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gram_talk_2.py
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gram_talk_2.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 WhatAboutNonParallelotopes(Scene):
def construct(self):
all_text = Tex(r"Gramian works ", r"for parallelotopes").scale(1.5)
all_text[1].set_color(RED)
but_text = Tex(r"What about a general shape?", color=YELLOW).scale(1.5).next_to(all_text, DOWN)
self.play(Write(all_text))
self.wait()
self.play(Write(but_text))
self.wait()
class SphereExample(ThreeDScene):
def construct(self):
axes = ThreeDAxes(z_min=-4.5, z_max=4.5)
self.set_camera_orientation(phi=60 * DEGREES, theta=-100 * DEGREES)
self.add(axes)
unit_sphere = Sphere()
self.play(Create(unit_sphere))
self.wait()
apply_text = Tex(r"Apply some nonlinear transformation into 4D").to_edge(UP).add_background_rectangle()
self.add_fixed_in_frame_mobjects(apply_text)
self.play(Write(apply_text[1]))
self.wait()
question_text = Tex(r"What is the new volume?", color=YELLOW).next_to(apply_text, DOWN).add_background_rectangle()
self.add_fixed_in_frame_mobjects(question_text)
self.play(Write(question_text[1]))
self.wait()
would_text = Tex(r"Would need some sort of ", r"integral...").next_to(question_text, DOWN).add_background_rectangle()
would_text[2].set_color(GREEN)
self.add_fixed_in_frame_mobjects(would_text)
self.play(Write(would_text[1:]))
self.wait()
question_text_2 = Tex(r"How to integrate over a 3D region in 4D?", color=YELLOW).next_to(would_text, DOWN).add_background_rectangle()
self.add_fixed_in_frame_mobjects(question_text_2)
self.play(Write(question_text_2[1]))
self.wait()
class ThePlan0(Scene):
def construct(self):
title_text = Tex(r"The Plan").scale(2).to_edge(UP)
self.play(Write(title_text))
self.wait()
line_1 = Tex(r"Recap ", r"integration", r" and ", r"change of variables", r" in 1D").shift(UP)
line_1[1].set_color(YELLOW)
line_1[3].set_color(YELLOW)
self.play(Write(line_1))
self.wait()
line_2 = Tex(r"See how to extend this to ", r"2D area integrals")
line_2[1].set_color(YELLOW)
self.play(Write(line_2))
self.wait()
line_3 = Tex(r"See how this extends to ", r"(generalised) surface integrals").shift(DOWN)
line_3[1].set_color(YELLOW)
self.play(Write(line_3))
self.wait()
box = SurroundingRectangle(line_1)
self.play(Create(box))
self.wait()
### Adding in this part for the "fake-out"
class HangOnAMinute(Scene):
def construct(self):
hold_text_1 = Tex(r"Actually, hold on...").scale(2.5).shift(UP)
hold_text_2 = Tex(r"Before the maths...", color=YELLOW).scale(2.5).next_to(hold_text_1, DOWN)
self.play(Write(hold_text_1))
self.wait()
self.play(Write(hold_text_2))
self.wait()
class MightNoticeIveBeenRecapping(Scene):
def construct(self):
might_text_1 = Tex(r"Might have noticed I've been").scale(1.6).shift(UP)
might_text_2 = Tex(r"recapping previous concepts", r" a lot").scale(1.6).next_to(might_text_1, DOWN)
might_text_2[0].set_color(YELLOW)
self.play(Write(might_text_1))
self.play(Write(might_text_2))
self.wait()
class MathsIsPrettyCumulative(Scene):
def construct(self):
title_text = Tex(r"Maths is ", r"Cumulative").scale(2.5).to_edge(UP)
title_text[1].set_color(YELLOW)
self.play(Write(title_text))
self.wait()
rects = list()
for count in range(15):
temp_rect = Rectangle(height=0.4, width=4, fill_opacity=1, fill_color=GREEN, stroke_color=WHITE, stroke_width=1)
if rects:
temp_rect.align_to(rects[count - 1], UP).shift(UP*0.4)
else:
temp_rect.to_edge(DOWN)
rects.append(temp_rect)
rects = VGroup(*rects)
rects.set_fill(opacity=0)
self.play(Create(rects))
self.wait()
for count in range(7):
self.play(rects[count].animate.set_fill(opacity=1), run_time=0.4)
self.wait(0.2)
rects[7].set_fill(color=GREY)
self.play(rects[7].animate.set_fill(opacity=1), run_time=0.5)
self.wait()
self.play(FadeOutAndShift(rects[7], LEFT))
for count in range(8, 15):
rects[count].set_fill(color=RED)
self.play(rects[count].animate.set_fill(opacity=1), run_time=0.1)
self.wait()
rects[7].set_fill(color=GREEN)
self.play(FadeIn(rects[7]))
self.wait()
for count in range(8, 15):
self.play(rects[count].animate.set_fill(color=GREEN), run_time=0.1)
self.wait()
class TheresOnlySoMuchICanDo(Scene):
def construct(self):
uh_oh_1 = Tex(r"There's only so much").scale(2).shift(UP)
uh_oh_2 = Tex(r"I can do!").scale(2).next_to(uh_oh_1, DOWN)
self.play(Write(uh_oh_1))
self.play(Write(uh_oh_2))
self.wait()
uh_oh_grp = VGroup(uh_oh_1, uh_oh_2)
self.play(uh_oh_grp.animate.to_edge(UP))
self.wait()
as_text = Tex(r"As \textbf{learners},", color=YELLOW).scale(2)
as_text_2 = Tex(r"what can we do?", color=YELLOW).scale(2).next_to(as_text, DOWN)
self.play(Write(as_text))
self.play(Write(as_text_2))
self.wait()
class ChangeOfVariables1D(GraphScene):
def __init__(self, **kwargs):
GraphScene.__init__(
self,
exclude_zero_label=False,
x_min=0,
x_max=1,
x_axis_config={"tick_frequency": 0.1},
x_labeled_nums=[0, 1],
y_min=0,
y_max=1.5,
y_axis_config={"tick_frequency": 0.1},
y_labeled_nums=[0,1],
**kwargs
)
@staticmethod
def test_func(x):
return 1 + 50 * x * (x - 0.1) * (x - 0.4) * (x - 0.7) * (x - 1)
@staticmethod
def test_func_u(u):
return ChangeOfVariables1D.test_func(2*u)
def construct(self):
self.setup_axes()
graph = self.get_graph(ChangeOfVariables1D.test_func, x_min=0, x_max=1)
self.play(Create(graph))
self.wait()
label = self.get_graph_label(graph)
self.play(Write(label))
self.wait()
integral = MathTex(r"\int_0^1 ", r"f(x)", r"dx", r" \approx \sum_1^n ", r"f(x_k)", "\Delta x")
integral[1].set_color(BLUE)
integral[4].set_color(BLUE)
integral.to_edge(UP)
self.play(Write(integral[:3]))
self.wait()
area_under = self.get_riemann_rectangles(graph, x_min=0, x_max=1, dx=0.01, start_color=GREEN, end_color=BLUE, stroke_width = 0, fill_opacity=1, stroke_color=GREEN)
self.play(Create(area_under))
self.wait()
self.play(FadeOut(area_under))
self.wait()
better_approxs = self.get_riemann_rectangles_list(
graph,
7,
max_dx=0.25,
x_min=0,
x_max=1,
input_sample_type="left",
start_color=GREEN,
end_color=BLUE
)
self.play(Write(integral[3:]))
self.wait()
previous_group = False
for group in better_approxs:
if previous_group:
self.play(ReplacementTransform(previous_group, group))
self.wait(0.2)
else:
self.play(Create(group))
self.wait(0.2)
previous_group = group
self.play(FadeOut(group), FadeOut(integral[3:]))
self.wait()
sub_text = MathTex(r"x", r"=", r"2u")
sub_text[0].set_color(BLUE)
sub_text[2].set_color(GREEN)
sub_text.next_to(integral, RIGHT)
self.play(Write(sub_text))
self.wait()
u_graph = self.get_graph(ChangeOfVariables1D.test_func_u, x_min=0, x_max=0.5)
u_label = self.get_graph_label(u_graph, label='f(2u)').shift(DOWN*2)
graph.save_state()
label.save_state()
self.play(Transform(graph, u_graph), Transform(label, u_label))
self.wait()
"""
u_label_2 = self.get_graph_label(u_graph, label='f(x(u))').shift(DOWN*2)
self.play(Transform(label, u_label_2))
"""
new_integral = MathTex(r"\int_0^1 ", r"f(x)", r"dx", r" = \int_0^{\frac 12} ", r"?", "du")
new_integral[1].set_color(BLUE)
new_integral[4].set_color(GREEN)
new_integral.to_edge(UP)
self.play(Transform(integral[:3], new_integral))
self.wait()
particular_approx = self.get_riemann_rectangles(u_graph, x_min=0, x_max=0.5, dx=0.1, start_color=GREEN, end_color=BLUE)
width_brace = Brace(particular_approx[0])
brace_tex = width_brace.get_tex("\Delta u")
x_approx = self.get_riemann_rectangles(graph, x_min=0, x_max=1, dx=0.2, start_color=GREEN, end_color=BLUE)
x_width_brace = Brace(x_approx[0])
x_brace_tex = x_width_brace.get_tex("\Delta x")
self.wait()
approx_group = VGroup(particular_approx, width_brace, brace_tex)
x_approx_group = VGroup(x_approx, x_width_brace, x_brace_tex)
approx_group.save_state()
self.play(Write(particular_approx))
self.wait()
self.play(Create(width_brace), Create(brace_tex))
self.wait()
self.play(Restore(graph), Restore(label), Transform(approx_group, x_approx_group))
self.wait()
self.wait(2)
class ChangeOfVariables1DDecomposingEffect(Scene):
def construct(self):
top_text = MathTex(r"\int_0^1", r"f(x)", r"dx")
top_text[0].set_color(PURPLE)
top_text[1].set_color(BLUE)
top_text[2].set_color(YELLOW)
top_text.to_edge(UP).scale(1.5)
self.play(Write(top_text))
self.wait()
bottom_mid = (top_text.get_corner(DL) + top_text.get_corner(DR))/2
domain = MathTex(r"\int_0^1", color=PURPLE)
domain.to_edge(LEFT).shift(RIGHT)
integrand = MathTex(r"f(x)", color=BLUE)
differential = MathTex(r"dx", color=YELLOW)
differential.to_edge(RIGHT).shift(LEFT)
domain_arrow = Arrow(bottom_mid, domain.get_corner(UR), color=PURPLE)
integrand_arrow = Arrow(bottom_mid, (integrand.get_corner(UR) + integrand.get_corner(UL))/2, color=BLUE)
differential_arrow = Arrow(bottom_mid, differential.get_corner(UL), color=YELLOW)
domain_brace = Brace(domain, direction=DOWN)
domain_brace_text = domain_brace.get_text("Domain")
domain_brace_text.set_color(PURPLE)
integrand_brace = Brace(integrand, direction=DOWN)
integrand_brace_text = integrand_brace.get_text("Integrand")
integrand_brace_text.set_color(BLUE)
differential_brace = Brace(differential, direction=DOWN)
differential_brace_text = differential_brace.get_text("Width element")
differential_brace_text.set_color(YELLOW)
self.play(Create(domain_arrow), Write(domain))
self.wait()
self.play(Create(domain_brace), Create(domain_brace_text))
self.wait()
self.play(Create(integrand_arrow), Write(integrand))
self.wait()
self.play(Create(integrand_brace), Create(integrand_brace_text))
self.wait()
self.play(Create(differential_arrow), Write(differential))
self.wait()
self.play(Create(differential_brace), Create(differential_brace_text))
self.wait()
sub_text = MathTex(r"x", r"=", r"2u").scale(1.5)
sub_text[0].set_color(BLUE)
sub_text[2].set_color(GREEN)
sub_text.next_to(top_text, RIGHT).to_edge(RIGHT)
self.play(Write(sub_text))
self.wait()
sub_text_rect = SurroundingRectangle(sub_text, buff=0.1)
self.play(Create(sub_text_rect))
self.wait()
new_domain = MathTex(r"\int_0^{\frac 12}", color=PURPLE)
new_domain.to_edge(LEFT).shift(RIGHT)
new_domain_rect = SurroundingRectangle(new_domain, buff=0.1)
new_integrand = MathTex(r"f(2u)", color=GREEN)
new_integrand_rect = SurroundingRectangle(new_integrand, buff=0.1)
new_differential = MathTex(r"?", color=YELLOW)
new_differential.move_to(differential.get_center())
new_differential_rect = SurroundingRectangle(new_differential, buff=0.2)
