Piecewise Ruled Approximation for Freeform Mesh Surfaces

A ruled surface is a shape swept out by moving a line in 3D space. Due to their simple geometric forms, ruled surfaces have applications in various domains such as architecture and engineering. In the past, various approaches have been proposed to approximate a target shape using developable surfaces, which are special ruled surfaces with zero Gaussian curvature. However, methods for shape approximation using general ruled surfaces remain limited and often require the target shape to be either represented as parametric surfaces or have non-positive Gaussian curvature. In this paper, we propose a method to compute a piecewise ruled surface that approximates an arbitrary freeform mesh surface. We first use a group-sparsity formulation to optimize the given mesh shape into an approximately piecewise ruled form, in conjunction with a tangent vector field that indicates the ruling directions. Afterward, we utilize the optimization result to extract seams that separate smooth families of rulings, and use the seams to construct the initial rulings. Finally, we further optimize the positions and orientations of the rulings to improve the alignment with the input target shape. We apply our method to a variety of freeform shapes with different topologies and complexity, demonstrating its effectiveness in approximating arbitrary shapes.