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Computer graphics, computer-aided design, and computer-aided manufacturing are tools that have become indispensable to a wide array of activities in contemporary society. Euclidean processing provides the basis for these computer-aided design systems although it contains elements that inevitably lead to an inaccurate, non-robust, and complex system. The primary cause of the deficiencies of Euclidean processing is the division operation, which becomes necessary if an n-space problem is to be processed in n-space. The difficulties that accompany the division operation may be avoided if processing is conducted entirely in (n+1)-space. The paradigm attained through the logical extension of this approach, totally four-dimensional processing, is the subject of this book. This book offers a new system of geometric processing techniques that attain accurate, robust, and compact computations, and allow the construction of a systematically structured CAD system.
List of contents
Preface.- I : Problem Definitions: 1. Problems of Euclidean Approach; 2. Homogeneous Coordinates; 3. Classical Projective Space; 4. Two-Sided Space; 5. Processing in Projective Spaces; 6. Transformations.- II : Presentation of a Solution - Four-Dimensional Processing: 7. A Solution; 8. Geometric Definitions in General; 9. Geometric Element Definitions; 10. Homogeneous Curves; 11. Homogeneous Surfaces; 12. Basic Intersections; 13. 4 Determinant Methods; 14. The Homogeneous Determinant Method; 15. Geometric Newton-Raphson Method; 16. Exact Arithmetic; 17. Fundamentals of Topology; 18. Data Structures; 19. Euclidean and Affine; 20. Set Operations; 21. Polyhedron Subdivisions; 22. Reconstruction of 3-D Points; 23. Display Process; 24. Classical or Two-Sided?- III : Test Results: 25. Various Tests.- IV : Conclusions: 26. Evaluations; 27. Role Division Operations; 28. Various Interpretations.- Afterword.- Bibliography.- Appendix: A: Fundamentals on Euclidean Processing; B: Various Geometries; C: Plane Determinant Method; D: Analytical Geometry in Vector Space.- Index.
Summary
Computer graphics, computer-aided design, and computer-aided manufacturing are tools that have become indispensable to a wide array of activities in contemporary society. Euclidean processing provides the basis for these computer-aided design systems although it contains elements that inevitably lead to an inaccurate, non-robust, and complex system. The primary cause of the deficiencies of Euclidean processing is the division operation, which becomes necessary if an n-space problem is to be processed in n-space. The difficulties that accompany the division operation may be avoided if processing is conducted entirely in (n+1)-space. The paradigm attained through the logical extension of this approach, totally four-dimensional processing, is the subject of this book. This book offers a new system of geometric processing techniques that attain accurate, robust, and compact computations, and allow the construction of a systematically structured CAD system.
