Ball’s point construction revisited

Abstract

Two well known graphical methods based on Bobillier’s construction of the inflection pole and Bereis’ construction of Ball’s point on the inflection circle are used for many decades. In this paper a new general-purpose method of step-by-step vectorization of constructions like these is introduced. It is based on symplectic geometry in its simplest possible 2D case and is making use of loop closure equations exclusively. The vectorization process is coordinate and trigonometry free. The formulas found by this method are new and their correctness is easily verified by comparison with results of the corresponding graphical methods.