self.play(ReplacementTransform(sub_text_rect, new_domain_rect))
self.wait()
self.play(Transform(domain, new_domain))
self.wait()
self.play(ReplacementTransform(new_domain_rect, new_integrand_rect))
self.wait()
self.play(Transform(integrand, new_integrand))
self.wait()
self.play(ReplacementTransform(new_integrand_rect, new_differential_rect))
self.wait()
self.play(Transform(differential, new_differential))
self.wait()
know_text = Tex(r"Know how ", r"domain", r" and ", r"integrand", r" change").next_to(integrand_brace_text, DOWN).shift(DOWN*0.5)
know_text[1].set_color(PURPLE)
know_text[3].set_color(GREEN)
self.play(Write(know_text))
self.wait()
harder_text = Tex(r"Need some thinking to find how ", r"width element", r" changes").next_to(know_text, DOWN)
harder_text[1].set_color(YELLOW)
self.play(Write(harder_text))
self.wait()
class HowWidthElementTransforms1D(ZoomedScene):
def get_dot_mapping_animation(self, dot_group, func, input_line, output_line):
animations = list()
for dot in dot_group:
new_coords = output_line.n2p(func(input_line.p2n(dot.get_center())))
animations.append(dot.animate.move_to(new_coords))
return animations
def zoom_in_on_input(self, input_num, input_line, animate=False):
zoomed_camera = self.zoomed_camera
if animate:
self.play(zoomed_camera.frame.animate.move_to(input_line.n2p(input_num)))
else:
zoomed_camera.frame.move_to(input_line.n2p(input_num))
def construct(self):
note_text = Tex(r"Write ", r"$x$ ", r"$=$ ", r"$g($", r"$u$", r"$)$ ", r"$\longrightarrow$ mapping from ", r"$u$", r" to ", r"$x$")
note_text.set_color_by_tex("$x$", BLUE)
note_text.set_color_by_tex("u$", GREEN)
note_text.to_edge(UP)
fit_text = Tex(r"Construct ", r"$u$", r"-integral that's ", r"equivalent", r" to ", r"$x$", r"-integral")
fit_text.set_color_by_tex("$x$", BLUE)
fit_text.set_color_by_tex("u$", GREEN)
fit_text[3].set_color(YELLOW)
fit_text.next_to(note_text, DOWN)
specific_func_text_1 = MathTex(r"2u", color=GREEN)
specific_func_text_1.move_to(note_text[3:6].get_center())
domain_text = MathTex(r"[0, 1] \text{ in }", r"x\text{-world }", r"\longrightarrow", r"[0, \frac 12] \text{ in }", r"u\text{-world}")
domain_text[1].set_color(BLUE)
domain_text[-1].set_color(GREEN)
domain_text.next_to(fit_text, DOWN)
apply_brace = Brace(domain_text[2], DOWN)
inverse_brace_text = apply_brace.get_text(r"Apply $g^{-1}($", r"$x$", r"$) =$ ", r"$\frac{1}{2} x$ ", r"$=$ ", r"$u$")
inverse_brace_text.set_color_by_tex("x$", BLUE)
inverse_brace_text.set_color_by_tex("u$", GREEN)
inverse_brace_text_simple = apply_brace.get_tex(r"g^{-1}")
normal_brace_text = apply_brace.get_tex(r"g")
self.play(Write(note_text[:6]))
self.wait()
self.play(Write(note_text[6:]))
self.wait()
self.play(Write(fit_text))
self.wait()
note_text.save_state()
self.play(Transform(note_text[3:6], specific_func_text_1))
self.wait()
self.play(Write(domain_text))
self.wait()
self.play(Create(apply_brace), Write(inverse_brace_text))
self.wait()
self.play(ReplacementTransform(inverse_brace_text, inverse_brace_text_simple))
self.wait()
self.play(FadeOut(apply_brace), FadeOut(inverse_brace_text_simple))
self.play(CyclicReplace(domain_text[:2], domain_text[3:]))
self.wait()
self.play(FadeOut(domain_text[3:]))
self.wait()
self.play(Create(apply_brace), Write(normal_brace_text))
self.wait()
question_brace = Brace(domain_text[3:], DOWN)
question_brace_text = question_brace.get_text(r"Equivalent domain?")
