Evolute
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In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. It is equivalent to the envelope of the normals. Opposite to evolute is involute. Equations of an evolute of a parametrically defined curve are:
![X[x,y]=x+y'\frac{x'^2+y'^2}{x''y'-y''x'}](../../../math/c/8/e/c8e238b46813629cab3df47183a5bd50.png)
![Y[x,y]=y+x'\frac{x'^2+y'^2}{y''x'-x''y'}](../../../math/c/9/4/c94a285074edb6fc5f149c3d16c39551.png)
If r is the curve parametrized by arc length (i.e. | r'(s) | = 1; see natural parametrization) then the center of curvature at s is
Such parametrization is usually between difficult and impossible, but it's still feasible to access r". If x is any (reasonably differentiable) parametrization, and s gives arc length over the same parameter, then the desired r would give r(s(t)) = x(t) which if differentiated twice gives
- r'(s(t))s'(t) = x'(t)
- r''(s(t))s'(t)2 + r'(s(t))s''(t) = x''(t)
which we rearrange to
Recognising that
- s'(t) = | x'(t) |
eliminates the need to know s itself, thus eliminating the integration in which the analytic impossibilities lie.
Intrinsic equation of the evolute of a curve defined by an intrinsic equation r=f(s) is
![R[y]=\frac{rr'}{(\mbox{inv }r)'}](../../../math/d/a/4/da4ed9e4c0f70003e061d0034a63c049.png)
where inv f is the inverse function.
The evolute will have a cusp when the curve has a vertex, that is when the curvature has a local maximum or minimum. When parallel curves are constructed they will have cusps when the distance from the curve matches the radius of curvature.
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| Differential transforms of plane curves |
| Parallel curve | Evolute | Involute | Pedal curve | Contrapedal curve | Negative pedal curve | Dual curve |
