Evolute

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A ellipse (red) and its evolute (blue), the dots are the vertices of the curve, each vertex corresponds to a cusp on the 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. Opposite to evolute is involute. Equations of an evolute of a parametrically defined curve are:

$X[x,y]=x+y'\frac{x'^2+y'^2}{x''y'-y''x'}$

$Y[x,y]=y+x'\frac{x'^2+y'^2}{y''x'-x''y'}$

If r is the curve parametrized 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 parametrization is usually between difficult and impossible, but it's still feasible to access r". If x is any (reasonably differentiable) parametrization, 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.

An ellipse (red), its evolute (blue) and some parallel curves. Note how the parallel curves have cusps when they touch the evolute

Intrinsic equation of the evolute of a curve defined by an intrinsic equation r=f(s) is

$R[y]=\frac{rr'}{(\mbox{inv }r)'}$

where inv f is the inverse function.

The evolute will have a cusp when the curve has a vertex, that is when the curvature has a local maximum or minimum. When parallel curves are constructed they will have cusps when the distance from the curve matches the radius of curvature.