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.

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

r(s)+{r''(s)\over|r''(s)|^2}

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

r''(s(t))={x''(t)s'(t)-x'(t)s''(t)\over s'(t)^3}

Recognising that

s'(t) = | x'(t) |

eliminates the need to know s itself, thus eliminating the integration in which the analytic impossibilities lie.

See also: Evolute, Arc length, Curvature, Curve, Differentiable, Differential geometry of curves, Envelope (mathematics), Mathematics, Perpendicular