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Evolute


A Hypotrochoid (red) and its evolute (black). Note that vertexes of a curve corresponds to its evolute cusps. The lines are radiuses of tangent circles. Mathematica icon graphics code.

Mathematica icon 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.

movie icon Tangent Circle in Motion gsp icon 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

Base Curve Evolute
cardioid cardioid scaled by 1/3
nephroid nephroid 1/2
astroid astroid 2
deltoid deltoid 3
epicycloid epicycloid
hypocycloid hypocycloid
cycloid cycloid
Cayley's sextic nephroid
parabola semicubic parabola
limacon of Pascal catacaustic of a circle
equiangular spiral equiangular spiral
tractrix catenary

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

Robert Yates: Curves and Their Properties.

Wikipedia: evolute↗.

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© 1995-2008 by Xah Lee.
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