbb{R}^3$ $\longrightarrow$ ", "magnitude of cross product")
cross_text[1].set_color(GREEN)
self.play(Write(cross_text))
self.wait(4)
length_text = Tex(r"Given 1 vector in $\mathbb{R}^n$ $\longrightarrow$ ", "find its length").shift(DOWN)
length_text[1].set_color(GREEN)
self.play(Write(length_text))
self.wait(3)
box = SurroundingRectangle(determinant_text)
self.play(Create(box))
self.wait()
class ParallelogramArea(LinearTransformationScene):
def construct(self):
matrix = [[2, 2], [0, 3]]
m0 = Matrix(matrix, include_background_rectangle = True).shift(4.5*LEFT + 2*UP)
m0.set_column_colors("#83C167", "#FC6255")
self.add_unit_square()
self.add_foreground_mobject(m0)
self.wait()
self.moving_mobjects = []
self.apply_matrix(matrix)
bottom_brace = Brace(self.i_hat, DOWN)
right_brace = Brace(self.square, RIGHT)
width = Tex(str(2))
height = Tex(str(3))
width.next_to(bottom_brace, DOWN)
height.next_to(right_brace, RIGHT)
for mob in bottom_brace, width, right_brace, height:
mob.add_background_rectangle()
self.play(Create(mob, run_time = 0.5))
det = get_det_text(m0,
determinant=6,
initial_scale_factor=1)
self.play(Create(det))
self.wait()
class ParallelepipedVolume(ThreeDScene):
def construct(self):
axes = ThreeDAxes(z_min=-4.5, z_max=4.5).add_coordinates()
self.add(axes)
array_u = [2, 0, 1]
array_v = [0, 2, 1]
array_w = [0, 0, 1]
matrix = [array_u, array_v, array_w]
m0 = Matrix(matrix)
m0.set_column_colors(GREEN, RED, BLUE)
det_text = get_det_text(m0, determinant=4, initial_scale_factor=1)
det_grp = VGroup(m0, det_text)
det_grp.to_edge(UP).to_edge(LEFT)
i_hat = Arrow3D([0, 0, 0], [1, 0, 0], color=GREEN)
j_hat = Arrow3D([0, 0, 0], [0, 1, 0], color=RED)
k_hat = Arrow3D([0, 0, 0], [0, 0, 1], color=BLUE)
self.set_camera_orientation(phi = 60 * DEGREES, theta = -100 * DEGREES)
cube = Cube(side_length=1).move_to([0.5, 0.5, 0.5])
vectors = VGroup(i_hat, j_hat, k_hat)
self.add(vectors)
self.play(Create(cube))
self.add_fixed_in_frame_mobjects(m0)
self.play(Create(m0))
self.wait()
self.play(FadeOut(vectors))
self.play(ApplyMatrix(matrix, cube))
self.wait()
self.move_camera(phi=90 * DEGREES, theta = -90 * DEGREES)
self.add_fixed_in_frame_mobjects(det_text)
self.play(Write(det_text))
self.wait()
class WhatWeCanDoSoFar1(Scene):
def construct(self):
title_text = Tex("What we know so far").scale(2)
title_text.to_edge(UP)
self.add(title_text)
determinant_text = Tex(r"Given $n$ vectors in $\mathbb{R}^n$ $\longrightarrow$ ", "take the determinant").shift(UP)
determinant_text[1].set_color(GREEN)
self.add(determinant_text)
cross_text = Tex(r"Given 2 vectors in $\mathbb{R}^3$ $\longrightarrow$ ", "magnitude of cross product")
cross_text[1].set_color(GREEN)
self.add(cross_text)
length_text = Tex(r"Given 1 vector in $\mathbb{R}^n$ $\longrightarrow$ ", "find its length").shift(DOWN)
length_text[1].set_color(GREEN)
self.add(length_text)
box = SurroundingRectangle(cross_text)
self.play(Create(box))
self.wait()
class ParallelogramAreaIn3D(ThreeDScene):
def construct(self):
axes = ThreeDAxes(z_min=-4.5, z_max=4.5)
