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Finished RelatedRatesExample of eoc5
This commit is contained in:
Grant Sanderson
parent
6dab7c958f
commit
c239f1af98
2 changed files with 503 additions and 28 deletions
513
eoc/chapter5.py
513
eoc/chapter5.py
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@ -573,7 +573,7 @@ class Ladder(VMobject):
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CONFIG = {
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"height" : 4,
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"width" : 1,
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"n_rungs" : 6,
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"n_rungs" : 7,
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}
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def generate_points(self):
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left_line, right_line = [
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@ -585,34 +585,39 @@ class Ladder(VMobject):
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left_line.point_from_proportion(a),
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right_line.point_from_proportion(a),
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)
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for a in np.linspace(0, 1, self.n_rungs+2)[1:-1]
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for a in np.linspace(0, 1, 2*self.n_rungs+1)[1:-1:2]
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]
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self.add(left_line, right_line, *rungs)
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self.center()
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class RelatedRatesExample(ThreeDScene):
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CONFIG = {
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"start_x" : 3.0,
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"start_y" : 4.0,
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"wall_dimensions" : [0.3, 5, 5],
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"wall_color" : color_gradient([GREY_BROWN, BLACK], 4)[1],
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"wall_center" : LEFT,
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"wall_center" : 1.5*LEFT+0.5*UP,
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}
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def construct(self):
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should_skip_animations = self.skip_animations
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self.skip_animations = True
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self.introduce_ladder()
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self.write_related_rates()
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self.measure_ladder()
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self.skip_animations = should_skip_animations
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self.slide_ladder()
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self.write_equation()
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self.isolate_x_of_t()
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self.discuss_lhs_as_function()
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self.let_dt_pass()
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self.take_derivative_of_rhs()
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self.take_derivative_of_lhs()
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self.bring_back_velocity_arrows()
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self.replace_terms_in_final_form()
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self.write_final_solution()
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def introduce_ladder(self):
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ladder = Ladder(height = self.get_ladder_length())
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wall = Prism(
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dimensions = [0.5, 6, 5],
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dimensions = self.wall_dimensions,
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fill_color = self.wall_color,
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fill_opacity = 1,
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)
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@ -626,7 +631,7 @@ class RelatedRatesExample(ThreeDScene):
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for ladder_copy in ladder.target, ladder.fallen:
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ladder_copy.rotate(-5*np.pi/12, UP)
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ladder_copy.next_to(wall, LEFT, 0, DOWN)
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ladder_copy.shift(LARGE_BUFF*RIGHT)
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ladder_copy.shift(0.8*RIGHT) ##BAD!
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self.play(
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@ -649,6 +654,8 @@ class RelatedRatesExample(ThreeDScene):
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self.play(Write(words))
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self.dither()
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self.related_rates_words = words
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def measure_ladder(self):
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ladder = self.ladder
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ladder_brace = self.get_ladder_brace(ladder)
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@ -664,10 +671,7 @@ class RelatedRatesExample(ThreeDScene):
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x_label.next_to(x_line, UP)
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x_label.highlight(x_line.get_color())
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self.play(
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GrowFromCenter(ladder_brace),
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Write(ladder_brace.length_label),
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)
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self.play(Write(ladder_brace))
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self.dither()
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self.play(ShowCreation(y_line), Write(y_label))
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self.dither()
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@ -677,36 +681,501 @@ class RelatedRatesExample(ThreeDScene):
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self.ladder_brace = ladder_brace
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self.x_and_y_lines = x_and_y_lines
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self.numerical_x_and_y_labels = VGroup(x_label, y_label)
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def slide_ladder(self):
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ladder = self.ladder
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brace = self.ladder_brace
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x_and_y_lines = self.x_and_y_lines
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x_line, y_line = x_and_y_lines
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down_arrow, left_arrow = [
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Arrow(ORIGIN, vect, color = YELLOW, buff = 0)
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for vect in DOWN, LEFT
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]
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down_arrow.shift(y_line.get_start()+MED_SMALL_BUFF*RIGHT)
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left_arrow.shift(x_line.get_start()+SMALL_BUFF*DOWN)
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# speed_label = TexMobject("1 \\text{m}/\\text{s}")
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speed_label = TexMobject("1 \\frac{\\text{m}}{\\text{s}}")
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speed_label.next_to(down_arrow, RIGHT, buff = SMALL_BUFF)
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q_marks = TexMobject("???")
