A Projective Framework for Polyhedral Mesh Modeling

We present a novel framework for polyhedral mesh editing with face-based projective maps, that preserves planarity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar-preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar-quadrilateral mesh subdivision, a resulting multi-resolution editing algorithm, and novel shape-space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate-preserving projective transformation fields on surfaces. Our shape space directly addresses planar-quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.