question_brace_text.set_color(YELLOW)
self.play(Create(question_brace), Write(question_brace_text))
self.wait()
self.play(
FadeOut(note_text),
FadeOut(domain_text[:3]),
FadeOut(question_brace),
FadeOut(question_brace_text),
FadeOut(apply_brace),
FadeOut(normal_brace_text),
FadeOut(fit_text)
)
self.wait()
transform_text = MathTex(r"x", r"=", r"2u").scale(1.5)
transform_text[0].set_color(BLUE)
transform_text[2].set_color(GREEN)
transform_text.to_edge(UP)
self.play(Write(transform_text))
self.wait()
u_line = NumberLine(
unit_size=12,
x_min=0,
x_max=0.5,
tick_frequency=0.05,
numbers_with_elongated_ticks=[0, 0.5]
).shift(UP).to_edge(LEFT)
x_line = NumberLine(
unit_size=12,
x_min=0,
x_max=1,
tick_frequency=0.1,
numbers_with_elongated_ticks=[0, 1]
).shift(DOWN).to_edge(LEFT)
u_label = MathTex("u", color=GREEN)
u_label.next_to(u_line, RIGHT)
x_label = MathTex("x", color=BLUE)
x_label.next_to(x_line, RIGHT)
self.play(Create(u_line), Create(u_label))
self.wait()
self.play(Create(x_line), Create(x_label))
self.wait()
evenly_spaced_dots = VGroup(*[Dot(u_line.n2p(number)) for number in np.linspace(0, 0.5, num=21)])
evenly_spaced_dots.set_color_by_gradient(RED, YELLOW)
evenly_spaced_dots.save_state()
self.play(Create(evenly_spaced_dots))
self.wait()
ghosts = evenly_spaced_dots.copy()
self.add(ghosts)
self.play(*self.get_dot_mapping_animation(evenly_spaced_dots, lambda x : 2 * x, u_line, x_line))
self.wait()
linear_text = Tex(r"Linear ", r"$\Leftrightarrow$", r" evenly spaced dots remain evenly spaced")
linear_text[0].set_color(YELLOW)
linear_text[2].set_color(YELLOW)
linear_text.next_to(x_line, DOWN).shift(DOWN)
self.play(Write(linear_text))
self.wait()
obvious_text = Tex(r"Width ", r"$\longrightarrow$", r" $2($", r"Width", r"$)$", r" $ = $ ", r'Correct Width')
obvious_text[0].set_color(GREEN)
obvious_text[3].set_color(GREEN)
obvious_text[-1].set_color(BLUE)
obvious_text.next_to(transform_text, DOWN)
self.play(Write(obvious_text[:-2]))
self.wait()
self.play(Write(obvious_text[-2:]))
self.wait()
obvious_math_text = MathTex(r"\Delta u", r"\longrightarrow", r"2", r"\Delta u", r"=", r"\Delta x", r"= \text{Correct Width}")
obvious_math_text[0].set_color(GREEN)
obvious_math_text[2].set_color(YELLOW)
obvious_math_text[3].set_color(GREEN)
obvious_math_text[5:].set_color(BLUE)
obvious_math_text.next_to(transform_text, DOWN)
self.play(ReplacementTransform(obvious_text, obvious_math_text))
self.wait()
obvious_scaling_brace = Brace(obvious_math_text[2], DOWN)
obvious_scaling_brace_text = obvious_scaling_brace.get_text("Multiply by correction factor")
self.play(Create(obvious_scaling_brace), Write(obvious_scaling_brace_text))
self.wait()
self.play(
FadeOut(obvious_scaling_brace),
FadeOut(obvious_scaling_brace_text),
FadeOut(obvious_math_text),
FadeOut(transform_text),
FadeOut(ghosts),
Restore(evenly_spaced_dots)
)
self.play(FadeOut(evenly_spaced_dots))
self.wait()
new_u_line = NumberLine(
unit_size=12,
x_min=0,
x_max=1,
tick_frequency=0.1,
numbers_with_elongated_ticks=[0, 1]
).shift(UP).to_edge(LEFT)
self.play(ReplacementTransform(u_line, new_u_line), u_label.animate.next_to(new_u_line, RIGHT))
self.wait()
u_line = new_u_line
transform_text = MathTex(r"x", r"=", r"u^2").scale(1.5)
transform_text[0].set_color(BLUE)
transform_text[2].set_color(GREEN)
transform_text.to_edge(UP)
self.play(Write(transform_text))
self.wait()
evenly_spaced_dots = VGroup(*[Dot(u_line.n2p(number)) for number in np.linspace(0, 1.0, num=21)])
evenly_spaced_dots.set_color_by_gradient(RED, YELLOW)
evenly_spaced_dots.save_state()
self.play(Create(evenly_spaced_dots))
self.wait()
ghosts = evenly_spaced_dots.copy()
self.add(ghosts)
self.play(*self.get_dot_mapping_animation(evenly_spaced_dots, lambda x : x * x, u_line, x_line))
self.wait()
nonlinear_text = Tex(r"Nonlinear").scale(1.5)
nonlinear_text.next_to(transform_text, DOWN)
rainbowify(nonlinear_text[0])
self.play(Write(nonlinear_text))
self.wait()
self.play(FadeOut(ghosts), Restore(evenly_spaced_dots))
self.wait()
# Fade out the dot corresponding to 0.75
self.play(FadeOut(evenly_spaced_dots[15]))
new_list = list(evenly_spaced_dots)
new_list.pop(15)
evenly_spaced_dots = VGroup(*new_list)
# Now need to cover up the number line, otherwise when we zoom in it gets too large
# 0.15 here is the zoom factor, 3 is the side length of the display square
# So 3 * 0.15 should be side length of zoom square
covering_rectangle = Rectangle(
color=BLACK, height = 3 * 0.15, width = 3 * 0.15, fill=BLACK, fill_opacity=1
).move_to(u_line.n2p(0.75))
self.bring_to_front(covering_rectangle)
self.add(covering_rectangle)
# Then, we add a dot cluster near the point
# We need to get the left and right endpoints
# So, we get the point corresponding to 0.75
# Subtract off half the width of the camera frame
# And get the corresponding point on the number line
left_endpoint = u_line.p2n(u_line.n2p(0.75) - np.array([1.5 * 0.15, 0, 0]))
right_endpoint = u_line.p2n(u_line.n2p(0.75) + np.array([1.5 * 0.15, 0, 0]))
dot_cluster = VGroup(*[Dot(u_line.n2p(number), radius=0.01) for number in np.linspace(left_endpoint, right_endpoint, num=11)])
dot_cluster.set_color_by_gradient(RED, YELLOW)
cluster_spacing = dot_cluster[1].get_center() - dot_cluster[0].get_center()
center_dot = dot_cluster[5]
# But also need a miniature number line for this
mini_line = Line(u_line.n2p(left_endpoint), u_line.n2p(right_endpoint), stroke_width = 0.5, color=WHITE)
self.add(mini_line)
# Also need to add ticks
ticks = list()
for dot in dot_cluster:
ticks.append(Line(dot.get_center() - np.array([0, 0.1 * 0.15, 0]), dot.get_center() + np.array([0, 0.1 * 0.15, 0]), stroke_width=0.5))
ticks = VGroup(*ticks)
self.add(ticks)
center_mark = DecimalNumber(number=0.75).move_to(ticks[5].get_start()).scale(0.1).shift(DOWN*0.04)
self.add(center_mark)
self.play(Create(dot_cluster))
self.wait()
self.activate_zooming()
self.zoom_in_on_input(0.75, u_line)
self.wait()
self.zoomed_camera.frame.add_updater(lambda m: m.move_to(center_dot.get_center()))
covering_rectangle.add_updater(lambda m: m.move_to(center_dot.get_center()))
mini_line.add_updater(lambda m: m.move_to(center_dot.get_center()))
ticks.add_updater(lambda m: m.move_to(center_dot.get_center()))
self.add(self.zoomed_camera.frame, covering_rectangle, mini_line, ticks) # Must add after adding any updater
self.bring_to_front(dot_cluster) # To stop them getting covered by covering rectangle
self.play(
*self.get_dot_mapping_animation(evenly_spaced_dots, lambda x: x**2, u_line, x_line),
*self.get_dot_mapping_animation(dot_cluster, lambda x: x**2, u_line, x_line)
)
self.wait()
scale_text = Tex(r"Scale by ", r"$1.5$").scale(0.9).to_edge(UP).to_edge(RIGHT).align_to(self.zoomed_camera.frame, alignment_vect=LEFT).shift(DOWN*0.2).shift(LEFT*0.2)
scale_text[1].set_color(YELLOW)
self.add_foreground_mobject(scale_text)
self.play(Write(scale_text))
self.wait()
onepointfive_copy = scale_text[1].copy()
scale_math_text = MathTex(r"\frac{dx}{du}(0.75) = ", r"1.5").to_edge(UP).to_edge(LEFT)
scale_math_text[1].set_color(YELLOW)
self.add_foreground_mobject(onepointfive_copy)
self.play(ReplacementTransform(onepointfive_copy, scale_math_text[1]), Write(scale_math_text[0]))
self.wait()
locally_linear = Tex(r"Locally linear", color=YELLOW).scale(1.5).next_to(transform_text, DOWN)
self.play(ReplacementTransform(nonlinear_text, locally_linear))
self.wait()
self.play(FadeOut(scale_math_text), FadeOut(scale_text))
self.wait()
# Gotta move locally linear out of the way
self.play(locally_linear.animate.to_edge(UP).to_edge(LEFT))
"""
self.play(
FadeOut(locally_linear),
