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. [itex]|r'(s)|=1[itex]; see natural parametrization) then the center of curvature at s is

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

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 [itex]r(s(t))=x(t)[itex] which if differentiated twice gives

[itex]r'(s(t))s'(t)=x'(t)[itex]
[itex]r''(s(t))s'(t)^2+r'(s(t))s''(t)=x''(t)[itex]

which we rearrange to

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

Recognising that

[itex]s'(t)=|x'(t)|[itex]

eliminates the need to know s itself, thus eliminating the integration in which the analytic impossibilities lie. Template:Math-stubde:Evolute pl:Ewoluta

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