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TransitionFromPathsToBoundaries
This commit is contained in:
Grant Sanderson
parent
1972874115
commit
ad330cc95b
1 changed files with 394 additions and 133 deletions
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@ -2127,6 +2127,188 @@ class PathContainingZero(InputOutputScene, PiCreatureScene):
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self.play(morty.change, "hooray")
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self.wait(3)
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class TransitionFromPathsToBoundaries(ColorMappedObjectsScene):
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CONFIG = {
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"func" : plane_func_by_wind_spec(
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(-2, 0, 2), (2, 0, 1)
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)
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}
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def construct(self):
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ColorMappedObjectsScene.construct(self)
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#Setup paths
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squares, joint_rect = self.get_squares_and_joint_rect()
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left_square, right_square = squares
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path1, path2 = paths = VGroup(*[
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Line(square.get_corner(UP+LEFT), square.get_corner(UP+RIGHT))
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for square in squares
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])
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joint_path = Line(path1.get_start(), path2.get_end())
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for mob in it.chain(paths, [joint_path]):
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mob.set_stroke(WHITE, 4)
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mob.color_using_background_image(self.background_image_file)
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dot = self.get_dot_and_add_continual_animations()
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#Setup path braces
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for mob, tex in (path1, "x"), (path2, "y"), (joint_path, "x+y"):
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mob.brace = Brace(mob, DOWN)
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label = TextMobject("Winding =", "$%s$"%tex)
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label.next_to(mob.brace, DOWN)
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mob.brace.add(label)
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#Setup region labels
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for square, tex in (left_square, "x"), (right_square, "y"), (joint_rect, "x+y \\, ?"):
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square.label = TextMobject("Winding = ", "$%s$"%tex)
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square.label.move_to(square)
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#Add paths
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self.position_dot(path1.get_start())
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for path in path1, path2:
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self.position_dot(path.get_start())
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self.play(
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MoveAlongPath(dot, path.copy()),
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ShowCreation(path),
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run_time = 2
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)
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self.play(GrowFromCenter(path.brace))
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self.wait()
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self.position_dot(joint_path.get_start())
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self.play(
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MoveAlongPath(dot, joint_path, run_time = 3),
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FadeOut(VGroup(path1.brace, path2.brace)),
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FadeIn(joint_path.brace),
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)
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self.wait()
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#Add regions
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self.play(
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FadeOut(paths),
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FadeOut(joint_path.brace),
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dot.move_to, path1.get_start()
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)
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for square in squares:
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self.position_dot(square.points[0])
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kwargs = {
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"run_time" : 4,
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"rate_func" : bezier([0, 0, 1, 1]),
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}
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self.play(
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MoveAlongPath(dot, square.copy(), **kwargs),
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ShowCreation(square, **kwargs),
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Write(square.label, run_time = 2),
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)
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self.wait()
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self.play(
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dot.move_to, joint_rect.points[0],
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FadeOut(squares),
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FadeIn(joint_rect),
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)
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self.position_dot(joint_rect.points[0])
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self.play(
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Transform(left_square.label[0], joint_rect.label[0]),
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Transform(
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left_square.label[1], joint_rect.label[1][0],
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path_arc = TAU/6
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),
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FadeIn(joint_rect.label[1][1]),
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FadeIn(joint_rect.label[1][3]),
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FadeOut(right_square.label[0]),
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Transform(
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right_square.label[1], joint_rect.label[1][2],
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path_arc = TAU/6
