2.4: 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 Xd(t) if d = 1/κg(t). Thus we may describe the evolute of a curve as
E(t) = X(t) + (1/κg(t))U(t).


This demo shows a curve X(t) (in red) and its evolute (in green), its parallel curve at distance d (in yellow), and its osculating circles (in purple). You can scroll through the t0 tapedeck to see these traced out. There is also a tapedeck controlling d.



1. For which values of t will the evolute curve E(t) have a singularity?

2. What can be said about the unit tangent vector to E'(t) at a point where the curve is not singular?

3.What is the length of the Evolute curve?
Given a plane curve, we want to find the circle which best approximates the curve in a neighborhood of a point t0. A circle is determined by three points, so we choose three points on the curve which are near the point in question. If the three points are not collinear, they determine a circle, with center of a circle given by the intersection of the perpendicular bisectors of any two pairs of them. If we take the limit as the points approach t0, these lines perpendicular to chords of the circle will approach the normal vectors to the curve near the point. The circle we obtain as the limit will have its center a distance 1kg along the normal vector to the curve at t0. This circle is the osculating circle, derived from the Latin for "kiss". Its radius is known as the radius of curvature and the inverse of the radius of curvature is the curvature.

Note that for any piece of curve which has monotonically increasing curvature (as the exponential spiral does), the osculating circles of lesser curvature contain the osculating circles of greater curvature. This demo shows a series of osculating circles of a curve. The centers of the osculating circles trace out the evolute of the curve as the parameter runs through its domain.



1. Enter a curve and its domain. Then hit play on the tape deck and watch the center of the osculating circles trace out the evolute. Note that the demo will draw the evolutes of some curves, such as X(t) = (t, t3) with singularities. Sometimes they are removable. Try to interpolate.

2. Type in a curve with an inflection point. Watch the osculating circle grow large on one side and then grow small again on the other side as the point is passed. Relate this phenomenon to the sign of the curvature.
Top of Page