vector_v = Arrow3D(np.array([0, 0, 0]), np.array([1, -2, 1]), color="#83C167")
vector_w = Arrow3D(np.array([0, 0, 0]), np.array([2, 0, 2]), buff=0, color="#FC6255")
vector_cross = Arrow3D([0,0,0], [-4,0,4], color=BLUE)
v_label = MathTex(r"\vec{\mathbf{v} }", color="#83C167")
w_label = MathTex(r"\vec{\mathbf{w} }", color="#FC6255")
cross_label = MathTex(r"\vec{\mathbf{v} } \times \vec{\mathbf{w} }", color=BLUE)
parallelogram = Polygon([0, 0, 0], [2, 0, 2], [3, -2, 3], [1, -2, 1], color="white")
parallelogram.set_fill(WHITE, opacity=0.5)
mobjects = [axes, vector_v, vector_w, vector_cross, parallelogram]
for mobj in mobjects:
mobj = mobj.scale(0.7, about_point=[0,0,0])
self.add(vector_v)
self.add(vector_w)
self.add(axes)
self.set_camera_orientation(phi=60 * DEGREES, theta=-100 * DEGREES)
v_label.next_to(vector_v, RIGHT*2 + UP*0)
v_label.shift(UP*0.4)
w_label.next_to(vector_w, UP*3)
self.add_fixed_in_frame_mobjects(v_label, w_label)
self.wait()
self.play(Create(parallelogram))
question_text = Tex(r"Area of parallelogram ", r"$= |\vec{\mathbf{v} } \times \vec{\mathbf{w} }|$")
question_text[1].set_opacity(0)
question_text[0].set_color(YELLOW)
question_text[1].set_color(BLUE)
question_text.to_edge(UP).to_edge(RIGHT)
self.add_fixed_in_frame_mobjects(question_text)
self.play(Write(question_text))
self.wait()
self.add(vector_cross)
cross_label.next_to(vector_cross, UP*6)
self.add_fixed_in_frame_mobjects(cross_label)
self.wait()
self.play(question_text[1].animate.set_opacity(1))
self.wait()
class ParallelogramAreaIn3DAlgebra(Scene):
def construct(self):
coord_text = MathTex(r"\vec{\mathbf{v} } = \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix} \space", r"\vec{\mathbf{w} } = \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix}")
coord_text[0].set_color(GREEN)
coord_text[1].set_color(RED)
coord_text.shift(UP*2)
self.play(Write(coord_text))
cross_text = MathTex(r"\vec{\mathbf{v} }", r"\times", r"\vec{\mathbf{w} }", r"=", r"\begin{bmatrix} -4 \\ 0 \\ 4 \end{bmatrix}")
cross_text[0].set_color(GREEN)
cross_text[2].set_color(RED)
cross_text[4].set_color(BLUE)
self.play(Write(cross_text))
self.wait()
mag_text = MathTex(r"|", r"\vec{\mathbf{v} }", r"\times", r"\vec{\mathbf{w} }", r"| =", r"\sqrt{32} \approx 5.66")
mag_text[1].set_color(GREEN)
mag_text[3].set_color(RED)
mag_text[5].set_color(BLUE)
mag_text.next_to(cross_text, DOWN)
self.play(Write(mag_text))
self.wait()
class WhatWeCanDoSoFar2(Scene):
def construct(self):
title_text = Tex("What we know so far").scale(2)
title_text.to_edge(UP)
self.add(title_text)
determinant_text = Tex(r"Given $n$ vectors in $\mathbb{R}^n$ $\longrightarrow$ ", "take the determinant").shift(UP)
determinant_text[1].set_color(GREEN)
self.add(determinant_text)
cross_text = Tex(r"Given 2 vectors in $\mathbb{R}^3$ $\longrightarrow$ ", "magnitude of cross product")
cross_text[1].set_color(GREEN)
self.add(cross_text)
length_text = Tex(r"Given 1 vector in $\mathbb{R}^n$ $\longrightarrow$ ", "find its length").shift(DOWN)
length_text[1].set_color(GREEN)
self.add(length_text)