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),
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MoveAlongPath(
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dot, joint_rect,
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run_time = 6,
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rate_func = bezier([0, 0, 1, 1])
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)
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)
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self.wait()
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###
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def get_squares_and_joint_rect(self):
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squares = VGroup(*[
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Square(side_length = 4).next_to(ORIGIN, vect, buff = 0)
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for vect in LEFT, RIGHT
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])
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joint_rect = SurroundingRectangle(squares, buff = 0)
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for mob in it.chain(squares, [joint_rect]):
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mob.set_stroke(WHITE, 4)
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mob.color_using_background_image(self.background_image_file)
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return squares, joint_rect
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def get_dot_and_add_continual_animations(self):
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#Define important functions for updates
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get_output = lambda : self.func(tuple(dot.get_center()[:2]))
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get_output_color = lambda : rgba_to_color(point_to_rgba(get_output()))
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get_output_rev = lambda : -point_to_rev(get_output())
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self.get_output_rev = get_output_rev
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self.start_rev = 0
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self.curr_winding = 0
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def get_total_winding(dt = 0):
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rev = (get_output_rev() - self.start_rev)%1
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possible_windings = [
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np.floor(self.curr_winding)+k+rev
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for k in -1, 0, 1
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]
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i = np.argmin([abs(pw - self.curr_winding) for pw in possible_windings])
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self.curr_winding = possible_windings[i]
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return self.curr_winding
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#Setup dot, arrow and label
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dot = self.dot = Dot(radius = 0.1)
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dot.set_stroke(WHITE, 1)
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update_dot_color = ContinualUpdateFromFunc(
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dot, lambda d : d.set_fill(get_output_color())
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)
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label = DecimalNumber(0, num_decimal_points = 1)
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label_upadte = ContinualChangingDecimal(
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label, get_total_winding,
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position_update_func = lambda l : l.next_to(dot, UP+LEFT, SMALL_BUFF)
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)
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arrow_length = 0.75
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arrow = Vector(arrow_length*RIGHT)
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arrow.set_stroke(WHITE, 1)
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def arrow_update_func(arrow):
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arrow.set_fill(get_output_color(), 1)
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arrow.rotate(-TAU*get_output_rev() - arrow.get_angle())
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arrow.scale(arrow_length/arrow.get_length())
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arrow.shift(dot.get_center() - arrow.get_start())
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return arrow
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update_arrow = ContinualUpdateFromFunc(arrow, arrow_update_func)
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self.add(update_arrow, update_dot_color, label_upadte)
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return dot
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def position_dot(self, point):
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self.dot.move_to(point)
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self.start_rev = self.get_output_rev()
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self.curr_winding = 0
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class BreakDownLoopWithNonzeroWinding(TransitionFromPathsToBoundaries):
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def construct(self):
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zero_point = 2*LEFT
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squares, joint_rect = self.get_squares_and_joint_rect()
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left_square, right_square = squares
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VGroup(squares, joint_rect).shift(SMALL_BUFF*UP)
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dot = self.get_dot_and_add_continual_animations()
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for rect in (left_square, "x"), (right_square, "y"), (joint_rect, "3"):
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pass
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class BackToEquationSolving(AltTeacherStudentsScene):
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def construct(self):
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self.teacher_says(
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@ -2139,6 +2321,218 @@ class BackToEquationSolving(AltTeacherStudentsScene):
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])
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self.wait(3)
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class MonomialTerm(PathContainingZero):
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CONFIG = {
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"non_renormalized_func" : plane_func_from_complex_func(lambda z : z**5),
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"full_func_label" : "f(x) = x^5",
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"func_label" : "x^5",
