Evolute

evoluteStar
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. graphics code.

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.

evoluteEllipseF1 evoluteEllipseF2 evoluteEllipseF3 evoluteEllipseF4 evoluteEllipseF5 evoluteEllipseF6
evoluteEllipseF
Tangent circles of a ellipse. evoluteEllipse.gsp
evoluteEllipse1 evoluteEllipse2 evoluteEllipse3
Evolute of a curve can also be defined as the envelope of its normal.

Formula

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'')}

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

Related Web Sites

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

Robert Yates: Curves and Their Properties.

evolute.

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