Evolute

Encyclopedia : E : EV : EVO : 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)+.]

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))=.]

Recognising that
[s'(t)=|x'(t)|]

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

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