Volume xx (200y), Number z, pp. 1–12 Implicit Hierarchical Quad-Dominant Meshes Daniele Panozzo † and Enrico Puppo ‡ DISI, University of Genoa, Genoa, Italy Abstract We present a method for producing quad-dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half-edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening.We show two possible applications of this method: we define an adaptive subdi- vision surface scheme that is topologically and geometrically consistent with the Catmull-Clark subdivision; and we present a remeshing method that produces semi-regular adaptive meshes. Categories and Subject Descriptors (according to ACM CCS): Computer Graphics [I.3.5]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations; Computer Graphics [I.3.6]: Methodology and Techniques—Graphics data structures and data types 1. Introduction Polygonal modeling is the main modeling paradigm for ap- plications that require computational intensive tasks other then rendering, such as video games and finite element meth- ods. In this context, quad-based meshes are often preferred to triangle-based ones, since they provide a more stable and better controllable framework for texturing, modeling and geometric computations. One notable property of quads is the possibility to be naturally aligned to anisotropic design features, as well as to line fields, or cross fields, such as those corresponding to principal curvatures [BZK09, DBG ∗ 06, KNP07, RLL ∗ 06]. A standard approach to polygonal modeling consists of starting from a coarse base mesh, which is then interac- tively edited and refined to model the features of the de- sired shape. One main goal is to obtain a meshe having a controlled budget of polygons, while being close to an ideal smooth surface. Mesh subdivision is often used to this pur- pose [MS01]. Non-trivial shapes may require adaptive sub- † e-mail: panozzo@disi.unige.it ‡ e-mail: puppo@disi.unige.it division to model different parts: tiny but relevant features will require a much finer mesh than large uniform areas. But subdivision is generally meant as a global process, while adaptive refinement of quad meshes is non trivial: local re- finement of quads produces non-quad faces and this process, if performed in an uncontrolled manner, can soon destroy the regular structure of a mesh. On the other hand, several authors have remarked that quad-dominant meshes containing a small amount of non- quad elements can be more flexible and more effective than purely quad meshes in capturing surface features, and they may enrich the design space [MNP08, SL03]. For instance, triangular and pentagonal elements can be used to collapse, split and merge lineal features and line fields, as well as to model the surface in the proximity of singularities. In this paper, we propose a method to adaptively refine a quad mesh through local operators for both mesh refinement and mesh coarsening (see Figures 10 and 1). Our method generates an implicit hierarchy of adaptively refined quad- dominant meshes, each containing a small amount of trian- gular and pentagonal transition elements. The adaptive sub- division patterns preserve the surface flows and lineal fea- submitted to COMPUTER GRAPHICS Forum (12/2010).