This book presents conventions, mathematical concepts, definitions and aplications central to the geometric modeling of solids, shells and wires (engineering jargon) embedded in 3D space. Currently, software is available for geometric modeling of technical or natural scenarios. However, the models built frequently fail in supporting the subsequent engineering computing and simulations. Only after repeated iterations and re-desing, the models permit engineering / science simulations. This book endows the reader with the basic mathematical concepts needed to ensure that the geometric models indeed stand the stages of physics simulation, manufacturing, etc., and rapidly applies the concepts on engineering and science geometric models.
- Title
- Copyright
- Contents
- Dedication
- 1 Introduction
- 2 Mathematical Background
- 2.1 Definition. Bijective Functions and Isomorphisms
- 2.2 Definition. Metric or Distance Function
- 2.3 Definition. Open Ball
- 2.4 Definition. Open Set vs. Closed Set
- 2.5 Definition. Boundary, Interior, Regular Sets
- 2.6 Definition. Topology and Topological Space
- 2.7 Definition. Neighborhood
- 2.8 Definition. Neighborhood Topology
- 2.9 Definition. Disks and Half-disks S2 and S1
- 2.10 Definition. 2-Manifold in ℝ3
- 2.11 Definition. 2-Manifold with Border in ℝ3
- 2.12 Definition. 1-Manifold in ℝ3
- 2.13 Definition. 1-Manifold with Border in ℝ3
- 2.14 Examples of Non-manifold Topologies
- 2.15 Definition. Connected Set in ℝ3
- 3 B-Reps
- 3.1 Solid (BODY) in ℝ3
- 3.2 LUMP in ℝ3
- 3.3 SHELL
- 3.4 FACE
- 3.5 LOOP
- 3.6 EDGE
- 3.7 VERTEX
- 3.8 Example. Pre-Columbian Shape
- 3.9 Examples Non-Manifold
- 3.10 Example. Manifold Preservation in Parametric CAD
- 3.10.1 Questions on Boundary - Representation
- 3.11 Example. Repair of Extrusion of Non-manifold Sections.
- 3.11.1 Extrusion of Non-Manifold Cross Section
- 3.11.2 Manifold Repair
- 3.12 Exam Example. Repair of Non-manifold Pyramid
- 3.12.1 Context
- 3.12.2 B-Rep example
- 3.12.3 Solution
- 3.13 2D Elements
- 4 Triangular B-Reps
- 4.1 Definition. Convex Set
- 4.2 Definition. Convex Hull
- 4.3 Definition. k - simplices
- 4.3.1 Low Dimension Simplices
- 4.3.2 Inventory of i-Faces in a k-Simplex
- 4.3.3 Example. Simplicial Complex
- 4.4 Boundary Representation for Triangle-Based Meshes
- 5 CSG Modeling
- 5.1 Constructive Solid Geometry tree
- 5.2 Primitives operands
- 5.3 Unary Operations
- 5.4 Binary operators
- 5.4.1 Regularized Set Operations
- 5.5 Example CSG
- 5.5.1 Primitive operands
- 5.5.2 Unary operations
- 5.5.3 Binary operations
- 5.5.4 CSG Tree structure
- 5.6 Observations
- 6 Enumerations
- 6.1 Scalar Fields
- 6.2 Uniform Cell Decompositions
- 6.2.1 Fixed Grids
- 6.2.2 Medical Imaging
- 6.3 Non-Uniform Cell Decompositions. Quadtrees
- 6.3.1 Example Quadtree 3 Levels
- 6.3.2 Example Quadtree 4 Levels
- 6.3.3 Observations on Quadtrees
- 6.3.4 Quadtree Boolean Operations
- 6.3.5 Theoretical Discussion
- 6.3.6 Octrees
- 7 Acknowledgments
- Bibliography
- Backcover
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