Modeling with discrete equivalence classes of planar quads

Abstract

We propose a novel method to model shapes using a small set of discrete equivalence classes of planar quads. Unlike previous modular modeling methods that only optimize vertex positions of an input quad mesh without modifying the mesh connectivity, we first generate a quadrilateral mesh considering the equivalence requirement and then optimize the geometry to be modular. Our meshing approach is field-driven, and the key is to transform the requirements of the quad mesh into those of the field. Specifically, the frame field is required to belong to a small set of discrete equivalence classes while being smooth, integrable, and nearly conjugate. After alternately optimizing the requirements to obtain the desired field, we use it to generate a quad mesh, which is further improved to be modeled with a small set of template planar quads. We demonstrate the practicability and effectiveness of our algorithm over various examples.