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. ; 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 which if differentiated twice gives
:
:
which we rearrange to
:
Recognising that
:
eliminates the need to know s itself, thus eliminating the integration in which the analytic impossibilities lie.
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