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.

de:Evolute