Feature-sensitive parameterization of polygonal meshes S. Morigi Department of Mathematics, University of Bologna, P.zza di Porta San Donato 5, 40127 Bologna, Italy article info Keywords: Parameterization Polygonal meshes Discrete curvature abstract This paper investigates a parameterization method for polygonal meshes which is able to improve map distortion exploiting local curvature of the surface mesh. Parametrization is an important tool in geometric modelling and computer graphics applications such as, for example, texture mapping, shape morphing, and remeshing. The paper presents a feature- sensitive scheme which is based on a stretch-minimizing strategy which iterates the shape-preserving parameterization taking into account the influence of the curvature on the mesh stretch. In particular, the main goal is to provide a distortion in the parametriza- tion driven by the curvature directions, in such a way that strong distortion correspond to area with mean curvature approaching to zero. Computed examples demonstrate the effec- tiveness of the proposed method using geometric measures on angle, edge and area stretching, and illustrate the high quality of the mesh parameterizations produced. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction A parameterization of a surface can be defined as a one-to-one mapping from a suitable domain to the surface. This comes naturally for surfaces that are defined parametrically, such as spline, but less naturally for other surfaces such as polygon meshes which have no inherent parameterization. Usually, in computer graphics and geometric modelling, a smooth surface is often approximated by a piecewise linear surface or triangular mesh M ¼ðT; V Þ, i.e. the union of a set T ¼fT 1 ; ... T M g of triangles T i such as triangles intersect only at common vertices or edges. We consider a triangular mesh with a boundary, and we denote by V the vertex set of M, which consists of V B , the set of vertices on the boundary, and V I , the set of interior vertices. The goal of mesh parameterization is to map a given mesh M, representing a surface that is homeomorphic to a disk, into the plane. That is to find a suitable piecewise linear mapping / : M ! D  R 2 such that it maps the boundary ver- tices V B onto the boundary of a simple planar region D, and the inner vertices V I inside D with the condition that the vertex connectivity of M is maintained inside the parametric region D. Basically, this one-to-one mapping provides un unfolding of a surface from 3D down to 2D, which inevitably introduces some form of distortion. This effect in the parameterization is due to the triangle dimensions in the parametric domain which are non-proportional to the associated mesh triangle sizes. In case the mesh is not homeomorphic to a disk, the mesh must be partitioned into a set of charts. Each chart can be parameterized on a planar domain (e.g. a rectangular region) using different methods, whose selection depends on its genus and number of boundary components, and these parametrizations collectively form an atlas. Disk-like charts (0-genus with one boundary component) can be parameterized using the barycentric coordinates method introduced by Floater [3]. Charts with an arbitrary genus or 0-genus charts with more than one boundary component are converted to disk-like regions by cutting them along cut-graphs and then embedded on the plane using the barycentric coordinates method, see [9]. Parameterization of polygonal meshes is an important computational tool in computer graphics, where it is required, for example, in texture mapping or remeshing tasks. In case of texture mapping, the source image (texture) is mapped onto a 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.07.014 E-mail address: morigi@dm.unibo.it Applied Mathematics and Computation 215 (2009) 1561–1572 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc