def get_circles_and_points(self, min_input, max_input):
input_left, input_right = [
self.interval.number_to_point(num)
for num in min_input, max_input
]
input_circle = Circle(
radius = np.linalg.norm(input_left-input_right)/2,
color = WHITE
)
input_circle.shift((input_left+input_right)/2)
input_points = Line(
input_left, input_right,
color = self.input_color
)
output_points = Mobject(color = self.output_color)
n = self.output.get_num_points()
output_points.add_points(
self.output.points[int(min_input*n):int(max_input*n)]
)
output_center = output_points.points[int(0.5*output_points.get_num_points())]
max_distance = np.linalg.norm(output_center-output_points.points[-1])
output_circle = Circle(
radius = max_distance,
color = WHITE
)
output_circle.shift(output_center)
return (
input_circle,
input_points,
output_circle,
output_points
)
def get_circles_and_points(self, min_input, max_input):
input_left, input_right = [
self.interval.number_to_point(num)
for num in min_input, max_input
]
input_circle = Circle(
radius = np.linalg.norm(input_left-input_right)/2,
color = WHITE
)
input_circle.shift((input_left+input_right)/2)
input_points = Line(
input_left, input_right,
color = self.input_color
)
output_points = Mobject(color = self.output_color)
n = self.output.get_num_points()
output_points.add_points(
self.output.points[int(min_input*n):int(max_input*n)]
)
output_center = output_points.points[int(0.5*output_points.get_num_points())]
max_distance = np.linalg.norm(output_center-output_points.points[-1])
output_circle = Circle(
radius = max_distance,
color = WHITE
)
output_circle.shift(output_center)
return (
input_circle,
input_points,
output_circle,
output_points
)
class FormalDefinitionOfContinuity(Scene):
def construct(self):
self.setup()
self.label_spaces()
self.move_dot()
self.label_jump()
self.draw_circles()
self.vary_circle_sizes()
self.discontinuous_point()
def setup(self):
self.input_color = YELLOW_C
self.output_color = RED
def spiril(t):
theta = 2*np.pi*t
return t*np.cos(theta)*RIGHT+t*np.sin(theta)*UP
self.spiril1 = ParametricFunction(
lambda t : 1.5*RIGHT + DOWN + 2*spiril(t),
density = 5*DEFAULT_POINT_DENSITY_1D,
)
self.spiril2 = ParametricFunction(
lambda t : 5.5*RIGHT + UP - 2*spiril(1-t),
density = 5*DEFAULT_POINT_DENSITY_1D,
)
Mobject.align_data(self.spiril1, self.spiril2)
self.output = Mobject(self.spiril1, self.spiril2)
self.output.ingest_submobjects()
self.output.highlight(GREEN_A)
self.interval = UnitInterval()
self.interval.scale_to_fit_width(SPACE_WIDTH-1)
self.interval.to_edge(LEFT)
self.input_dot = Dot(color = self.input_color)
self.output_dot = self.input_dot.copy().highlight(self.output_color)
left, right = self.interval.get_left(), self.interval.get_right()
self.input_homotopy = lambda (x, y, z, t) : (x, y, t) + interpolate(left, right, t)
output_size = self.output.get_num_points()-1
output_points = self.output.points
self.output_homotopy = lambda (x, y, z, t) : (x, y, z) + output_points[int(t*output_size)]
def get_circles_and_points(self, min_input, max_input):
input_left, input_right = [
self.interval.number_to_point(num)
for num in min_input, max_input
]
input_circle = Circle(
radius = np.linalg.norm(input_left-input_right)/2,
color = WHITE
)
input_circle.shift((input_left+input_right)/2)
input_points = Line(
input_left, input_right,
color = self.input_color
)
output_points = Mobject(color = self.output_color)
n = self.output.get_num_points()
output_points.add_points(
self.output.points[int(min_input*n):int(max_input*n)]
)
output_center = output_points.points[int(0.5*output_points.get_num_points())]
max_distance = np.linalg.norm(output_center-output_points.points[-1])
output_circle = Circle(
radius = max_distance,
color = WHITE
)
output_circle.shift(output_center)
