A minimal and complete set of operators for the development of robust manifold mesh modelers

In this paper, we identify a minimal and complete set of fundamental operators, which is necessary and sufficient for performing all homeomorphic and topological operations on 2-manifold mesh structures. Efficient algorithms are developed for the implementation of these operators. We also developed a set of powerful, userfriendly, and effective operators at the level of user-interface. Using these operators, we have developed a prototype system for robust, interactive and user friendly modeling of orientable 2-manifold meshes. Users of our system can perform a large set of homeomorphic and topological changes with these user-interface level operators. Our system is topologically robust in the sense that users will never create invalid 2-manifold mesh structure with these operators. In our system, the homeomorphic and topological surgery operations can be applied alternatively on 2-manifold meshes. With our system,users can blend surfaces, construct rinds and open holes on these rind shapes. With our system, the shapes that look like solid, non-manifold, or 2-manifold with boundary can be manipulated. The system also provides automatic texture mapping during topology changes. ∗ Corresponding Author. Email: [email protected]. phone: +(979) 845-6599. fax: +(979) 845-4491. 1 Supported in part by Research Council of College of Architecture and Interdisciplinary Program of Texas A&M University. Preprint submitted to Elsevier Science 17 January 2003 1 Motivation and Introduction Orientable 2-manifold meshes [15,17] can describe functional 3D shapes with arbitrary topological structure (i.e. having any number of holes, handles and ”disconnected” surfaces) [16,14]. A 2-manifold mesh does not include unwanted mesh artifacts such as wrongly-oriented or missing polygons, cracks, and T-junctions [4,25]. These properties of 2-manifold shapes are essential for physical simulations. For instance, in order to simulate pouring water from a computer generated teapot shape, the artificial teapot must have a functional (not just apparent) spout. Similarly, for heat transfer, there should be no cracks, otherwise there can be an energy leak. In rendering, a model that has a crack or missing polygon can ruin the radiosity computation. In raytracing, a transparent shape with wrongly-oriented polygons and cracks can cause unwanted artifacts in the resulting image. Orientable 2-manifold meshes are also essential for subdivision schemes since most subdivision algorithms such as Catmull-Clark and Doo-Sabin require that the local topological region must correspond to a simple disk (topologically) [8,12,30]. This implies that the underlying structure must be a valid 2-manifold. Since the quality and topology of the smooth surface resulting from subdivision depends greatly on the initial control mesh, theoretical assurance of the quality of initial control meshes is extremely important. In other words, the process of obtaining the initial control mesh must be robust and guarantee a valid 2-manifold structure. This paper presents a prototype system for robust, interactive and user-friendly modeling of 2-manifold meshes based on topological graph theory. The main contributions of this paper is the following. • Minimal and complete set of operators. We have identified four operators that can create any 2-manifold mesh. These operators can be implemented very efficiently over our Dubly Linked Face List (DLFL) data structure [1]. Once these operators are implemented, the development of a 2-manifold modeling system with high-level operators becomes extremely easy. • User-Interface operators. We have also developed and implemented a set of user-friendly high-level operators based on the minimal operator set. By using these high-level operators, users can intuitively perform topological and homeomorphic surgery operations on a given 2-manifold mesh structure. • New design methods. One of the most important contributions of this paper is the introduction of two new design methods, rind modeling and surface blending. rind modeling method is useful to design shapes such as teapots. Surface blending helps to create shapes that resemble blobby shapes that are generally created by implicit methods.