Constructing self-supporting surfaces with planar quadrilateral elements

Abstract

We present a simple yet effective method for constructing 3D self-supporting surfaces with planar quadrilateral (PQ) elements. Starting with a triangular discretization of a self-supporting surface, we first compute the principal curvatures and directions of each triangular face using a new discrete differential geometry approach, yielding more accurate results than existing methods. Then, we smooth the principal direction field to reduce the number of singularities. Next, we partition all faces into two groups in terms of principal curvature difference. For each face with small curvature difference, we compute a stretch matrix that turns the principal directions into a pair of conjugate directions. For the remaining triangular faces, we simply keep their smoothed principal directions. Finally, applying a mixed-integer programming solver to the mixed principal and conjugate direction field, we obtain a planar quadrilateral mesh. Experimental results show that our method is computationally efficient and can yield high-quality PQ meshes that well approximate the geometry of the input surfaces and maintain their self-supporting properties.