return (
input_circle,
input_points,
output_circle,
output_points
)
def label_spaces(self):
input_space = TextMobject("Input Space")
input_space.to_edge(UP)
input_space.shift(LEFT*SPACE_WIDTH/2)
output_space = TextMobject("Output Space")
output_space.to_edge(UP)
output_space.shift(RIGHT*SPACE_WIDTH/2)
line = Line(
UP*SPACE_HEIGHT, DOWN*SPACE_HEIGHT,
color = WHITE
)
self.play(
ShimmerIn(input_space),
ShimmerIn(output_space),
ShowCreation(line),
ShowCreation(self.interval),
)
self.wait()
def move_dot(self):
kwargs = {
"rate_func" : None,
"run_time" : 3
}
self.play(
Homotopy(self.input_homotopy, self.input_dot, **kwargs),
Homotopy(self.output_homotopy, self.output_dot, **kwargs),
ShowCreation(self.output, **kwargs)
)
self.wait()
def label_jump(self):
jump_points = Mobject(
Point(self.spiril1.points[-1]),
Point(self.spiril2.points[0])
)
self.brace = Brace(jump_points, RIGHT)
self.jump = TextMobject("Jump")
self.jump.next_to(self.brace, RIGHT)
self.play(
GrowFromCenter(self.brace),
ShimmerIn(self.jump)
)
self.wait()
self.remove(self.brace, self.jump)
def draw_circles(self):
input_value = 0.45
input_radius = 0.04
for dot in self.input_dot, self.output_dot:
dot.center()
kwargs = {
"rate_func" : lambda t : interpolate(1, input_value, smooth(t))
}
self.play(
Homotopy(self.input_homotopy, self.input_dot, **kwargs),
Homotopy(self.output_homotopy, self.output_dot, **kwargs)
)
A, B = map(Mobject.get_center, [self.input_dot, self.output_dot])
A_text = TextMobject("A")
A_text.shift(A+2*(LEFT+UP))
A_arrow = Arrow(
A_text, self.input_dot,
color = self.input_color
)
B_text = TextMobject("B")
B_text.shift(B+2*RIGHT+DOWN)
B_arrow = Arrow(
B_text, self.output_dot,
color = self.output_color
)
tup = self.get_circles_and_points(
input_value-input_radius,
input_value+input_radius
)
input_circle, input_points, output_circle, output_points = tup
for text, arrow in [(A_text, A_arrow), (B_text, B_arrow)]:
self.play(
ShimmerIn(text),
ShowCreation(arrow)
)
self.wait()
self.remove(A_text, A_arrow, B_text, B_arrow)
self.play(ShowCreation(input_circle))
self.wait()
self.play(ShowCreation(input_points))
self.wait()
input_points_copy = input_points.copy()
self.play(
Transform(input_points_copy, output_points),
run_time = 2
)
self.wait()
self.play(ShowCreation(output_circle))
self.wait()
self.wait()
self.remove(*[
input_circle, input_points,
output_circle, input_points_copy
])
def vary_circle_sizes(self):
input_value = 0.45
radius = 0.04
vary_circles = VaryCircles(
self, input_value, radius,
run_time = 5,
)
self.play(vary_circles)
self.wait()
text = TextMobject("Function is ``Continuous at A''")
text.shift(2*UP).to_edge(LEFT)
arrow = Arrow(text, self.input_dot)
self.play(
ShimmerIn(text),
ShowCreation(arrow)
)
self.wait()
self.remove(vary_circles.mobject, text, arrow)
def discontinuous_point(self):
point_description = TextMobject(
"Point where the function jumps"
)
point_description.shift(3*RIGHT)
discontinuous_at_A = TextMobject(
"``Discontinuous at A''",
size = "\\Large"
)
discontinuous_at_A.shift(2*UP).to_edge(LEFT)
text = TextMobject("""
Circle around ouput \\\\
points can never \\\\
be smaller than \\\\
the jump
""")
text.scale(0.75)
text.shift(3.5*RIGHT)
input_value = 0.5
input_radius = 0.04
vary_circles = VaryCircles(
self, input_value, input_radius,
run_time = 5,
)
for dot in self.input_dot, self.output_dot:
dot.center()
kwargs = {
"rate_func" : lambda t : interpolate(0.45, input_value, smooth(t))
}
self.play(
Homotopy(self.input_homotopy, self.input_dot, **kwargs),
Homotopy(self.output_homotopy, self.output_dot, **kwargs)
)
discontinuous_arrow = Arrow(discontinuous_at_A, self.input_dot)
arrow = Arrow(
point_description, self.output_dot,
buff = 0.05,
color = self.output_color
)
self.play(
