Overview  Animation  Classic Surfaces  Parm Surfaces  Curves on Surfaces  Discrete Geodesics  ODE  Platonic Solids  Cycloid  Root Finder  Harmonic Maps  Rivara Bisection  Scalar Field  Weierstrass  Closed Polygon  Elastic Curve  Billiard in an Ellipse  LIC Visualization  Discrete VF  Hodge Splitting  Textured Surface  Surfaces of Rotation  Mean Curvature Flow  Pythagoraen Tree  Julia Sets 

Cycloid Curves

Among the famous planar curves is the cycloid. A cycloid is defined as the trace of a point on a disk when this disk rolls along a line. The disk is not allowed to slide.

The shape of the cycloid depends on two parameters, the radius r of the circle and the distance d of the point generating the cycloid to the center of rolling disk. The mathematical expression of a cycloid is

Cycloid[r, d](t) = (r t + d sin(t), r - d cos(t)).

We scale our experiment such that the radius of the circle is normalized to 1 and cannot be changed.


Slider "Cycloid Discr" Number of vertices of discretely drawn cycloid.
Slider "Cycloid Length" Length of definition interval for x.
Slider "Distance d" Depending on the relation between distance d and radius r, the cycloid has different shapes. For d<r the cycloids waves up and down, and are embedded like a sine curve. For d=r the curve looks roughly like a collection of half-circles, and for d>r they have self-intersections.
Checkbox "Show Circle" Switch evolving circle to be visible or invisible in display.
Checkbox "Show Cycloid" Switch cycloid curve to be visible or invisible in display.
Checkbox "Show Surface" Switch cycloid rotational surface to be visible or invisible in display. To see the surface in 3d, switch the display to 3d by F1->Inspector->Camera->Perspective.
Button "Animate" Start and stop animation of evolving circle.
Button "Reset" Set all sliders to their default values.

By F4 or ctrl-a you can also get an Animation Panel .

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