Apollonius's Ellipse and Evolute Revisited

Editor’s note: This article was published in February 2008.

The problem of finding the distance from a point to a curve is a standard exercise in calculus. If the curve is differentiable and has no endpoints, the connecting line segment from the given point to the nearest (farthest) point on the curve must be perpendicular to the tangent line there. One can therefore ask the more general question of how many such points on the curve have this geometric property, i.e. how many "normal line segments" can be drawn from the given point to the curve. If one adopts a parametrization of the curve, then these points correspond to critical points of the distance function.

Since a continuous real-valued function on a closed interval always has a maximum and minimum value, there are at least two such normals for any simple differentiable closed curve like the ellipse. For example, if the curve is a circle and the point is not at the center of the circle, then precisely two normal line segments can be drawn (in the line through the given point and the center), while if the point is at the center of the circle an infinite number of normals can be drawn. Similarly if the curve is a proper ellipse (not a circle) and the point is at the center of the ellipse, then clearly four normals can be drawn to the center from the points on the ellipse lying on its symmetry axes. It is natural to ask how does the number of such normals vary as the given point moves away from the center.

The question of determining the minimum and if it exists, the maximum distance to a conic section was addressed and answered some twenty-two hundred years ago by Apollonius of Perga, "The Great Geometer." Apollonius is most famous for his Conics series which originally consisted of eight Books. Only Books I-IV survive in the original Greek [3], but Books I-VII of the Conics exist in an Arabic translation [6]. Apollonius used the "normal" approach in addressing this problem in Book V, and although his original work apparently contains the resolution of the problem, both his proofs and results are very difficult to follow even in their annotated translated form [6,7]. He lacked even an appropriate mathematical language to discuss quadratic relationships, which makes his results all the more remarkable as well as difficult to translate into modern notation. His approach to the problem relies on the determination of the evolutes of the conics whose equations "can be easily deduced from the results obtained by Apollonius ... and it is a veritable geometric tour de force" [2, p. 159].

The purpose of this paper is to develop Apollonius's results using present day mathematics suitable for a second year college mathematics student in a way that demonstrates the geometry through visualization while avoiding excessive formula manipulation that computer algebra systems are well suited to handle. The approach used here is based on a remark made without detail by Heath in [1, p. cxxvii]. In addition we connect the results to an application of the discriminant of a quartic polynomial, illustrating the no longer well known fact that a familiar and widely used idea for the quadratic case extends to the less familiar cubics and quartics. In the following we restrict our attention to the ellipse and note that a similar approach would work for the remaining conics, leaving some doable problems for similar exploration by interested advanced undergraduates based on our Maple worksheet, where the otherwise tedious details are made manageable even for such students.

For a graphic illustration of the results, see http://www3.villanova.edu/maple/misc/ellipse/images/frenetellipse499.gif