ShimmerIn(point_description),
ShowCreation(arrow)
)
self.wait()
self.remove(point_description, arrow)
tup = self.get_circles_and_points(
input_value-input_radius,
input_value+input_radius
)
input_circle, input_points, output_circle, output_points = tup
input_points_copy = input_points.copy()
self.play(ShowCreation(input_circle))
self.play(ShowCreation(input_points))
self.play(
Transform(input_points_copy, output_points),
run_time = 2
)
self.play(ShowCreation(output_circle))
self.wait()
self.play(ShimmerIn(text))
self.remove(input_circle, input_points, output_circle, input_points_copy)
self.play(vary_circles)
self.wait()
self.play(
ShimmerIn(discontinuous_at_A),
ShowCreation(discontinuous_arrow)
)
self.wait(3)
self.remove(vary_circles.mobject, discontinuous_at_A, discontinuous_arrow)
def continuous_point(self):
pass
class FormalDefinitionOfContinuity(Scene):
def construct(self):
self.setup()
self.label_spaces()
self.move_dot()
self.label_jump()
self.draw_circles()
self.vary_circle_sizes()
self.discontinuous_point()
def setup(self):
self.input_color = YELLOW_C
self.output_color = RED
def spiril(t):
theta = 2*np.pi*t
return t*np.cos(theta)*RIGHT+t*np.sin(theta)*UP
self.spiril1 = ParametricFunction(
lambda t : 1.5*RIGHT + DOWN + 2*spiril(t),
density = 5*DEFAULT_POINT_DENSITY_1D,
)
self.spiril2 = ParametricFunction(
lambda t : 5.5*RIGHT + UP - 2*spiril(1-t),
density = 5*DEFAULT_POINT_DENSITY_1D,
)
Mobject.align_data(self.spiril1, self.spiril2)
self.output = Mobject(self.spiril1, self.spiril2)
self.output.ingest_sub_mobjects()
self.output.highlight(GREEN_A)
self.interval = UnitInterval()
self.interval.scale_to_fit_width(SPACE_WIDTH-1)
self.interval.to_edge(LEFT)
self.input_dot = Dot(color = self.input_color)
self.output_dot = self.input_dot.copy().highlight(self.output_color)
left, right = self.interval.get_left(), self.interval.get_right()
self.input_homotopy = lambda (x, y, z, t) : (x, y, t) + interpolate(left, right, t)
output_size = self.output.get_num_points()-1
output_points = self.output.points
self.output_homotopy = lambda (x, y, z, t) : (x, y, z) + output_points[int(t*output_size)]
def get_circles_and_points(self, min_input, max_input):
input_left, input_right = [
self.interval.number_to_point(num)
for num in min_input, max_input
]
input_circle = Circle(
radius = np.linalg.norm(input_left-input_right)/2,
color = WHITE
)
input_circle.shift((input_left+input_right)/2)
input_points = Line(
input_left, input_right,
color = self.input_color
)
output_points = Mobject(color = self.output_color)
n = self.output.get_num_points()
output_points.add_points(
self.output.points[int(min_input*n):int(max_input*n)]
)
output_center = output_points.points[int(0.5*output_points.get_num_points())]
max_distance = np.linalg.norm(output_center-output_points.points[-1])
output_circle = Circle(
radius = max_distance,
color = WHITE
)
output_circle.shift(output_center)
return (
input_circle,
input_points,
output_circle,
output_points
)
def label_spaces(self):
input_space = TextMobject("Input Space")
input_space.to_edge(UP)
input_space.shift(LEFT*SPACE_WIDTH/2)
output_space = TextMobject("Output Space")
output_space.to_edge(UP)
output_space.shift(RIGHT*SPACE_WIDTH/2)
line = Line(
UP*SPACE_HEIGHT, DOWN*SPACE_HEIGHT,
color = WHITE
)
self.play(
ShimmerIn(input_space),
ShimmerIn(output_space),
ShowCreation(line),
ShowCreation(self.interval),
)
self.dither()
def move_dot(self):
kwargs = {
"rate_func" : None,
"run_time" : 3
}
self.play(
Homotopy(self.input_homotopy, self.input_dot, **kwargs),
Homotopy(self.output_homotopy, self.output_dot, **kwargs),
ShowCreation(self.output, **kwargs)
)
self.dither()
def label_jump(self):
jump_points = Mobject(
Point(self.spiril1.points[-1]),
Point(self.spiril2.points[0])
)
self.brace = Brace(jump_points, RIGHT)
self.jump = TextMobject("Jump")
self.jump.next_to(self.brace, RIGHT)
self.play(
GrowFromCenter(self.brace),
ShimmerIn(self.jump)
)
self.dither()
self.remove(self.brace, self.jump)
def draw_circles(self):
input_value = 0.45
input_radius = 0.04
for dot in self.input_dot, self.output_dot:
dot.center()
kwargs = {
"rate_func" : lambda t : interpolate(1, input_value, smooth(t))
}
self.play(
Homotopy(self.input_homotopy, self.input_dot, **kwargs),
Homotopy(self.output_homotopy, self.output_dot, **kwargs)
)
A, B = map(Mobject.get_center, [self.input_dot, self.output_dot])
A_text = TextMobject("A")
A_text.shift(A+2*(LEFT+UP))
A_arrow = Arrow(
A_text, self.input_dot,
color = self.input_color
)
B_text = TextMobject("B")
B_text.shift(B+2*RIGHT+DOWN)
B_arrow = Arrow(
B_text, self.output_dot,
color = self.output_color
)
tup = self.get_circles_and_points(
input_value-input_radius,
input_value+input_radius
)
input_circle, input_points, output_circle, output_points = tup
for text, arrow in [(A_text, A_arrow), (B_text, B_arrow)]:
self.play(
ShimmerIn(text),
ShowCreation(arrow)
)
self.dither()
self.remove(A_text, A_arrow, B_text, B_arrow)
self.play(ShowCreation(input_circle))
self.dither()
self.play(ShowCreation(input_points))
self.dither()
input_points_copy = input_points.copy()
self.play(
Transform(input_points_copy, output_points),
run_time = 2
)
self.dither()
self.play(ShowCreation(output_circle))
self.dither()
self.dither()
self.remove(*[
input_circle, input_points,
output_circle, input_points_copy
])
def vary_circle_sizes(self):
input_value = 0.45
radius = 0.04
vary_circles = VaryCircles(
self, input_value, radius,
run_time = 5,
)
self.play(vary_circles)
self.dither()
text = TextMobject("Function is ``Continuous at A''")
text.shift(2*UP).to_edge(LEFT)
arrow = Arrow(text, self.input_dot)
self.play(
ShimmerIn(text),
ShowCreation(arrow)
)
self.dither()
self.remove(vary_circles.mobject, text, arrow)
def discontinuous_point(self):
point_description = TextMobject(
"Point where the function jumps"
)
point_description.shift(3*RIGHT)
discontinuous_at_A = TextMobject(
"``Discontinuous at A''",
size = "\\Large"
)
discontinuous_at_A.shift(2*UP).to_edge(LEFT)
text = TextMobject("""
Circle around ouput \\\\
points can never \\\\
be smaller than \\\\
the jump
""")
text.scale(0.75)
text.shift(3.5*RIGHT)
input_value = 0.5
input_radius = 0.04
vary_circles = VaryCircles(
self, input_value, input_radius,
run_time = 5,
)
for dot in self.input_dot, self.output_dot:
dot.center()
kwargs = {
"rate_func" : lambda t : interpolate(0.45, input_value, smooth(t))
}
self.play(
Homotopy(self.input_homotopy, self.input_dot, **kwargs),
Homotopy(self.output_homotopy, self.output_dot, **kwargs)
)
discontinuous_arrow = Arrow(discontinuous_at_A, self.input_dot)
arrow = Arrow(
point_description, self.output_dot,
buff = 0.05,
color = self.output_color
)
self.play(
ShimmerIn(point_description),
ShowCreation(arrow)
)
self.dither()
self.remove(point_description, arrow)
tup = self.get_circles_and_points(
input_value-input_radius,
input_value+input_radius
)
input_circle, input_points, output_circle, output_points = tup
input_points_copy = input_points.copy()
self.play(ShowCreation(input_circle))
self.play(ShowCreation(input_points))
self.play(
Transform(input_points_copy, output_points),
run_time = 2
)
self.play(ShowCreation(output_circle))
self.dither()
self.play(ShimmerIn(text))
self.remove(input_circle, input_points, output_circle, input_points_copy)
self.play(vary_circles)
self.dither()
self.play(
ShimmerIn(discontinuous_at_A),
ShowCreation(discontinuous_arrow)
)
self.dither(3)
self.remove(vary_circles.mobject, discontinuous_at_A, discontinuous_arrow)
def continuous_point(self):
pass