question_text = Tex(r"What about the general case?", color=YELLOW).scale(2)
question_text.next_to(length_text, DOWN)
self.play(Write(question_text))
self.wait()
class TheGeneralCase(Scene):
def construct(self):
title_text = Tex(r"Problem statement").scale(2).to_edge(UP)
self.play(Write(title_text))
self.wait()
given_text = Tex(r"Given $n$ vectors ", r"$\{\vec{\mathbf{u} }_1, \dots, \vec{\mathbf{u} }_n\}$", r" in $\mathbb{R}^d$").next_to(title_text, DOWN).shift(DOWN*0.5)
given_text[1].set_color(BLUE)
self.play(Write(given_text))
self.wait()
want_text = Tex(r"Want to compute ", r"volume of $n$-dimensional parallelotope").next_to(given_text, DOWN)
want_text[1].set_color(YELLOW)
self.play(Write(want_text))
self.wait()
note_text = Tex(r"Assume $n < d$ - if equal use determinant, if greater then just $0$").next_to(want_text, DOWN)
self.play(Write(note_text))
self.wait()
self.play(FadeOut(note_text))
self.wait()
hard_text = Tex(r"Hard problem - ", r"can't just take determinant", color=RED).next_to(want_text, DOWN)
self.play(Write(hard_text))
self.wait()
example_matrix = Matrix([[1, 2], [-2, 0], [1, 2]]).next_to(hard_text, DOWN)
example_matrix.set_column_colors(GREEN, RED)
det_text = get_det_text(example_matrix, r"???", initial_scale_factor=1)
self.play(Create(example_matrix))
self.wait()
self.play(Write(det_text))
self.wait()
self.play(FadeOut(example_matrix), FadeOut(det_text))
self.wait()
illus_text = Tex(r"Simpler problem - ", r"other way to do for 2 3D vectors?").next_to(hard_text, DOWN)
illus_text[1].set_color(YELLOW)
self.play(Write(illus_text))
self.wait()
class RotateDownTo2D(ThreeDScene):
def construct(self):
axes = ThreeDAxes(z_min=-4.5, z_max=4.5).add_coordinates()
vector_v = [1, -2, 1]
vector_w = [2, 0, 2]
arrow_v = Arrow3D(np.array([0, 0, 0]), np.array([1, -2, 1]), color="#83C167")
arrow_w = Arrow3D(np.array([0, 0, 0]), np.array([2, 0, 2]), buff=0, color="#FC6255")
parallelogram = Polygon([0, 0, 0], [2, 0, 2], [3, -2, 3], [1, -2, 1], color="white")
parallelogram.set_fill(WHITE, opacity=0.5)
mobjects = [axes, arrow_v, arrow_w, parallelogram]
for mobj in mobjects:
mobj = mobj.scale(0.7, about_point=[0,0,0])
self.add(arrow_v)
self.add(arrow_w)
self.add(axes)
self.set_camera_orientation(phi=60 * DEGREES, theta=-100 * DEGREES)
self.play(Create(parallelogram))
known_text = Tex(r"Know: Area of parallelogram ", r"$= |\vec{\mathbf{v} } \times \vec{\mathbf{w} }|$")
known_text[0].set_color(YELLOW)
known_text[1].set_color(BLUE)
question_text = Tex(r"How else could we find the area?", color=YELLOW)
idea_text = Tex(r"Idea: Rotate into a lower dimension", color=BLUE)
coord_text = MathTex(r"\vec{\mathbf{v} } = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \space", r"\vec{\mathbf{w} } = \begin{pmatrix} 2 \\ 0 \\ 2 \end{pmatrix}")
coord_text[0].set_color("#83C167")
coord_text[1].set_color("#FC6255")
magic_text = Tex("Find a rotation matrix... ", "by magic")
rainbowify(magic_text[1])
magic_text[1].set_opacity(0)
mid_axis = normalize([0, 0, 1] + get_unit_normal(vector_v, vector_w))
matrix = DecimalMatrix(rotation_matrix(180 * DEGREES, mid_axis), element_to_mobject_config={'num_decimal_places': 2}).scale(0.8)
apply_text = Tex("Apply the matrix")
perspective_text = Tex("Change our perspective", color=YELLOW)
known_text.to_edge(UP).to_edge(RIGHT)
question_text.to_edge(UP).to_edge(RIGHT)
idea_text.to_edge(UP).to_edge(RIGHT)
coord_text.to_edge(UP).to_edge(RIGHT)
magic_text.to_edge(UP).to_edge(RIGHT)
matrix.to_edge(UP).to_edge(RIGHT)
apply_text.to_edge(UP).to_edge(RIGHT)
perspective_text.to_edge(UP).to_edge(RIGHT)
self.wait(2)
self.add_fixed_in_frame_mobjects(known_text)
self.play(Write(known_text))
self.wait(3)
self.play(FadeOut(known_text))
self.add_fixed_in_frame_mobjects(question_text)
self.play(Write(question_text))
self.wait(2)
self.play(FadeOut(question_text))
self.add_fixed_in_frame_mobjects(idea_text)
self.play(Write(idea_text))
self.wait(2)
self.play(FadeOut(idea_text))
self.add_fixed_in_frame_mobjects(coord_text)
self.play(Write(coord_text))
self.wait(2)
self.play(FadeOut(coord_text))
self.add_fixed_in_frame_mobjects(magic_text)
self.play(Write(magic_text))
self.wait(3)
self.play(magic_text[1].animate.set_opacity(1))
self.wait(3)
self.play(FadeOut(magic_text))
magic_text.set_opacity(0)
magic_text[1].set_opacity(0)
self.add_fixed_in_frame_mobjects(matrix)
self.play(Create(matrix))
self.wait(2)
self.play(FadeOut(matrix))
self.add_fixed_in_frame_mobjects(apply_text)
self.play(Write(apply_text))
self.wait(2)
self.play(Rotate(arrow_v, axis=mid_axis, about_point=[0,0,0]), Rotate(arrow_w, axis=mid_axis, about_point=[0,0,0]), Rotate(parallelogram, axis=mid_axis, about_point=[0,0,0]))
self.play(FadeOut(apply_text))
self.add_fixed_in_frame_mobjects(perspective_text)
self.play(Write(perspective_text))
self.wait(3)
self.move_camera(phi=0, gamma=0, theta=-90 * DEGREES)
self.wait(2)
self.play(FadeOut(perspective_text))
result = Tex(r"Now regard as 2D vectors", color=BLUE)
result.to_edge(UP).to_edge(RIGHT)
self.play(Write(result))
self.wait()
class RotateDownTo2DAlgebra(Scene):
def construct(self):
array_v = [1, -2, 1]
array_w = [2, 0, 2]
coord_text = MathTex(r"\vec{\mathbf{v} } = \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix} \space", r"\vec{\mathbf{w} } = \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix}")
coord_text[0].set_color(GREEN)
coord_text[1].set_color(RED)
coord_text.to_edge(UP)
self.play(Write(coord_text))
self.wait()
mid_axis = normalize([0, 0, 1] + get_unit_normal(array_v, array_w))
matrix = DecimalMatrix(rotation_matrix(180 * DEGREES, mid_axis), element_to_mobject_config={'num_decimal_places': 2}).scale(0.8).to_edge(UP).to_edge(LEFT)
self.play(Create(matrix))
self.wait()
target_matrix = np.matmul(rotation_matrix(180 * DEGREES, mid_axis), np.transpose([array_v, array_w]))[:2]
target_matrix = DecimalMatrix(target_matrix, element_to_mobject_config={'num_decimal_places': 2}).scale(0.8).to_edge(UP).to_edge(RIGHT)
target_matrix.set_column_colors(GREEN, RED)
for i in range(2):
for j in range(2):
target_matrix[0][i + 2*j].set_opacity(0)
self.play(Create(target_matrix))
self.wait()
matrix_1 = matrix.copy().next_to(matrix, DOWN)
start_v = DecimalMatrix([[elem] for elem in array_v], element_to_mobject_config={'num_decimal_places': 2}).scale(0.8).next_to(matrix_1, RIGHT).set_color(GREEN)
equals = MathTex(r"=", color=BLUE).next_to(start_v, RIGHT)
result_v = DecimalMatrix([[elem] for elem in np.matmul(rotation_matrix(180 * DEGREES, mid_axis), array_v)], element_to_mobject_config={'num_decimal_places': 2}).scale(0.8).next_to(equals, RIGHT).set_color(GREEN)
application_1 = VGroup(matrix_1, start_v, equals, result_v)
self.play(Create(application_1))
self.wait()
temp = result_v.copy()
self.play(temp.animate.move_to(target_matrix[0][0].get_center()).align_to(target_matrix, UP))
self.wait()
self.play(FadeOut(temp), target_matrix[0][0].animate.set_opacity(1), target_matrix[0][2].animate.set_opacity(1))
self.wait()
start_w = DecimalMatrix([[elem] for elem in array_w], element_to_mobject_config={'num_decimal_places': 2}).scale(0.8).next_to(matrix_1, RIGHT).set_color(RED)
result_w = DecimalMatrix([[elem] for elem in np.matmul(rotation_matrix(180 * DEGREES, mid_axis), array_w)], element_to_mobject_config={'num_decimal_places': 2}).scale(0.8).next_to(equals, RIGHT).set_color(RED)
self.play(FadeOut(start_v), FadeOut(result_v), FadeIn(start_w), FadeIn(result_w))
self.wait()
temp_2 = result_w.copy()
self.play(temp_2.animate.move_to(target_matrix[0][1].get_center()).align_to(target_matrix, UP))
self.wait()
self.play(FadeOut(temp_2), target_matrix[0][1].animate.set_opacity(1), target_matrix[0][3].animate.set_opacity(1))
self.wait()
self.play(FadeOut(matrix_1), FadeOut(start_w), FadeOut(equals), FadeOut(result_w))
self.wait()
self.play(target_matrix.animate.move_to([0, 0, 0]))
self.wait()
det_text = get_det_text(target_matrix, determinant=5.66, initial_scale_factor=1)
self.play(Write(det_text))
self.wait()
class MagicIsBad(Scene):
def construct(self):
big_text = Tex(r"Magic").scale(4)
rainbowify(big_text[0])
self.play(Write(big_text))
self.wait()
cross_0 = Line(big_text.get_corner(DL), big_text.get_corner(UR), color=RED, stroke_width=20)
cross_1 = Line(big_text.get_corner(UL), big_text.get_corner(DR), color=RED, stroke_width=20)
self.play(Create(cross_0), Create(cross_1))
self.play(FadeOut(big_text), FadeOut(cross_0), FadeOut(cross_1))
math_text = Tex(r"Mathematics", color=YELLOW).scale(4)
self.play(Write(math_text))
self.wait()
class IdeaChangeBetweenOrthonormalBases(Scene):
def construct(self):
idea_text = Tex(r"Instead of rotation, can use orthogonal matrix").to_edge(UP).shift(DOWN*1.5)
self.play(Write(idea_text))
self.wait()
recap_text = Tex(r"Orthogonal matrix $\Leftrightarrow$ ", r"$\mathbf{P}^T \mathbf{P} = \mathbf{I}$").next_to(idea_text, DOWN)
recap_text[1].set_color(PURPLE)
self.play(Write(recap_text))
self.wait()
might_text = Tex(r"(Might flip area - doesn't affect size though)").next_to(recap_text, DOWN)
self.play(Write(might_text))
self.wait()
idea_text_2 = Tex(r"Orthogonal matrix $\Leftrightarrow$ ", r"Change between orthonormal bases").next_to(might_text, DOWN)
