Previous: Parallel Curves
Evolutes and Osculating CirclesConsider 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?
The Evolute |