Anisotropic Surface Meshing Siu-Wing Cheng ∗ Tamal K. Dey † Edgar A. Ramos ‡ Rephael Wenger § Abstract We study the problem of triangulating a smooth closed implicit surface Σ endowed with a 2D metric tensor that varies over Σ. This is commonly known as the anisotropic surface meshing problem. We extend the 2D metric tensor naturally to 3D and employ the 3D anisotropic Voronoi diagram of a set P of samples on Σ to triangulate Σ. We prove that a restricted dual, Mesh P , is a valid triangulation homeomorphic to Σ under appropriate conditions. We also develop an algorithm for constructing P and Mesh P . In addition to being homeomorphic to Σ, each triangle in Mesh P is well-shaped when measured using the 3D metric tensors of its vertices. Users can set upper bounds on the anisotropic edge lengths and the angles between the surface normals at vertices and the normals of incident triangles (measured both isotropically and anisotropically). 1 Introduction Many applications in science and engineering need to mesh a smooth closed surface Σ for numerical simu- lations, prototyping, rendering and various other pur- poses. A variety of algorithms and software systems based on them have been developed for the problem by the mesh generation, computer graphics, and image pro- cessing communities, e.g. [4, 12, 15, 17, 19, 21]. A subset of them focus on implicit surfaces, e.g. [4, 15, 17, 19]. An implicit surface representation is appealing for the relative ease in combining surfaces, computing line- surface intersection, and carrying out the inside/outside test. In this representation, there is an implicit function E : R 3 → R and Σ is its zero-set E(x) = 0. The surface Σ can have arbitrary genus. Despite the experimental success, it is not until re- cently that some algorithms have been designed with a guarantee about output quality: specifically, the out- ∗ Supported by the Research Grant Council, Hong Kong, China (HKUST6181/04E). Department of Computer Science, HKUST, Hong Kong. Email: scheng@cs.ust.hk. † Supported by NSF CCR-0430735 and NSF DMS-0310642. Department of Computer Science and Engineering, Ohio State University, USA. Email: tamaldey@cse.ohio-state.edu. ‡ Department of Computer Science, University of Illinois at Urbana-Champaign, USA. Email: eramosn@cs.uiuc.edu. § Department of Computer Science and Engineering, Ohio State University, USA. Email: wenger@cse.ohio-state.edu. put triangulation being homeomorphic to the input sur- face Σ and being geometrically close to it. Cheng, Dey, Edelsbrunner, and Sullivan [10] designed a meshing al- gorithm for skin surfaces in molecular modeling. Bois- sonnat and Oudot [6] and Cheng, Dey, Ramos, and Ray [11] proposed algorithms for general implicit sur- faces. Although both algorithms sample points and compute a restricted Delaunay triangulation, they dif- fer in many ways. In particular, a lower bound estima- tion of the local feature size is needed in [6] where crit- ical point computations replacing local feature size are required in [11]. Boissonnat, Cohen-Steiner, and Vet- ger [7] designed an algorithm to approximate the level sets of the implicit function, which yields an isotopic surface triangulation. The above certified meshing algorithms are de- signed for the isotropic setting in which nearly equi- lateral triangles are used. Sometimes, long and skinny triangles offer better approximation because they adapt to the principal curvatures of the surface. They are also preferred in numerical simulations where the phys- ical phenomena are strongly directional. These applica- tions call for an anisotropic triangulation. The general formulation is to associate a 2D metric tensor M x ,a 2 ×2 positive definite symmetric matrix, with each point x ∈ Σ. Its eigenvalues 0 <k x,1 ≤ k x,2 and correspond- ing orthogonal unit eigenvectors d x,1 and d x,2 specify the following deformation on the tangent plane at x. Any point z on the tangent plane is mapped to the point x+〈 z -x, d x,1 〉·  k x,1 d x,1 +〈 z -x, d x,2 〉·  k x,2 d x,2 . So the distance components along d x,1 and d x,2 are scaled by factors  k x,1 and  k x,2 , respectively. We call  k x,1 and  k x,2 the scaling factors and d x,1 and d x,2 the prin- cipal directions. The points on the tangent plane at dis- tance 1 from x (measured using M x ) lie on an ellipse centered at x with d x,1 as the major axis. To adapt to M x , the surface mesh triangles near x should be elon- gated in direction d x,1 . Few theoretical results are known for the anisotropic triangulation problem. Algorithms were obtained re- cently for the problem of computing a 2D conformal anisotropic triangulation of a planar straight line graph with no angle much less π/2 [8, 16]. The definition of anisotropic Voronoi diagram originates from the work of Labelle and Shewchuk [16]. A surface associated with This is the Pre-Published Version