Evolute

Mathematica Notebook for This Page.

History

Studied by Huygens in 1673.

Description

Evolute is a method of deriving a new curve based on a given curve. It is the locus of the centers of tangent circles of the given curve.

Evolute of a ellipse.

Tangent Circle in Motion evoluteEllipse.gsp

Evolute of a curve can also be defined as the envelope of its normal.

Formulas

Given a curve in parametric form {x[t], y[t]}, its evolute is

{x + (y'*(x'^2 + y'^2)) / (  y'*x''  - x'*y''), 
 y + (x'*(x'^2 + y'^2)) / (-(y'*x'') + x'*y'')}

The trailing [t] is ommited for easy reading.

Properties

Parallels and Evolute

Theorem: The locus of Cusps of a curve C's parallel curves is the evolute of C. This is a alternative definition of evolute. See the Parallel page.

Evolute and Involute

If curve A is the involute of curve B, then curve B is the evolute of curve A. The converse is true locally, that is: If curve B is the evolute of curve A, then any part of curve A is the involute of some parts of B.

Curves relations by evolute and involute

Related Web Sites

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

Robert Yates: Curves and Their Properties.

evolute.