Boolean Operations on 3D Selective Nef Complexes Data Structure, Algorithms, and Implementation Miguel Granados Peter Hachenberger Susan Hert Lutz Kettner Kurt Mehlhorn Michael Seel Max-Planck Institut f¨ ur Informatik, Saarbr¨ ucken 1 Introduction Partitions of three space into cells are a common theme of solid modeling and computational geometry. We re- strict ourselves to partitions induced by planes. A set of planes partitions space into cells of various dimen- sions. Each cell may carry a label. We call such a partition together with the labelling of its cells a selec- tive Nef complex (SNC). When the labels are boolean ( in out ) the complex describes a set, a so-called Nef polyhedron [Nef78]. Nef polyhedra can be obtained from halfspaces by boolean operations union, intersec- tion, and complement. Nef complexes slightly gener- alize Nef polyhedra through the use of a larger set of labels. Figure 1 shows a Nef polyhedron. Nef polyhedra and complexes are quite general. They can model non-manifold solids, unbounded solids, and objects comprising parts of different dimensionality. Is this generality needed? 1. Nef polyhedra are the smallest family of solids containing the half-spaces and being closed under boolean operations. In particular, boolean opera- tions may generate non-manifold solids (see, e.g., the symmetric difference of two cubes as shown in Figure 1) and lower dimensional features. The latter can be avoided by regularized boolean oper- ations. 2. We may want to model a box which is divided into two parts by a membrane. The membrane is to be modeled as a two-dimensional object without thickness. 3. Similarly, two-dimensional surfaces bound differ- ent material compositions in semi-conductor de- Email: mgranado@cad4.eafit.edu.co, [hachenb|hert|kettner|mehlhorn]@mpi- sb.mpg.de, michael.seel@sap.com Figure 1: A Nef polyhedron with non-manifold edges, a dangling facet, two isolated vertices, and an open boundary in the tunnel. sign. The different properties can be distinguish with the labels. In a three-dimensional earth model with different layers, reservoirs, faults, etc. one can use labels to distinguish between different soil types. Furthermore, in this application we en- counter complex topology, for example, non man- ifold edges. 4. In machine tooling we may want to generate a polyhedron Q by a cutting tool M. When the tool is placed at a point p in the plane all points in p M are removed. Observe, when the cutting tool is modeled as a closed polyhedron and moved along a path L (including its endpoints) an open polyhe- dron is generated. Thus open and closed polyhedra need to be modeled. The set of legal placements for M is the set C p; p M Q / 0 ; C may contain lower dimensional features as shown in Figure 2. This is one of the examples where Mid- dleditch [Mid94] argues that we need more than regularized boolean operations, i.e., here the need 1