Quasi-Developable Mesh Surface Interpolation via Mesh Deformation

Abstract

We present a new algorithm for finding a most "developable" smooth mesh surface to interpolate a given set of arbitrary points or space curves. Inspired by the recent progress in mesh editing that employs the concepts of preserving the Laplacian coordinates and handle-based shape editing, we formulate the interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, such as a planar figure, to a final mesh surface that interpolates the given points and/or curves. During the deformation, the developability of the intermediate mesh is maintained by means of preserving the zero-valued Gaussian curvature on the mesh. To treat the high nonlinearity of the geometric constrains owing to the preservation of Gaussian curvature, we linearize those nonlinear constraints using Taylor expansion and eventually construct a sparse and over-determined linear system which is subsequently solved by a robust least-squares solution. By iteratively performing this procedure, the initial mesh is gradually and smoothly "dragged" to the given points and/or curves. The initial experimental data has shown some promising aspects of the proposed algorithm as a general quasi-developable surface interpolation tool.