FadeOut(self.zoomed_camera.frame),
FadeOut(self.zoomed_display),
FadeOut(u_line),
FadeOut(x_line),
FadeOut(u_label),
FadeOut(x_label),
FadeOut(evenly_spaced_dots),
FadeOut(dot_cluster),
FadeOut(mini_line),
FadeOut(ticks),
FadeOut(linear_text)
)
"""
# And gotta get rid of camera stuff
self.zoom_activated = False
self.play(
FadeOut(self.zoomed_camera.frame),
FadeOut(self.zoomed_display),
)
self.wait()
nonobvious_text = Tex(r"Width ", r"$\longrightarrow$", r" $2(0.75)$", r"$($", r"Width", r"$)$ ", r"$ = $", r' Correct Width')
nonobvious_text[0].set_color(GREEN)
nonobvious_text[2].set_color(GREEN)
nonobvious_text[-1].set_color(BLUE)
nonobvious_text.next_to(transform_text, DOWN)
general_scale_factor = MathTex(r"2u").set_color(GREEN).move_to(nonobvious_text[2].get_center())
self.play(Write(nonobvious_text[:-2]))
self.wait()
self.play(Write(nonobvious_text[-2:]))
self.wait()
self.play(Transform(nonobvious_text[2], general_scale_factor))
self.wait()
nonobvious_math_text = MathTex(r"\Delta u", r"\longrightarrow", r"2u", r"\Delta u", r"=", r"\Delta x", r"= \text{Correct Width}")
nonobvious_math_text[0].set_color(GREEN)
nonobvious_math_text[2].set_color(YELLOW)
nonobvious_math_text[3].set_color(GREEN)
nonobvious_math_text[5:].set_color(BLUE)
nonobvious_math_text.next_to(transform_text, DOWN)
self.play(ReplacementTransform(nonobvious_text, nonobvious_math_text))
self.wait()
nonobvious_scaling_brace = Brace(nonobvious_math_text[2], DOWN)
nonobvious_scaling_brace_text = nonobvious_scaling_brace.get_text("Multiply by correction factor")
nonobvious_scaling_brace_text_2 = nonobvious_scaling_brace.get_text(r"Multiply by ", r"width distortion factor")
nonobvious_scaling_brace_text_2[1].set_color(YELLOW)
self.play(Create(nonobvious_scaling_brace), Write(nonobvious_scaling_brace_text))
self.wait()
self.play(Transform(nonobvious_scaling_brace_text, nonobvious_scaling_brace_text_2))
self.wait()
class LocalWidthDistortionFactor(Scene):
def construct(self):
general_text = Tex(r"In general, have ", r"$x$ ", r"$=$ ", r"$g(u)$").scale(1.5).to_edge(UP)
general_text[1].set_color(BLUE)
general_text[3].set_color(GREEN)
self.play(Write(general_text))
self.wait()
width_transformation_law = MathTex(r"\Delta u", r"\longrightarrow", r"g'(u)", r"\Delta u", r"=", r"\Delta x").scale(2)
width_transformation_law[0].set_color(GREEN)
width_transformation_law[2].set_color(YELLOW)
width_transformation_law[3].set_color(GREEN)
width_transformation_law[5].set_color(BLUE)
width_brace = Brace(width_transformation_law[0], UP)
width_brace_text = width_brace.get_text(r"Width").scale(1.8)
width_brace_text.set_color(GREEN)
correct_width_brace = Brace(width_transformation_law[-1], UP)
correct_width_brace_text = correct_width_brace.get_text(r"Correct width").set_color(BLUE).scale(1.8)
distort_brace = Brace(width_transformation_law[2], DOWN)
distort_brace_text = distort_brace.get_text(r"Local width distortion factor").set_color(YELLOW).scale(2)
self.play(Write(width_transformation_law))
self.wait()
self.play(Create(width_brace), Write(width_brace_text))
self.wait()
self.play(Create(correct_width_brace), Write(correct_width_brace_text))
self.wait()
self.play(Create(distort_brace), Write(distort_brace_text))
self.wait()
fade_grp = VGroup(width_brace, width_brace_text, correct_width_brace, correct_width_brace_text, distort_brace, distort_brace_text)
self.play(FadeOut(fade_grp))
self.wait()
approx_box = SurroundingRectangle(width_transformation_law[4])
approx_sign = MathTex(r"\approx").scale(2).move_to(width_transformation_law[4].get_center())
width_transformation_law[4].save_state()
self.play(Create(approx_box))
self.play(Transform(width_transformation_law[4], approx_sign))
self.wait()
equals_brace = Brace(width_transformation_law[4], DOWN)
