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This book explains the Beta-spline representation for curves and surfaces. This topic is important for the modeling of objects delineated by free-form curved surfaces. Such models are needed in the design and manufacture of three-dimensional parts, as well as in the generation of synthetic computer images. The Beta-spline technique follows from the earlier techniques of B zier and B-spline curves and surfaces, and forms the latest method in this field. The utility of this technique is the capability of representing complex curved surface objects using natural input specifications. The Beta-spline technique combines powerful and general mathematics with an intuitive specification method. Mathematical underpinnings are sufficiently flexible to allow the representation of a wide class of complex shapes.
At the same time, description of these shapes is achieved through the use of intuitive and meaningful specification parameters. In addition to a set of points in three-space, called control vertices, shape parameters are introduced which are parameters whose values indicate the shape of the object in terms of "sharpness" or "flatness".
List of contents
Contents: Introduction.- The Parametric Piecewise Representation.- The Application of Tension to a Curve.- Elementary Differential Geometry Concepts.- Fundamental Geometric Measures.- Geometric Continuity and Shape Parameters.- Derivation of the Beta-spline Curve Representation.- Curve Evaluation and Perturbation with Uniform Shape Parameters.- Generalizing to Continuous Shape Parameters for Curves.- Curve Evaluation and Perturbation with Continous Shape Parameters.- Classification and Analysis of Beta-spline Curve End Conditions.- Explanation of the Surface Representation.- Surface Evaluation and Perturbation with Uniform Shape Parameters.- Generalizing to Continuous Shape Parameters for Surfaces.- Surface Evaluation and Perturbation with Continuous Shape Parameters.- Classification and Analysis of Beta-spline Surface Boundary Conditions.- Geometrical Interpretation of the Shape Parameters.- Controlling Surfaces Using the Shape Parameters.- Synthetic Images of Beta-spline Objects.- Conclusion and Future Research Directions.- Appendix: REDUCE Programs.- References.- Bibliography on Curves and Surfaces.- Subject Index.
