Cycloid

In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette.

About

Setup

Create a new Colab notebook at colab.new and run the cells below.

The session will restart to apply changes.

%pip install mscene
import mscene
%mscene -l manim plugins

from mscene.manim import *
from mscene.plugins import *

Actions

The trochoid_scene function animates cycloid or trochoid scenes. A cycloid is a special case of trochoid where the tracing point is on the circumference of the rolling circle.

def trochoid_scene(scene, title, markers):
    length = config.frame_width * 0.75
    radius = length / (2 * TAU)

    num_line = NumberLine(
        x_range=[-TAU, TAU, PI], length=length, color=ManimColor("#6F828A")
    ).shift(DOWN * radius)

    point = num_line.n2p(-TAU) + radius * UP

    wheel = Wheel(radius=radius, color=ManimColor("#04D9FF"), point=point)
    scene.play(GrowFromCenter(wheel), Create(num_line, lag_ratio=0))
    scene.wait()

    scene.play(wheel.draw_markers(markers), Write(title))
    scene.wait()

    path = wheel.trace_paths()
    scene.add(path)

    scene.play(wheel.roll(length * RIGHT, run_time=4))
    scene.wait()

    path.clear_updaters()
    scene.play(path.animate.fade(2 / 3))

    new_path = wheel.trace_paths(dissipating_time=2)
    scene.add(new_path)

    scene.play(wheel.roll(length * LEFT, run_time=4))
    scene.wait()

    scene.play(Unwrite(title), FadeOut(path))

    new_path.clear_updaters()
    scene.wait()

    scene.remove(new_path)
    scene.play(wheel.undraw_markers(), Unwrite(title))
    scene.wait()

    scene.play(ShrinkToCenter(wheel), Uncreate(num_line, lag_ratio=0))
    scene.wait(0.5)

Scenes

Cycloid

%%manim -qm Cycloid
class Cycloid(Scene):
    def construct(self):

        title = Text("Cycloid").to_edge(DOWN, buff=1.25)
        marker = [(1, PI / 2, ManimColor("#6019E3"))]

        trochoid_scene(self, title, marker)

Prolate Cycloid

%%manim -qm ProlateCycloid
class ProlateCycloid(Scene):
    def construct(self):

        title = Text("Prolate Cycloid").to_edge(DOWN, buff=1.25)
        marker = [(1.5, PI / 2, ManimColor("#E31937"))]

        trochoid_scene(self, title, marker)

Curtate Cycloid

%%manim -qm CurtateCycloid
class CurtateCycloid(Scene):
    def construct(self):

        title = Text("Curtate Cycloid").to_edge(DOWN, buff=1.25)
        marker = [(0.5, PI / 2, ManimColor("#19E360"))]

        trochoid_scene(self, title, marker)

Two Markers

%%manim -qm TwoMarkers
class TwoMarkers(Scene):
    def construct(self):

        title = VMobject()
        marker = [
            (1, PI / 2, ManimColor("#6019E3")),
            (0.5, -PI / 2, ManimColor("#E31937")),
        ]

        trochoid_scene(self, title, marker)

Trochoids

%%manim -qm Trochoids
class Trochoids(Scene):
    def construct(self):

        colors = [ManimColor(hex) for hex in ("#19E360", "#6019E3", "#E31937")]

        cycloids = [
            ("Curtate Cycloid", (0.5, PI / 2, colors[0])),
            ("Cycloid", (1, PI / 2, colors[1])),
            ("Prolate Cycloid", (1.5, PI / 2, colors[2])),
        ]

        title = VGroup()
        markers = []

        for i in cycloids:
            text = Text(i[0], font_size=40)
            dot = Dot(radius=0.125, color=i[1][2]).next_to(text[0], LEFT)
            title.add(VGroup(dot, text))
            markers.append(i[1])

        title.arrange(RIGHT, buff=0.5).to_edge(DOWN, buff=1.25)
        markers.reverse()

        trochoid_scene(self, title, markers)