Tanvolutes: Generalized Involutes

Abstract

The classical involute of a plane base curve intersects every tangent line at a right angle. This paper introduces a tanvolute, which intersects every tangent line at any given fixed angle. This minor change in the definition of a classical concept leads to a wealth of new examples and phenomena that go far beyond the original situation. Our treatment is based on two differential equations relating arclength functions for the base curve and its tanvolute, the tangent-length function from the base to the tanvolute, and the fixed angle. The parameters in the differential equations contribute many essential features to the solution curves. Even when the base curve is relatively simple, for example a circle, the variety in the shapes of the tanvolutes is remarkably rich. To illustrate, as a circle shrinks to a single point, its tanvolute becomes a logarithmic spiral! An application is given to a generalized pursuit problem in which a missile is fired at constant speed in an unknown tangent direction from an unknown point on a base curve. Surprisingly, it can always be intercepted by a faster constant-speed missile that follows a specific tanvolute of the base curve.