Computer Graphics and Geometric Modeling Using Beta-splines

Front Matter....Pages I-IX
Introduction....Pages 1-2
The Parametric Piecewise Representation....Pages 3-4
The Application of Tension to a Curve....Pages 5-9
Elementary Differential Geometry Concepts....Pages 11-19
Fundamental Geometric Measures....Pages 21-25
Geometric Continuity and Shape Parameters....Pages 27-28
Derivation of the Beta-spline Curve Representation....Pages 29-41
Curve Evaluation and Perturbation with Uniform Shape Parameters....Pages 43-49
Generalizing to Continuous Shape Parameters for Curves....Pages 51-52
Curve Evaluation and Perturbation with Continuous Shape Parameters....Pages 53-58
Classification and Analysis of Beta-spline Curve End Conditions....Pages 59-68
Explanation of the Surface Representation....Pages 69-69
Surface Evaluation and Perturbation with Uniform Shape Parameters....Pages 71-79
Generalizing to Continuous Shape Parameters for Surfaces....Pages 81-83
Surface Evaluation and Perturbation with Continuous Shape Parameters....Pages 85-88
Classification and Analysis of Beta-spline Surface Boundary Conditions....Pages 89-98
Geometrical Interpretation of the Shape Parameters....Pages 99-107
Controlling Surfaces Using the Shape Parameters....Pages 109-117
Synthetic Images of Beta-spline Objects....Pages 119-123
Conclusion and Future Research Directions....Pages 125-126
Back Matter....Pages 127-157