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Just beginning FTOC animations of eoc8
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Grant Sanderson
parent
326e7fc76f
commit
18855c0522
1 changed files with 307 additions and 23 deletions
330
eoc/chapter8.py
330
eoc/chapter8.py
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@ -1,4 +1,5 @@
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from helpers import *
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import scipy
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from mobject.tex_mobject import TexMobject
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from mobject import Mobject
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@ -361,8 +362,11 @@ class PlotVelocity(GraphScene):
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return graph, graph_label
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class Chapter2Wrapper(Scene):
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CONFIG = {
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"title" : "Chapter 2: The paradox of the derivative",
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}
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def construct(self):
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title = TextMobject("Chapter 2: The paradox of the derivative")
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title = TextMobject(self.title)
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title.to_edge(UP)
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rect = Rectangle(width = 16, height = 9, color = WHITE)
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rect.scale_to_fit_height(1.5*SPACE_HEIGHT)
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@ -546,13 +550,12 @@ class AreaUnderVGraph(PlotVelocity):
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class ConstantVelocityCar(Scene):
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def construct(self):
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car = Car()
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car.scale(2)
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car.move_to(3*LEFT + 3*DOWN)
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car.move_to(5*LEFT + 3*DOWN)
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self.add(car)
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self.dither()
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self.play(MoveCar(
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car, 6*RIGHT+3*DOWN,
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car, 7*RIGHT+3*DOWN,
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run_time = 5,
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rate_func = None,
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))
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@ -759,6 +762,7 @@ class PiecewiseConstantPlot(PlotVelocity):
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self.write_integral_symbol()
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self.roles_of_dt()
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self.what_does_sum_approach()
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self.label_integral()
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def setup_graph(self):
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self.setup_axes()
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@ -1035,8 +1039,9 @@ class PiecewiseConstantPlot(PlotVelocity):
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self.dither()
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for x in range(2):
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self.play(
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rect.scale_to_fit_height, v_lines[1].get_height(),
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rect.stretch_to_fit_height, v_lines[1].get_height(),
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rect.move_to, rect.get_bottom(), DOWN,
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Animation(v_lines),
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run_time = 4,
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rate_func = there_and_back
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)
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@ -1263,6 +1268,16 @@ class PiecewiseConstantPlot(PlotVelocity):
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brace = Brace(self.integral, DOWN)
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dt_to_0 = brace.get_text("$dt \\to 0$")
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distance_words = TextMobject(
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"Area", "= Distance traveled"
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)
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distance_words.next_to(rects, UP)
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arrow = Arrow(
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distance_words[0].get_bottom(),
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rects.get_center(),
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color = WHITE
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)
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self.play(PiCreatureSays(
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morty, "Why not $\\Sigma$?",
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target_mode = "sassy"
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@ -1293,6 +1308,51 @@ class PiecewiseConstantPlot(PlotVelocity):
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)
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self.dither()
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self.play(
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Write(distance_words),
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ShowCreation(arrow),
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morty.change_mode, "pondering",
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morty.look_at, distance_words,
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)
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self.dither()
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self.play(Blink(morty))
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self.dither()
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self.area_arrow = arrow
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def label_integral(self):
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words = TextMobject("``Integral of $v(t)$''")
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words.to_edge(UP)
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arrow = Arrow(
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words.get_right(),
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self.integral.get_left()
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)
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self.play(Indicate(self.integral))
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self.play(Write(words, run_time = 2))
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self.play(ShowCreation(arrow))
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self.dither()
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self.play(*[
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ApplyFunction(
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lambda r : r.shift(0.2*UP).set_fill(None, 1),
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rect,
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run_time = 3,
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rate_func = squish_rate_func(
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there_and_back,
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alpha, alpha+0.2,
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)
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)
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for rect, alpha in zip(
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self.rects,
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np.linspace(0, 0.8, len(self.rects))
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)
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]+[
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Animation(self.area_arrow),
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self.morty.change_mode, "happy",
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self.morty.look_at, self.rects,
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])
