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Evolutes and Osculating Circles

Consider the parallel curves of the ellipse. For small values of the distance d, the parallel curves are smooth, but once d reaches a certain value, the interior parallel curves develop cusps . The greater the curvature in a portion of the curve, the sooner the cusps appear. The collection of cusps of parallel curves of the ellipse forms a new curve called the evolute of the ellipse.

We have shown that the parallel curve at distance d has a cusp at X{_d}(t) if d = 1/{kappa}{_g}(t). Thus we may describe the evolute of a curve as E(t) = X(t) + (1/{kappa}{_g}(t)) U(t).

Exercise 1: For which values of t will the evolute curve E(t) have a singularity?
Exercise 2: What can be said about the unit tangent vector to E'(t) at a point where the curve is not singular?
Exercise 3: What is the length of the Evolute curve?

The Evolute
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Next: Osculating Circles