Geodesic curvature preservation in surface flattening through constrained global optimization

Abstract

In this paper, two methods for generating a planar development of a three-dimensional (3D) surface are proposed. The first method is based on the solution of a global optimization problem without constraints taking into consideration the geodesic curvature of the surface isoparametric curves. The second method is based on the solution of a global optimization problem subject to constraints which are used to control the local accuracy in the derived planar development. Contrarily to previous approaches the proposed second method does not depend on the surface parameterization, since the formed energy function and the proposed constraints are based only on the triangulation of the object surface. Several experiments illustrate that when the 3D surface is approximated with satisfying accuracy, modifications in the triangulation do not have any significant influence to the result of the proposed method. The proposed second method can be used without any modifications with trimmed surfaces, while the generated planar developments can also be used for minimally distorted texture mapping. Finally, the effectiveness of the proposed second method is illustrated through an indicative application in shoe designing.