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q_marks.next_to(left_arrow, DOWN, buff = SMALL_BUFF)
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added_anims = [
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UpdateFromFunc(brace, self.update_brace),
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UpdateFromFunc(x_and_y_lines, self.update_x_and_y_lines),
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Animation(down_arrow),
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]
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self.play(ShowCreation(down_arrow))
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self.play(Write(speed_label))
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self.let_ladder_fall(ladder, *added_anims)
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self.dither()
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self.reset_ladder(ladder, *added_anims)
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self.play(ShowCreation(left_arrow))
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self.play(Write(q_marks))
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self.dither()
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self.let_ladder_fall(ladder, *added_anims)
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self.dither()
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self.reset_ladder(ladder, *added_anims)
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self.dither()
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self.dy_arrow = down_arrow
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self.dy_label = speed_label
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self.dx_arrow = left_arrow
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self.dx_label = q_marks
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def write_equation(self):
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self.x_and_y_labels = self.get_x_and_y_labels()
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x_label, y_label = self.x_and_y_labels
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equation = TexMobject(
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"x(t)", "^2", "+", "y(t)", "^2", "=", "5^2"
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)
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equation[0].highlight(GREEN)
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equation[3].highlight(RED)
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equation.next_to(self.related_rates_words, DOWN, buff = MED_LARGE_BUFF)
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equation.to_edge(RIGHT, buff = LARGE_BUFF)
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self.play(Write(y_label))
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self.dither()
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self.let_ladder_fall(
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self.ladder,
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y_label.shift, self.start_y*DOWN/2,
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*self.get_added_anims_for_ladder_fall()[:-1],
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rate_func = lambda t : 0.2*there_and_back(t),
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run_time = 3
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)
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self.play(FocusOn(x_label))
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self.play(Write(x_label))
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self.dither(2)
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self.play(
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Transform(
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ladder, ladder.fallen,
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rate_func = None,
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run_time = self.start_y,
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),
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ReplacementTransform(x_label.copy(), equation[0]),
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ReplacementTransform(y_label.copy(), equation[3]),
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Write(VGroup(*np.array(equation)[[1, 2, 4, 5, 6]]))
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)
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self.dither(2)
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self.let_ladder_fall(
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self.ladder,
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*self.get_added_anims_for_ladder_fall(),
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rate_func = there_and_back,
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run_time = 6
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)
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self.dither()
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self.equation = equation
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def isolate_x_of_t(self):
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alt_equation = TexMobject(
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"x(t)", "=", "\\big(5^2", "-", "y(t)", "^2 \\big)", "^{1/2}",
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)
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alt_equation[0].highlight(GREEN)
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alt_equation[4].highlight(RED)
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alt_equation.next_to(self.equation, DOWN, buff = MED_LARGE_BUFF)
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alt_equation.to_edge(RIGHT)
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randy = Randolph()
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randy.next_to(
