Apollonius's Ellipse and Evolute Revisited - The Evolute

The evolute of an ellipse may be defined in terms of the curvature at a point on the ellipse. Suppose that the ellipse is parameterized by [(r)\vec](t)=áacos(t),bsin(t)ñ, 0 £ t < 2p. The curvature k(t) at [(r)\vec](t) can be evaluated using the standard calculus formula in parametric form

k(t) = |(-acos(t))(bcos(t))-(-bsin(t))(-asin(t))| [a2sin2(t)+b2cos2(t)]3/2

and from it one obtains the radius of curvature, R(t)=1/k(t). By definition the center of the osculating circle is located a distance R(t) along the inward pointing unit normal    N(t) from its point of origin on the ellipse, so the position vector of this center is simply C(t)=r(t)+R(t) N(t). By direct calculation and simplification one finds

® C   (t) = a2-b2 a cos3(t), b2-a2 b sin3(t)  .

(4)

If one adopts the definition that the evolute is the locus of the centers of the osculating circles, then for the ellipse it is this parametrized curve C(t) as t varies from 0 to 2p. Since the focal distance from the center along the major axis is c = Ö{a²-b²}, the two cusps of the evolute at a distance c²/a=c(c/a) < c < a along that axis must fall short of the foci inside the ellipse, while the other two cusps exit the ellipse along the minor axis when c²/b > b or a > Ö2 b.

 

FIGURE 3:  The center of the osculating circle traces out the evolute of the ellipse.