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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.

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

<math>r(s)+{r(s)\over|r(s)|^2}.</math>

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 <math>r(s(t))=x(t)</math> which if differentiated twice gives

<math>r'(s(t))s'(t)=x'(t)</math>

<math>r(s(t))s'(t)^2+r'(s(t))s(t)=x(t)</math>

which we rearrange to

<math>r(s(t))={x(t)s'(t)-x'(t)s(t)\over s'(t)^3}.</math>

Recognising that

<math>s'(t)=|x'(t)|</math>

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.

de:Evolute

fr:Développée pl:Ewoluta