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alt_equation, DOWN,
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buff = MED_LARGE_BUFF,
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aligned_edge = LEFT,
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)
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randy.look_at(alt_equation)
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find_dx_dt = TexMobject("\\text{Find } \\,", "\\frac{dx}{dt}")
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find_dx_dt.next_to(randy, RIGHT, aligned_edge = UP)
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find_dx_dt[1].highlight(GREEN)
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self.play(FadeIn(randy))
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self.play(
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randy.change_mode, "raise_right_hand",
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randy.look_at, alt_equation,
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*[
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ReplacementTransform(
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self.equation[i].copy(),
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alt_equation[j],
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path_arc = np.pi/2,
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run_time = 3,
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rate_func = squish_rate_func(
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smooth, j/12.0, (j+6)/12.0
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)
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)
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for i, j in enumerate([0, 6, 3, 4, 5, 1, 2])
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]
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)
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self.play(Blink(randy))
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self.dither()
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self.play(
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Write(find_dx_dt),
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randy.change_mode, "pondering",
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randy.look_at, find_dx_dt,
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)
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self.let_ladder_fall(
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self.ladder, *self.get_added_anims_for_ladder_fall(),
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run_time = 8,
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rate_func = there_and_back
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)
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self.play(*map(FadeOut, [
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randy, find_dx_dt, alt_equation
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]))
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self.dither()
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def discuss_lhs_as_function(self):
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equation = self.equation
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lhs = VGroup(*equation[:5])
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brace = Brace(lhs, DOWN)
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function_of_time = brace.get_text(
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"Function of time"
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)
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constant_words = TextMobject(
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"""that happens to
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be constant"""
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)
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constant_words.highlight(YELLOW)
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constant_words.next_to(function_of_time, DOWN)
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derivative = TexMobject(
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"\\frac{d\\left(x(t)^2 + y(t)^2 \\right)}{dt}"
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)
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derivative.next_to(equation, DOWN, buff = MED_LARGE_BUFF)
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derivative.shift( ##Align x terms
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equation[0][0].get_center()[0]*RIGHT-\
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derivative[2].get_center()[0]*RIGHT
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)
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derivative_interior = lhs.copy()
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derivative_interior.move_to(VGroup(*derivative[2:13]))
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derivative_scaffold = VGroup(
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*list(derivative[:2])+list(derivative[13:])
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)
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self.play(
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GrowFromCenter(brace),
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Write(function_of_time)
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)
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self.dither()
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self.play(Write(constant_words))
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self.let_ladder_fall(
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self.ladder, *self.get_added_anims_for_ladder_fall(),
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run_time = 6,
