The Implicit Equation of a Holditch Curve

Abstract

Holditch’s theorem is a classical geometrical result on the areas of a given closed curve and another one, its Holditch curve, which is constructed as the locus of a fixed point dividing a chord of constant length that moves with its endpoints over the given curve and that returns back to its original position after some full revolution. Holditch curves have already been studied from the parametric point of view, although numerical methods and approximations are often necessary for their computation. In this paper, implicit equations of Holditch curves of algebraic curves are studied. The implicit equations can be simply found from the computation of a resultant of two polynomials. With the same techniques, Holditch curves of two initial algebraic curves are also considered. Moreover, the use of implicit equations allows to find new and explicit parameterizations of non-trivial Holditch curves, such as in the case of having an ellipse as an initial curve.