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lecnotes14_fixed - 13.472J/1.128J/2.158J/16.940J...

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Unformatted text preview: 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lectures 14 and 15 Prof. N. M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright c 2003 Massachusetts Institute of Technology Contents Constructive Solid Geometry (CSG) 2 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 14.2 Primitives of CSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 14.3 Boolean operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 14.3.1 Regularized Boolean operators . . . . . . . . . . . . . . . . . . . . . . . 4 14.4 Set membership classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 14.5 Properties of CSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Boundary Representation 8 14.6 Two-manifold B-rep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 14.6.1 Information contained in a B-rep . . . . . . . . . . . . . . . . . . . . . . 10 14.6.2 Characteristics of domain for two-manifold solid object representations . 12 14.6.3 Euler-Poincar´ e equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 14.6.4 Sufficiency of a geometric modeling representation . . . . . . . . . . . . 16 14.6.5 Boundary representation model . . . . . . . . . . . . . . . . . . . . . . . 16 14.7 Data structures for manifold representations . . . . . . . . . . . . . . . . . . . . 16 14.7.1 Winged-edge data structure . . . . . . . . . . . . . . . . . . . . . . . . . 18 14.7.2 Vertex-edge data structure (V-E) . . . . . . . . . . . . . . . . . . . . . . 20 14.7.3 Face-edge data structure (FE) . . . . . . . . . . . . . . . . . . . . . . . 21 14.8 Operators for manipulating manifold topologies . . . . . . . . . . . . . . . . . . 21 14.8.1 Basic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 14.8.2 Building high level functions on the Euler operators . . . . . . . . . . . 28 14.9 Non Two-Manifold B-rep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 14.9.1 Topological elements in NTM topologies . . . . . . . . . . . . . . . . . . 29 14.9.2 Topological sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 14.10Radial edge data structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Bibliography 37 1 Constructive Solid Geometry (CSG) 14.1 Introduction CSG is a method in which an object is constructed from the standard primitives using regu- larized Boolean operations. The model is represented in the data structure as a CSG tree, see Figure 14.1, whose terminal nodes are primitives and non-terminal nodes are Boolean operators ( intersection, union and difference). Primitives are sized, positioned and oriented first....
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