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self.dither()
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#####
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def get_pw_constant_graph(self):
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@ -1320,25 +1380,249 @@ class PiecewiseConstantPlot(PlotVelocity):
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ticks.highlight(YELLOW)
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return ticks
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class CarJourneyApproximations(Scene):
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class CarJourneyApproximation(Scene):
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CONFIG = {
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"n_jumps" : 5,
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}
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def construct(self):
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pass
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points = [5*LEFT + v for v in UP, 2*DOWN]
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cars = [Car().move_to(point) for point in points]
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h_line = Line(LEFT, RIGHT).scale(SPACE_WIDTH)
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words = [
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TextMobject("Real motion (smooth)").shift(3*UP),
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TextMobject("Approximated motion (jerky)").shift(0.5*DOWN),
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]
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self.add(h_line, *cars + words)
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self.dither()
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self.play(*[
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MoveCar(
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car, point+10*RIGHT,
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run_time = 5,
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rate_func = rf
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)
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for car, point, rf in zip(cars, points, [
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s_rate_func,
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self.get_approximated_rate_func(self.n_jumps)
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])
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])
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self.dither()
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def get_approximated_rate_func(self, n):
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new_v_rate_func = lambda t : v_rate_func(np.floor(t*n)/n)
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max_integral, err = scipy.integrate.quad(
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v_rate_func, 0, 1
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)
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def result(t):
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integral, err = scipy.integrate.quad(new_v_rate_func, 0, t)
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return integral/max_integral
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return result
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class LessWrongCarJourneyApproximation(CarJourneyApproximation):
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CONFIG = {
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"n_jumps" : 20,
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}
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class HowDoesThisHelp(TeacherStudentsScene):
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def construct(self):
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self.student_says(
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"How does this help\\textinterrobang",
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target_mode = "angry",
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run_time = 1
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)
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self.change_student_modes(
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"confused", "angry", "confused",
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)
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self.dither(2)
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self.teacher_says(
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"You're right.",
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target_mode = "shruggie",
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run_time = 1
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)
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self.change_student_modes(*["sassy"]*3)
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self.dither(2)
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class AreaUnderACurve(GraphScene):
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CONFIG = {
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"y_max" : 4,
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"y_min" : 0,
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"num_iterations" : 7
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}
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def construct(self):
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self.setup_axes()
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graph = self.get_graph(self.func)
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rect_list = self.get_riemann_rectangles_list(
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graph, self.num_iterations
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)
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VGroup(*rect_list).set_fill(opacity = 0.8)
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rects = rect_list[0]
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self.play(ShowCreation(graph))
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self.play(Write(rects))
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for new_rects in rect_list[1:]:
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rects.align_submobjects(new_rects)
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for every_other_rect in rects[::2]:
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every_other_rect.set_fill(opacity = 0)
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self.play(Transform(
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rects, new_rects,
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run_time = 2,
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submobject_mode = "lagged_start"
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))
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self.dither()
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def func(self, x):
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return np.sin(x) + 1
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class AltAreaUnderCurve(AreaUnderACurve):
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CONFIG = {
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"graph_origin" : 2*DOWN,
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"x_min" : -3,
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"x_max" : 3,
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"x_axis_width" : 12,
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"y_max" : 2,
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"y_axis_height" : 4,
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}
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def func(self, x):
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return np.exp(-x**2)
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class Chapter1Wrapper(Chapter2Wrapper):
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CONFIG = {
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"title" : "Essence of calculus, chapter 1",
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}
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class AreaIsDerivative(PlotVelocity):
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CONFIG = {
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"y_axis_label" : "",
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# "num_rects" : 250,
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"num_rects" : 20,
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}
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def setup(self):
