Chapter 4: Geometric Modeling
نویسندگان
چکیده
2 Smoothing Polyhedra 13 2.1 C1 Continuity and Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Necessary and Sufficiency Conditions . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Compatibility and Non-Singularity Constraints . . . . . . . . . . . . . . . . . 14 2.2 Polyhedra Smoothing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Wireframe Construction for Implicit Algebraic Splines . . . . . . . . . . . . . . . . . 17 2.3.1 Choice of Vertex Normals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.2 Generation of a Conic Wireframe . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Assigning Normals along Edge curves . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Local Interpolatory Patch Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 Conditions for Low Degree Interpolants . . . . . . . . . . . . . . . . . . . . . 20 2.4.2 C1 Interpolation of a Conic Wireframe . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Surface Selection and Local Shape Control . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5.1 Solution of Interpolation and Least-Squares Matrices . . . . . . . . . . . . . . 22 2.5.2 Display of the Triangular Algebraic Patch . . . . . . . . . . . . . . . . . . . . 24 2.5.3 Smoothing a Convex Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Normals and the Simplicial Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Construction of a C1 Interpolatory Surface using Cubic A-Patches . . . . . . . . . . 31 2.7.1 The Construction of a Piecewise C1 Cubic Function . . . . . . . . . . . . . . 31 2.7.2 The Solvability of the Related System . . . . . . . . . . . . . . . . . . . . . . 35 2.8 Construction of Single Sheeted A-Patches . . . . . . . . . . . . . . . . . . . . . . . . 35 2.9 Shape Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.10 Curvilinear Patch Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.10.1 Constructions of Wire Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.10.2 Parametric Surface Patches Interpolation . . . . . . . . . . . . . . . . . . . . 43 2.10.3 G0 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.11 C1 Modeling with Hybrid Multiple-Sided A-patches . . . . . . . . . . . . . . . . . . 47 2.11.1 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.11.2 Finite Element Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.11.3 C1 Modeling of Surface by Rational A-patches . . . . . . . . . . . . . . . . . 51 2.11.4 Evaluate the Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.12 Adaptive Model Reconstruction by Triangular Prism A-Patches . . . . . . . . . . . . 58 2.12.1 Construction of the Triangular Surface Patches . . . . . . . . . . . . . . . . . 60
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