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"loop_radius" : 1.1,
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"label_buff" : 0.3,
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"label_move_to_corner" : ORIGIN,
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"should_end_with_rescaling" : True,
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}
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def construct(self):
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self.setup_planes()
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self.relabel_planes()
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self.add_function_label()
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self.show_winding()
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if self.should_end_with_rescaling:
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self.rescale_output_plane()
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def relabel_planes(self):
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for plane in self.input_plane, self.output_plane:
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for mob in plane:
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if isinstance(mob, TexMobject):
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plane.remove(mob)
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if hasattr(plane, "numbers_to_show"):
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_range = plane.numbers_to_show
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else:
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_range = range(-2, 3)
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for x in _range:
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if x == 0:
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continue
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label = TexMobject(str(x))
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label.scale(0.5)
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point = plane.coords_to_point(x, 0)
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label.next_to(point, DOWN, MED_SMALL_BUFF)
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plane.add(label)
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self.add_foreground_mobject(label)
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tick = Line(SMALL_BUFF*DOWN, SMALL_BUFF*UP)
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tick.move_to(point)
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plane.add(tick)
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for y in _range:
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if y == 0:
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continue
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label = TexMobject("%di"%y)
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label.scale(0.5)
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point = plane.coords_to_point(0, y)
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label.next_to(point, LEFT, MED_SMALL_BUFF)
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plane.add(label)
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self.add_foreground_mobject(label)
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tick = Line(SMALL_BUFF*LEFT, SMALL_BUFF*RIGHT)
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tick.move_to(point)
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plane.add(tick)
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self.add(self.input_plane, self.output_plane)
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def add_function_label(self):
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label = TexMobject(self.full_func_label)
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label.add_background_rectangle(opacity = 1, buff = SMALL_BUFF)
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arrow = Arrow(
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2*LEFT, 2*RIGHT, path_arc = -TAU/3,
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use_rectangular_stem = False
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)
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arrow.pointwise_become_partial(arrow, 0, 0.95)
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label.next_to(arrow, UP)
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VGroup(arrow, label).to_edge(UP)
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self.add(label, arrow)
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def show_winding(self):
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loop = Arc(color = WHITE, angle = 1.02*TAU, num_anchors = 42)
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loop.scale(self.loop_radius)
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loop.match_background_image_file(self.input_coloring)
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loop.move_to(self.input_plane.coords_to_point(0, 0))
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out_loop = loop.copy()
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out_loop.apply_function(self.point_function)
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out_loop.match_background_image_file(self.output_coloring)
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get_in_point = lambda : loop.points[-1]
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get_out_point = lambda : out_loop.points[-1]
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in_origin = self.input_plane.coords_to_point(0, 0)
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out_origin = self.output_plane.coords_to_point(0, 0)
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dot = Dot()
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update_dot = UpdateFromFunc(dot, lambda d : d.move_to(get_in_point()))
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out_dot = Dot()
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update_out_dot = UpdateFromFunc(out_dot, lambda d : d.move_to(get_out_point()))
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buff = self.label_buff
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def generate_label_update(label, point_func, origin):
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return UpdateFromFunc(
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label, lambda m : m.move_to(
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(1+buff)*point_func() - buff*origin,
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self.label_move_to_corner
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)
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)
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x = TexMobject("x")
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fx = TexMobject(self.func_label)
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update_x = generate_label_update(x, get_in_point, in_origin)
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update_fx = generate_label_update(fx, get_out_point, out_origin)
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morty = self.pi_creature
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kwargs = {
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"run_time" : 15,
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"rate_func" : None,
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}
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self.play(
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ShowCreation(loop, **kwargs),
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ShowCreation(out_loop, **kwargs),
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update_dot,
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update_out_dot,
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update_x,
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update_fx,
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ApplyMethod(morty.change, "pondering", out_dot),
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)
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self.play(
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FadeOut(VGroup(dot, out_dot, x, fx))
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)
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self.loop = loop
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self.out_loop = out_loop
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def rescale_output_plane(self):
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output_stuff = VGroup(self.output_plane, self.output_coloring)
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self.play(*map(FadeOut, [self.loop, self.out_loop]))
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self.play(
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output_stuff.scale, 3.0/50, run_time = 2
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)
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self.wait()
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###
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def func(self, coords):
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return self.non_renormalized_func(coords)
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class PolynomialTerms(MonomialTerm):
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CONFIG = {
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"non_renormalized_func" : plane_func_from_complex_func(lambda z : z**5 - z - 1),
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"full_func_label" : "f(x) = x^5 - x - 1",
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"func_label" : "x^5 + \\cdots",
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"loop_radius" : 2.0,
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"label_buff" : 0.15,
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"label_move_to_corner" : DOWN+LEFT,
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"should_end_with_rescaling" : False,
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}
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def construct(self):
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self.pi_creature.change("pondering", VectorizedPoint(ORIGIN))
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MonomialTerm.construct(self)
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self.cinch_loop()
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# self.sweep_through_loop_interior()
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def relabel_planes(self):
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self.output_plane.x_radius = 50
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self.output_plane.y_radius = 50
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self.output_plane.numbers_to_show = range(-45, 50, 15)
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MonomialTerm.relabel_planes(self)
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def sweep_through_loop_interior(self):
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loop = self.loop
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morty = self.pi_creature
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line, line_target = [
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Line(
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loop.get_left(), loop.get_right(),
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path_arc = u*TAU/2,
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n_arc_anchors = 40,
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background_image_file = self.input_coloring.background_image_file ,
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stroke_width = 4,
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)
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for u in -1, 1
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]
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out_line = line.copy()
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update_out_line = UpdateFromFunc(
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out_line,
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lambda m : m.set_points(line.points).apply_function(self.point_function),
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)
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self.play(
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Transform(
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line, line_target,
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run_time = 10,
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rate_func = there_and_back
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),
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update_out_line,
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morty.change, "hooray"
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)
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self.wait()
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def cinch_loop(self):
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loop = self.loop
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out_loop = self.out_loop
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morty = self.pi_creature
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update_out_loop = UpdateFromFunc(
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out_loop,
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lambda m : m.set_points(loop.points).apply_function(self.point_function)
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)
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self.add(
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loop.copy().set_stroke(width = 1),
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out_loop.copy().set_stroke(width = 1),
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)
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self.play(
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ApplyMethod(
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loop.scale, 0, {"about_point" : self.input_plane.coords_to_point(0.2, 1)},
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run_time = 12,
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rate_func = bezier([0, 0, 1, 1])
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),
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update_out_loop,
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morty.change, "hooray"
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)
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self.wait()
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class SearchSpacePerimeterVsArea(EquationSolver2d):
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CONFIG = {
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"func" : plane_func_by_wind_spec(
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@ -2206,139 +2600,6 @@ class SearchSpacePerimeterVsArea(EquationSolver2d):
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self.play(FadeOut(full_rect))
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self.wait()
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class MonomialTerm(PathContainingZero):
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CONFIG = {
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"non_renormalized_func" : plane_func_from_complex_func(lambda z : z**5),
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"func_label" : "x^5",
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"loop_radius" : 1.1,
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"label_buff" : 0.3,
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}
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def construct(self):
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self.setup_planes()
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self.relabel_planes()
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self.add_function_label()
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self.show_winding()
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def relabel_planes(self):
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for plane in self.input_plane, self.output_plane:
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for mob in plane:
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if isinstance(mob, TexMobject):
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plane.remove(mob)
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if hasattr(plane, "numbers_to_show"):
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_range = plane.numbers_to_show
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else:
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_range = range(-2, 3)
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for x in _range:
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if x == 0:
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continue
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label = TexMobject(str(x))
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label.scale(0.5)
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point = plane.coords_to_point(x, 0)
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label.next_to(point, DOWN, MED_SMALL_BUFF)
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plane.add(label)
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tick = Line(SMALL_BUFF*DOWN, SMALL_BUFF*UP)
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tick.move_to(point)
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plane.add(tick)
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for y in _range:
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if y == 0:
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continue
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label = TexMobject("%di"%y)
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label.scale(0.5)
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point = plane.coords_to_point(0, y)
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label.next_to(point, LEFT, MED_SMALL_BUFF)
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plane.add(label)
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tick = Line(SMALL_BUFF*LEFT, SMALL_BUFF*RIGHT)
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tick.move_to(point)
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plane.add(tick)
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def add_function_label(self):
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label = TexMobject("f(x) = %s"%self.func_label)
|
||||
label.add_background_rectangle(opacity = 1)
|
||||
arrow = Arrow(
|
||||
2*LEFT, 2*RIGHT, path_arc = -TAU/3,
|
||||
use_rectangular_stem = False
|
||||
)
|
||||
arrow.pointwise_become_partial(arrow, 0, 0.95)
|
||||
label.next_to(arrow, UP)
|
||||
VGroup(arrow, label).to_edge(UP)
|
||||
self.add(label, arrow)
|
||||
|
||||
def show_winding(self):
|
||||
loop = Arc(color = WHITE, angle = 1.02*TAU, num_anchors = 42)
|
||||
loop.scale(self.loop_radius)
|
||||
loop.match_background_image_file(self.input_coloring)
|
||||
loop.move_to(self.input_plane.coords_to_point(0, 0))
|
||||
|
||||
out_loop = loop.copy()
|
||||
out_loop.apply_function(self.point_function)
|
||||
out_loop.match_background_image_file(self.output_coloring)
|
||||
|
||||
get_in_point = lambda : loop.points[-1]
|
||||
get_out_point = lambda : out_loop.points[-1]
|
||||
in_origin = self.input_plane.coords_to_point(0, 0)
|
||||
out_origin = self.output_plane.coords_to_point(0, 0)
|
||||
|
||||
dot = Dot()
|
||||
update_dot = UpdateFromFunc(dot, lambda d : d.move_to(get_in_point()))
|
||||
|
||||
out_dot = Dot()
|
||||
update_out_dot = UpdateFromFunc(out_dot, lambda d : d.move_to(get_out_point()))
|
||||
|
||||
buff = self.label_buff
|
||||
def generate_label_update(label, point_func, origin):
|
||||
return UpdateFromFunc(
|
||||
label, lambda m : m.move_to(
|
||||
(1+buff)*point_func() - buff*origin,
|
||||
)
|
||||
)
|
||||
x = TexMobject("x")
|
||||
fx = TexMobject(self.func_label)
|
||||
update_x = generate_label_update(x, get_in_point, in_origin)
|
||||
update_fx = generate_label_update(fx, get_out_point, out_origin)
|
||||
|
||||
morty = self.pi_creature
|
||||
|
||||
kwargs = {
|
||||
"run_time" : 15,
|
||||
"rate_func" : None,
|
||||
}
|
||||
self.play(
|
||||
ShowCreation(loop, **kwargs),
|
||||
ShowCreation(out_loop, **kwargs),
|
||||
update_dot,
|
||||
update_out_dot,
|
||||
update_x,
|
||||
update_fx,
|
||||
Succession(
|
||||
morty.change, "pondering",
|
||||
Blink(morty),
|
||||
Animation(morty, run_time = 4),
|
||||
Blink(morty),
|
||||
Animation(morty, run_time = 4),
|
||||
Blink(morty),
|
||||
)
|
||||
)
|
||||
|
||||
###
|
||||
|
||||
def func(self, coords):
|
||||
return self.non_renormalized_func(coords)
|
||||
|
||||
class PolynomialTerms(MonomialTerm):
|
||||
CONFIG = {
|
||||
"non_renormalized_func" : plane_func_from_complex_func(lambda z : z**5 - z - 1),
|
||||
"func_label" : "x^5 - x - 1",
|
||||
"loop_radius" : 2.0,
|
||||
"label_buff" : 0.3,
|
||||
}
|
||||
|
||||
def relabel_planes(self):
|
||||
self.output_plane.x_radius = 50
|
||||
self.output_plane.y_radius = 50
|
||||
self.output_plane.numbers_to_show = range(-45, 50, 10)
|
||||
MonomialTerm.relabel_planes(self)
|
||||
|
||||
class EndingCredits(Scene):
|
||||
def construct(self):
|
||||
text = TextMobject(
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue