Blending, smoothing and interpolation of irregular meshes using N-sided Varady patches

Blending, smoothing and interpolation of irregular meshes are important modeling problems. Traditionally they are solved by using parametric (NURBS) or subdivision surfaces, despite NURBS representation imposes some topological restrictions, and subdivision surfaces are mostly used for smoothing and interpolation. N-sided Varady surface representation is not new, but its use in solving modeling problems has been limited. In this paper a unified method for blending, smoothing and interpolation of irregular meshes of arbitrary topological type using Varady patches is presented. The new approach treats all these problems as a surface fitting problem by first restructuring the given mesh to obtain a new mesh suitable for network curves construction, and then filling all the holes framed by the network curves with n-sided Varady patches. The smoothness of the resulting surface is achieved through a global energy minimization process. The resulting surface is C¹ continuous. Several new techniques are introduced; these include restructuring techniques for a given mesh regarding blending and interpolation problems, a network curves construction technique, an energy formulation technique for n-sided Varady patches, and an energy computation technique for Varady surfaces. Advantages of the new method include: (1) a unified solution to three important problems in geometric modeling; (2) carrying good features of Varady patches (such as no need in domain decomposition in the hole filling process and having an additional degree of freedom in adjusting the shape of an n-sided patch); (3) overall smoothness and more even distribution of curvature because of the global energy minimization process; (4) presenting new, effective way for solving modeling problems using Varady patches.