Conjugate shape simplification via precise algebraic planar sweeps toward gear design

Gears play a pivotal role in machine design. This paper proposes an algorithm to simplify the shapes of planar gears. This is achieved via iterative conjugation, using precise algebraic sweeps. The notion of shape simplification is introduced in a mathematically rigorous manner and it is shown that the conjugation process converges, yielding a pair of meshing gears that follow the desired motion. Simplified gear shapes may lead to improved mechanical characteristics and reduction in manufacturing costs. The generality of algebraic sweeps allows precise design of gears with freeform shapes and non-uniform motion transmission. Moreover, the computational framework proposed in this paper is versatile, with applications beyond gear design. A variety of examples from an implementation of our algorithm, that offers topological guarantees, are presented, which demonstrate the robustness and efficacy of our approach.