Evolute
![evoluteStar](evoluteStar.png)
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](evoluteEllipseF1.png)
![evoluteEllipseF2](evoluteEllipseF2.png)
![evoluteEllipseF3](evoluteEllipseF3.png)
![evoluteEllipseF4](evoluteEllipseF4.png)
![evoluteEllipseF5](evoluteEllipseF5.png)
![evoluteEllipseF6](evoluteEllipseF6.png)
![evoluteEllipseF](evoluteEllipseF.gif)
![evoluteEllipse1](evoluteEllipse1.png)
![evoluteEllipse2](evoluteEllipse2.png)
![evoluteEllipse3](evoluteEllipse3.png)
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 |