idea_text_2[1].set_color(YELLOW)
self.play(Write(idea_text_2))
self.wait()
class ChangeBetweenOrthonormalBases(ThreeDScene):
def construct(self):
axes = ThreeDAxes(z_min=-4.5, z_max=4.5)
array_v = [1, -2, 1]
array_w = [2, 0, 2]
mid_axis = normalize([0, 0, 1] + get_unit_normal(array_v, array_w))
rotate_matrix = rotation_matrix(180 * DEGREES, mid_axis)
array_1 = [1, 0, 0]
array_2 = [0, 1, 0]
array_3 = [0, 0, 1]
array_4 = np.transpose(np.matmul(rotate_matrix, np.transpose(array_1)))
array_5 = np.transpose(np.matmul(rotate_matrix, np.transpose(array_2)))
array_6 = np.transpose(np.matmul(rotate_matrix, np.transpose(array_3)))
arrays = [array_1, array_2, array_3, array_4, array_5, array_6]
vectors = list()
for array, color in zip(arrays, [BLUE, BLUE, BLUE, PURPLE, PURPLE, PURPLE]):
vectors.append(Arrow3D([0, 0, 0], array, color=color))
vectors = VGroup(*vectors)
self.set_camera_orientation(phi=60 * DEGREES, theta=-100 * DEGREES)
self.add(axes)
self.play(Create(vectors))
#self.add(vectors)
#self.begin_ambient_camera_rotation(rate=0.2)
self.wait()
question_text = Tex(r"How do we know change of basis matrix is orthogonal?").to_edge(UP).add_background_rectangle()
answer_text = Tex(r"Preserves dot product", color=YELLOW).next_to(question_text, DOWN).add_background_rectangle()
self.add_fixed_in_frame_mobjects(question_text)
self.play(Write(question_text[1]))
self.wait()
self.play(Rotate(vectors[3:], axis=mid_axis, about_point=[0,0,0]))
self.wait()
self.add_fixed_in_frame_mobjects(answer_text)
self.play(Write(answer_text[1]))
self.wait()
class ChangeBetweenOrthonormalBasesAlgebra(Scene):
def construct(self):
linear_text = Tex(r"$\mathbf{P}$", r" sends ", r"$\{\vec{\mathbf{e} }_i\}$", r" to ", r"$\{\vec{\mathbf{e} }_i'\}$").to_edge(UP).to_edge(LEFT)
linear_text[0].set_color(PURPLE)
linear_text[2].set_color(BLUE)
linear_text[4].set_color(GREEN)
line_1 = MathTex(r"\vec{\mathbf{e} }_i", r" \cdot ", r"\vec{\mathbf{e} }_j", r" = \delta_{ij}").next_to(linear_text, DOWN).to_edge(LEFT)
line_1[0].set_color(BLUE)
line_1[2].set_color(BLUE)
line_2 = MathTex(r"(", r"\mathbf{P}", r" \vec{\mathbf{e} }_i", r") \cdot (", r"\mathbf{P}", r"\vec{\mathbf{e} }_j", r") = ", r"\vec{\mathbf{e} }_i'", r"\cdot", r"\vec{\mathbf{e} }_j'", r"= \delta_{ij}").next_to(line_1, DOWN).to_edge(LEFT)
line_2[1].set_color(PURPLE)
line_2[2].set_color(BLUE)
line_2[4].set_color(PURPLE)
line_2[5].set_color(BLUE)
line_2[7].set_color(GREEN)
line_2[9].set_color(GREEN)
line_3 = Tex(r"$\mathbf{P}$", r" preserves dot product between basis vectors").next_to(line_2, DOWN).to_edge(LEFT)
line_3[0].set_color(PURPLE)
line_3[1].set_color(YELLOW)
line_4 = Tex(r"$\Rightarrow$", r"$\mathbf{P}$", r" preserves dot product in general").next_to(line_3, DOWN).to_edge(LEFT)
line_4[1].set_color(PURPLE)
line_4[2].set_color(YELLOW)
self.play(Write(linear_text))
self.wait()
self.play(Write(line_1))
self.wait()
self.play(Write(line_2))
self.wait()
self.play(Write(line_3))
self.wait()
self.play(Write(line_4))
self.wait()