Pedal of a Cycloid.

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radial evolute negative pedal pedal caustic parallel inversion derivative cissoid wrt a line conchoid strophoid


Differential Equations, Mechanics, and Computation


Cycloid (tautochrone, brachistochrone) is a member of cycloidal family of curves. (See: Curve Family Index) Prolate (extended) or curtate (contracted) cycloids are also known as trochoids. In this page, we use the narrowest definition of the term cycloid, defined as the trace of a point on the circumsference of a circle rolling on a line without slipping.

In the right figure, c is the rolling circle. P is the tracing point. A is the point of contact with line. PA is the normal at P. E is a reflection of P through A. The locus of E is the evolute of the cycloid.

cycloid cycloid
Tracing a Cycloid Tracing a Cycloid by Tangent




The catacaustic of a cycloid with respect to parallel rays coming beneath its arc are two smaller cycloids. (Or, the diacaustic of the cycloid with rays coming from above.)

cycloid cycloid
Catacaustic with vertical rays

Evolute and Involute

The evolute of a cycloid is another cycloid. The first figure show succesive evolutes of a cycloid. The second connect points on the curve with their center of tangent circles. cycloid_evolute.gcf

cycloid cycloid
Left: The green cycloid is the evolute of the red cycloid is the evolute of the blue cycloid. Right: The yellow cycloid's normals (green) are draw up to the center of osculating circle. The endings form its evolute curve, which is another cycloid. Constructing Evolute of a Cycloid

The involute of a cycloid is also a cycloid. cycloid_involute.gcf Both evolute and involute properties are easily proved by a direct application of the formula and simplify the result.


The radial of a cycloid is a circle.

Generating Radial Radial of a Cycloid

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive.


Alexander Bogomolny, with java applet.

Brombacher Aarnout, with GSP and QuickTime movies.

Dinoj Surendran.

Joseph Portney. A somewhat detailed exposition on the property of cycloid being the curve of fastest decent. (called brachistochrone)

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