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rate_func = lambda t : 0.5*there_and_back(t)
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)
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self.dither()
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self.play(*map(FadeOut, [
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brace, constant_words, function_of_time
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]))
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self.play(
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ReplacementTransform(lhs.copy(), derivative_interior),
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Write(derivative_scaffold),
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)
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self.dither()
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self.derivative = VGroup(
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derivative_scaffold, derivative_interior
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)
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def let_dt_pass(self):
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dt_words = TextMobject("After", "$dt$", "seconds...")
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dt_words.to_corner(UP+LEFT)
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dt = dt_words[1]
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dt.highlight(YELLOW)
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dt_brace = Brace(dt, buff = SMALL_BUFF)
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dt_brace_text = dt_brace.get_text("Think 0.01", buff = SMALL_BUFF)
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dt_brace_text.highlight(dt.get_color())
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shadow_ladder = self.ladder.copy()
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shadow_ladder.fade(0.5)
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x_line, y_line = self.x_and_y_lines
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y_top = y_line.get_start()
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x_left = x_line.get_start()
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self.play(Write(dt_words, run_time = 2))
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self.play(
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GrowFromCenter(dt_brace),
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Write(dt_brace_text, run_time = 2)
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)
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self.play(*map(FadeOut, [
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self.dy_arrow, self.dy_label,
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self.dx_arrow, self.dx_label,
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]))
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self.add(shadow_ladder)
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self.let_ladder_fall(
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self.ladder, *self.get_added_anims_for_ladder_fall(),
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rate_func = lambda t : 0.1*smooth(t),
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run_time = 1
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)
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new_y_top = y_line.get_start()
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new_x_left = x_line.get_start()
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dy_line = Line(y_top, new_y_top)
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dy_brace = Brace(dy_line, RIGHT, buff = SMALL_BUFF)
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dy_label = dy_brace.get_text("$dy$", buff = SMALL_BUFF)
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dy_label.highlight(RED)
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dx_line = Line(x_left, new_x_left)
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dx_brace = Brace(dx_line, DOWN, buff = SMALL_BUFF)
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dx_label = dx_brace.get_text("$dx$")
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dx_label.highlight(GREEN)
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VGroup(dy_line, dx_line).highlight(YELLOW)
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for line, brace, label in (dy_line, dy_brace, dy_label), (dx_line, dx_brace, dx_label):
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self.play(
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ShowCreation(line),
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GrowFromCenter(brace),
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Write(label),
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run_time = 1
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)
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self.dither()
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self.play(Indicate(self.derivative[1]))
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self.dither()
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self.dy_group = VGroup(dy_line, dy_brace, dy_label)
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self.dx_group = VGroup(dx_line, dx_brace, dx_label)
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self.shadow_ladder = shadow_ladder
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def take_derivative_of_rhs(self):
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derivative = self.derivative
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equals_zero = TexMobject("= 0")
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equals_zero.next_to(derivative)
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rhs = self.equation[-1]
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self.play(Write(equals_zero))