equals_brace_text = equals_brace.get_text(r"Only at finite zoom level")
self.play(Create(equals_brace), Write(equals_brace_text))
self.wait()
self.play(FadeOut(equals_brace), FadeOut(equals_brace_text))
self.wait()
du_symbol = MathTex(r"du", color=GREEN).scale(2).move_to(width_transformation_law[0].get_center())
du_symbol_2 = MathTex(r"du", color=GREEN).scale(2).move_to(width_transformation_law[3].get_center())
dx_symbol = MathTex(r"dx", color=BLUE).scale(2).move_to(width_transformation_law[-1].get_center())
self.play(Transform(width_transformation_law[0], du_symbol), Transform(width_transformation_law[3], du_symbol_2), Transform(width_transformation_law[-1], dx_symbol), Restore(width_transformation_law[4]))
self.play(FadeOut(approx_box))
self.wait()
just_text = Tex(r"Just integration by substitution formula!", color=YELLOW).scale(1.5).next_to(general_text, DOWN)
self.play(Write(just_text))
self.wait()
class ThePlan1(Scene):
def construct(self):
title_text = Tex(r"The Plan").scale(2).to_edge(UP)
self.add(title_text)
line_1 = Tex(r"Recap ", r"integration", r" and ", r"change of variables", r" in 1D").shift(UP)
line_1[1].set_color(YELLOW)
line_1[3].set_color(YELLOW)
self.add(line_1)
line_2 = Tex(r"See how to extend this to ", r"2D area integrals")
line_2[1].set_color(YELLOW)
self.add(line_2)
line_3 = Tex(r"See how this extends to ", r"(generalised) surface integrals").shift(DOWN)
line_3[1].set_color(YELLOW)
self.add(line_3)
box = SurroundingRectangle(line_2)
self.play(Create(box))
self.wait()
class ChangeOfVariables2D(Scene):
def construct(self):
title_text = Tex(r"2D Change of variables").scale(1.6).to_edge(UP)
self.play(Write(title_text))
self.wait()
heads_up = Tex(r"Will only focus on how ", r"area element", " changes").next_to(title_text, DOWN)
heads_up[1].set_color(YELLOW)
self.play(Write(heads_up))
self.wait()
heads_up_2 = Tex(r"(Know how ", r"domain", r" and ", r"integrand", r" change)").next_to(heads_up, DOWN)
heads_up_2[1].set_color(PURPLE)
heads_up_2[3].set_color(BLUE)
self.play(Write(heads_up_2))
self.wait()
sub_text = MathTex(r"\begin{bmatrix} x \\ y \end{bmatrix}", r"=", r"\begin{bmatrix} f_1(u, v) \\ f_2(u, v) \end{bmatrix}").scale(2)
sub_text[0].set_color(BLUE)
sub_text[2].set_color(YELLOW)
self.play(Write(sub_text))
self.wait()
map_text = Tex(r"Mapping from ", r"u-v", r" to ", r"x-y").scale(2).next_to(sub_text, DOWN)
map_text[1].set_color(YELLOW)
map_text[3].set_color(BLUE)
self.play(Write(map_text))
self.wait()
general_text = Tex(r"Generally ", r"nonlinear").scale(2).next_to(map_text, DOWN)
rainbowify(general_text[1])
self.play(Write(general_text))
self.wait()
class WeirdAndWonderfulNonlinearTransformations(Scene):
def construct(self):
grid = NumberPlane()
self.add(grid)
self.play(Create(grid))
self.wait()
grid.save_state()
grid.prepare_for_nonlinear_transform()
self.play(grid.animate(run_time=2).apply_function(lambda point: [point[0]**2 - point[1]**2, 4 * point[0] * point[1], 0]))
self.wait()
self.play(Restore(grid))
self.wait()
self.play(grid.animate(run_time=2).apply_function(lambda point: complex_to_R3(np.exp(R3_to_complex(point)))))
self.wait()
self.play(Restore(grid))
self.wait()
self.play(grid.animate.apply_function(lambda point: [point[0] + np.sin(point[1]), point[1] + np.sin(point[0]), 0]))
self.wait()
self.play(Restore(grid))
self.wait()
class LocallyLinear2DTransformation(ZoomedScene):
def __init__(self, **kwargs):
ZoomedScene.__init__(
self,
zoomed_camera_frame_starting_position=np.array([1.5, -0.5, 0]),
**kwargs
)
def construct(self):
grid = NumberPlane()
self.add(grid)
self.wait()
grid.save_state()
grid.prepare_for_nonlinear_transform()
linear_text = Tex(r"Linear ", r"$\Leftrightarrow$", r" Gridlines remain parallel and evenly spaced", color=YELLOW).scale(0.7)
linear_text[1].set_color(WHITE)