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PlotVelocity.setup(self)
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self.setup_axes()
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self.add(*self.get_v_graph_and_label())
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self.foreground_mobjects = []
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def construct(self):
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self.force_skipping()
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self.introduce_variable_area()
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self.write_integral()
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def introduce_variable_area(self):
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area = self.area = self.get_area(0, 6)
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x_nums = self.x_axis.numbers
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self.play(Write(area, run_time = 2))
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self.play(FadeOut(self.x_axis.numbers))
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self.add_T_label(6)
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self.change_area_bounds(
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new_t_max = 4,
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rate_func = there_and_back,
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run_time = 2
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)
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self.dither()
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def write_integral(self):
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integral = TexMobject("\\int", "^T", "_0", "v(t)", "\\,dt")
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integral.to_corner(UP+RIGHT)
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top_T = integral.get_part_by_tex("T")
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moving_T = self.T_label_group[0]
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s_T = TexMobject("s(T) = ")
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s_T.next_to(integral, LEFT)
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int_arrow, s_arrow = [
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Arrow(
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mob.get_left(), self.area.get_center(),
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color = WHITE
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),
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for mob in integral, s_T
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]
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self.revert_to_original_skipping_status()
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self.play(Write(integral))
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self.play(ShowCreation(int_arrow))
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self.foreground_mobjects.append(int_arrow)
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self.dither()
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self.change_area_bounds(
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new_t_max = 8,
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rate_func = there_and_back,
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run_time = 2,
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)
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self.play(Indicate(top_T))
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self.play(ReplacementTransform(
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top_T.copy(), moving_T
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))
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self.change_area_bounds(
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new_t_max = 3,
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rate_func = there_and_back,
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run_time = 2
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)
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self.dither()
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####
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def add_T_label(self, x_val):
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triangle = RegularPolygon(n=3, start_angle = np.pi/2)
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triangle.scale_to_fit_height(MED_SMALL_BUFF)
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triangle.move_to(self.coords_to_point(x_val, 0), UP)
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triangle.set_fill(WHITE, 1)
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triangle.set_stroke(width = 0)
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T_label = TexMobject("T")
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T_label.next_to(triangle, DOWN)
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v_line = self.get_vertical_line_to_graph(
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x_val, self.v_graph,
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color = YELLOW
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)
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self.play(
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DrawBorderThenFill(triangle),
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ShowCreation(v_line),
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Write(T_label, run_time = 1)
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)
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self.T_label_group = VGroup(T_label, triangle)
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self.right_v_line = v_line
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def get_area(self, t_min, t_max):
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dx = float(t_max - t_min) / self.num_rects
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return self.get_riemann_rectangles(
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self.v_graph,
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x_min = t_min,
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x_max = t_max,
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dx = dx,
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stroke_width = 0,
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).set_fill(opacity = 0.8)
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def change_area_bounds(self, new_t_min = None, new_t_max = None, **kwargs):
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curr_t_min = self.x_axis.point_to_number(self.area.get_left())
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curr_t_max = self.x_axis.point_to_number(self.area.get_right())
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new_t_min = new_t_min or curr_t_min
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new_t_max = new_t_max or curr_t_max
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group = VGroup(self.area, self.right_v_line, self.T_label_group)
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def update_group(group, alpha):
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area, v_line, T_label = group
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t_min = interpolate(curr_t_min, new_t_min, alpha)
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t_max = interpolate(curr_t_max, new_t_max, alpha)
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new_area = self.get_area(t_min, t_max)
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new_v_line = self.get_vertical_line_to_graph(
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t_max, self.v_graph
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)
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new_v_line.highlight(v_line.get_color())
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T_label.move_to(new_v_line.get_bottom(), UP)
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Transform(area, new_area).update(1)
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Transform(v_line, new_v_line).update(1)
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return group
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self.play(
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UpdateFromAlphaFunc(group, update_group),
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*map(Animation, self.foreground_mobjects),
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**kwargs
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)
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