Evolute

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A ellipse (red) and its evolute (blue), the dots are the vertices of the curve, each vertex corresponds to a cusp on the evolute.

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:

If r is the curve parametrised by arc length (i.e. | r'(s) | = 1; see natural parametrization) then the center of curvature at s is

Such parametrisation is usually between difficult and impossible, but it's still feasible to access r". If x is any (reasonably differentiable) parametrisation, 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.

The evolute will have a cusp when the curve has a vertex, that is when the curvature has a local maximum or minimum.

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