Computational Geometry

Curve and Surface Modeling

1st Edition - September 28, 1989
Authors: Su Bu-qing, Liu Ding-yuan
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 7 2 2 8 - 3

Computational Geometry: Curve and Surface Modeling provides information pertinent to the fundamental aspects of computational geometry. This book discusses the geometric properties… Read more

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Computational Geometry: Curve and Surface Modeling provides information pertinent to the fundamental aspects of computational geometry. This book discusses the geometric properties of parametric polynomial curves by using the theory of affine invariants for algebraic curves. Organized into eight chapters, this book begins with an overview of the objects studies in computational geometry, namely surfaces and curves. This text then explores the developments in the theory and application of spline functions, which began with cubic spline functions. Other chapters consider the mechanical background of the cubic spline functions, which is the wooden spline with small deflection. This book discusses as well that in mathematical lofting the information of a geometric shape is given by a set of data points, while in geometric design other ways of representations are available. The final chapter deals with the concepts in the theory of algebraic curves. This book is a valuable resource for mathematicians.

Preface to the English EditionChapter I Introduction 1 What is Computational Geometry? 2 Fitting and Fairing in Curves and Surfaces 3 Fitting and Fairing in Curves with Large Deflection 4 The Bézier Curve and Its Extensions 5 Bicubic Spline Functions and Their Applications to Surface Fairing 6 Intrinsic Affine Invariants of Parametric Curves in Affine HyperspaceChapter II Spline Functions 1 Cubic Spline Functions 2 Quadratic Spline Functions 3 The Spline Function in Tension and the Convexity Preserving PropertyChapter III Parametric Cubic Spline Curves 1 Background and Developments 2 Parametric Cubic Curves and the Related Affine Invariants 3 Necessary and Sufficient Conditions for Inflection Points 4 A Theorem Concerning Segments of Parametric Cubic Curves 5 Extending (λ,μ) to the Whole Plane 6 Parametric Cubic Spline Curves of Cumulative Chord Length 7 Composite Curves of Parametric Cubic SegmentsChapter IV Bézier Curves and B-Spline Curves 1 Background 2 Bézier Curves 3 B-Spline CurvesChapter V Spline Surfaces 1 Bicubic Spline Functions 2 Coons Patches 3 Bézier Patches 4 B-Spline PatchesChapter VI Nonlinear Splines 1 Geometric Spline Curves 2 Local Cubic Spline Curves 3 Mechanical Spline Curves 4 Biarc Interpolation 5 Interpolation by Pairs of Quadratic Curves 6 Arc Splines 7 Local Spline Curves in Tension 8 Determination of the Tangent Line at a Data PointChapter VII Curve and Net Fairing 1 Conditions for Fairing 2 Curve Fairing 3 Net Fairing 4 Boundary Conditions for FairingChapter VIII The Intrinsic Affine Invariants of Parametric Curves in Affine Hyperspace 1 Concepts and Conclusions in the Theory of Algebraic Curves 2 Certain Quintic Rational Integral Curves 3 Some Relative Affine Invariants of Rational Integral Curves of Degree n 4 The Intrinsic Affine Invariants of Parametric Curves in an Affine HyperspaceReferencesAdditional ReferencesIndex