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self.dither()
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self.play(FocusOn(rhs))
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self.play(Indicate(rhs))
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self.dither()
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self.reset_ladder(
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self.ladder,
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*self.get_added_anims_for_ladder_fall()+[
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Animation(self.dy_group),
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Animation(self.dx_group),
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],
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rate_func = there_and_back,
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run_time = 3
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)
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self.dither()
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self.equals_zero = equals_zero
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def take_derivative_of_lhs(self):
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derivative_scaffold, equation = self.derivative
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equals_zero_copy = self.equals_zero.copy()
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lhs_derivative = TexMobject(
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"2", "x(t)", "\\frac{dx}{dt}", "+",
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"2", "y(t)", "\\frac{dy}{dt}",
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)
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lhs_derivative[1].highlight(GREEN)
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VGroup(*lhs_derivative[2][:2]).highlight(GREEN)
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lhs_derivative[5].highlight(RED)
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VGroup(*lhs_derivative[6][:2]).highlight(RED)
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lhs_derivative.next_to(
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derivative_scaffold, DOWN,
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aligned_edge = RIGHT,
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buff = MED_LARGE_BUFF
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)
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equals_zero_copy.next_to(lhs_derivative, RIGHT)
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pairs = [
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(0, 1), (1, 0), #x^2 -> 2x
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(2, 3), (3, 5), (4, 4), #+y^2 -> +2y
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]
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def perform_replacement(index_pairs):
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self.play(*[
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ReplacementTransform(
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equation[i].copy(), lhs_derivative[j],
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path_arc = np.pi/2,
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run_time = 2
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)
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for i, j in index_pairs
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])
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perform_replacement(pairs[:2])
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self.play(Write(lhs_derivative[2]))
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self.dither()
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self.play(Indicate(
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VGroup(
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*list(lhs_derivative[:2])+\
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list(lhs_derivative[2][:2])
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),
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run_time = 2
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))
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self.play(Indicate(VGroup(*lhs_derivative[2][3:])))
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self.dither(2)
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perform_replacement(pairs[2:])
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self.play(Write(lhs_derivative[6]))
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self.dither()
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self.play(FocusOn(self.equals_zero))
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self.play(ReplacementTransform(
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self.equals_zero.copy(),
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equals_zero_copy
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))
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self.dither(2)
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lhs_derivative.add(equals_zero_copy)
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self.lhs_derivative = lhs_derivative
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def bring_back_velocity_arrows(self):
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dx_dy_group = VGroup(self.dx_group, self.dy_group)
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arrow_group = VGroup(
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self.dy_arrow, self.dy_label,
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self.dx_arrow, self.dx_label,
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)
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ladder_fall_args = [self.ladder] + self.get_added_anims_for_ladder_fall()
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self.reset_ladder(*ladder_fall_args + [
