Esempio n. 1
0
    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
        )
Esempio n. 2
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    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
        )
Esempio n. 3
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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
Esempio n. 4
0
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