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FadeOut(dx_dy_group),
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FadeOut(self.derivative),
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FadeOut(self.equals_zero),
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self.lhs_derivative.shift, 2*UP,
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])
|
||||
self.remove(self.shadow_ladder)
|
||||
self.play(FadeIn(arrow_group))
|
||||
self.let_ladder_fall(*ladder_fall_args)
|
||||
self.dither()
|
||||
self.reset_ladder(*ladder_fall_args)
|
||||
self.dither()
|
||||
|
||||
def replace_terms_in_final_form(self):
|
||||
x_label, y_label = self.x_and_y_labels
|
||||
num_x_label, num_y_label = self.numerical_x_and_y_labels
|
||||
|
||||
new_lhs_derivative = TexMobject(
|
||||
"2", "(%d)"%int(self.start_x), "\\frac{dx}{dt}", "+",
|
||||
"2", "(%d)"%int(self.start_y), "(1)",
|
||||
"= 0"
|
||||
)
|
||||
new_lhs_derivative[1].highlight(GREEN)
|
||||
VGroup(*new_lhs_derivative[2][:2]).highlight(GREEN)
|
||||
new_lhs_derivative[5].highlight(RED)
|
||||
new_lhs_derivative.next_to(
|
||||
self.lhs_derivative, DOWN,
|
||||
buff = MED_LARGE_BUFF,
|
||||
aligned_edge = RIGHT
|
||||
)
|
||||
def fill_in_equation_part(*indices):
|
||||
self.play(*[
|
||||
ReplacementTransform(
|
||||
self.lhs_derivative[i].copy(),
|
||||
new_lhs_derivative[i],
|
||||
run_time = 2
|
||||
)
|
||||
for i in indices
|
||||
])
|
||||
|
||||
self.play(FadeOut(y_label), FadeIn(num_y_label))
|
||||
fill_in_equation_part(3, 4, 5)
|
||||
self.play(FadeOut(x_label), FadeIn(num_x_label))
|
||||
for indices in [(0, 1), (6,), (2, 7)]:
|
||||
fill_in_equation_part(*indices)
|
||||
self.dither()
|
||||
self.dither()
|
||||
|
||||
self.new_lhs_derivative = new_lhs_derivative
|
||||
|
||||
def write_final_solution(self):
|
||||
solution = TexMobject(
|
||||
"\\frac{dx}{dt} = \\frac{-4}{3}"
|
||||
)
|
||||
for i in 0, 1, -1:
|
||||
solution[i].highlight(GREEN)
|
||||
solution[-3].highlight(RED)
|
||||
solution.next_to(
|
||||
self.new_lhs_derivative, DOWN,
|
||||
buff = MED_LARGE_BUFF,
|
||||
aligned_edge = RIGHT
|
||||
)
|
||||
|
||||
box = Rectangle(color = YELLOW)
|
||||
box.replace(solution)
|
||||
box.scale_in_place(1.3)
|
||||
|
||||
self.play(Write(solution))
|
||||
self.dither()
|
||||
self.play(ShowCreation(box))
|
||||
self.dither()
|
||||
|
||||
#########
|
||||
|
||||
def get_added_anims_for_ladder_fall(self):
|
||||
return [
|
||||
UpdateFromFunc(self.ladder_brace, self.update_brace),
|
||||
UpdateFromFunc(self.x_and_y_lines, self.update_x_and_y_lines),
|
||||
UpdateFromFunc(self.x_and_y_labels, self.update_x_and_y_labels),
|
||||
]
|
||||
|
||||
def let_ladder_fall(self, ladder, *added_anims, **kwargs):
|
||||
kwargs["run_time"] = kwargs.get("run_time", self.start_y)
|
||||
kwargs["rate_func"] = kwargs.get("rate_func", None)
|
||||
self.play(
|
||||
Transform(ladder, ladder.fallen),
|
||||
*added_anims,
|
||||
**kwargs
|
||||
)
|
||||
|
||||
def reset_ladder(self, ladder, *added_anims, **kwargs):
|
||||
kwargs["run_time"] = kwargs.get("run_time", 2)
|
||||
self.play(
|
||||
Transform(ladder, ladder.target),
|
||||
*added_anims,
|
||||
**kwargs
|
||||
)
|
||||
|
||||
def update_brace(self, brace):
|
||||
Transform(
|
||||
brace, self.get_ladder_brace(self.ladder)
|
||||
).update(1)
|
||||
return brace
|
||||
|
||||
def update_x_and_y_lines(self, x_and_y_lines):
|
||||
Transform(
|
||||
x_and_y_lines,
|
||||
self.get_x_and_y_lines(self.ladder)
|
||||
).update(1)
|
||||
return x_and_y_lines
|
||||
|
||||
def update_x_and_y_labels(self, x_and_y_labels):
|
||||
Transform(
|
||||
x_and_y_labels,
|
||||
self.get_x_and_y_labels()
|
||||
).update(1)
|
||||
return x_and_y_labels
|
||||
|
||||
def get_ladder_brace(self, ladder):
|
||||
vect = rotate_vector(LEFT, -self.get_ladder_angle())
|
||||
brace = Brace(ladder, vect)
|
||||
length_string = "%dm"%int(self.get_ladder_length())
|
||||
length_label = brace.get_text(length_string)
|
||||
length_label = brace.get_text(
|
||||
length_string, use_next_to = False
|
||||
)
|
||||
brace.add(length_label)
|
||||
brace.length_label = length_label
|
||||
return brace
|
||||
|
||||
def get_x_and_y_labels(self):
|
||||
x_line, y_line = self.x_and_y_lines
|
||||
|
||||
x_label = TexMobject("x(t)")
|
||||
x_label.highlight(x_line.get_color())
|
||||
x_label.next_to(x_line, DOWN, buff = SMALL_BUFF)
|
||||
|
||||
y_label = TexMobject("y(t)")
|
||||
y_label.highlight(y_line.get_color())
|
||||
y_label.next_to(y_line, LEFT, buff = SMALL_BUFF)
|
||||
|
||||
return VGroup(x_label, y_label)
|
||||
|
||||
def get_x_and_y_lines(self, ladder):
|
||||
top_point = ladder.get_corner(UP+RIGHT)
|
||||
bottom_point = ladder.get_corner(DOWN+LEFT)
|
||||
bottom_point, top_point = np.array(ladder[1].get_start_and_end())
|
||||
interim_point = top_point[0]*RIGHT + bottom_point[1]*UP
|
||||
interim_point += SMALL_BUFF*DOWN
|
||||
y_line = Line(top_point, interim_point)
|
||||
y_line.highlight(RED)
|
||||
x_line = Line(bottom_point, interim_point)
|
||||
|
|
|
|||
|
|
@ -171,12 +171,18 @@ class Brace(TexMobject):
|
|||
for mob in mobject, self:
|
||||
mob.rotate(angle)
|
||||
|
||||
def put_at_tip(self, mob, **kwargs):
|
||||
mob.next_to(
|
||||
self.get_tip(),
|
||||
np.round(self.get_direction()),
|
||||
**kwargs
|
||||
)
|
||||
def put_at_tip(self, mob, use_next_to = True, **kwargs):
|
||||
if use_next_to:
|
||||
mob.next_to(
|
||||
self.get_tip(),
|
||||
np.round(self.get_direction()),
|
||||
**kwargs
|
||||
)
|
||||
else:
|
||||
mob.move_to(self.get_tip())
|
||||
buff = kwargs.get("buff", DEFAULT_MOBJECT_TO_MOBJECT_BUFFER)
|
||||
shift_distance = mob.get_width()/2.0+buff
|
||||
mob.shift(self.get_direction()*shift_distance)
|
||||
return self
|
||||
|
||||
def get_text(self, *text, **kwargs):
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue