The chapter describes the type of curves and surfaces often used in modern CAD systems. The de facto industry standard here are NURBS which stands for Non Uniform Rational B-Splines. Even though the animation industry has largely switched to subdivision surfaces NURBS are still relevant: They are a very flexible tool for the representation of smooth surfaces, allow for exact representation of conic surfaces, and the CAD business has a lot of software and know-how pertaining to B-Splines. For further reading and proofs we refer to the vast literature on the subject.
Splines are piecewise polynomial or rational functions and B-splines or NURBS is a particular nice basis for a spline space. We describe knot insertions and the de Boor algorithm to evaluate a spline curve, we give formulas for the differentiation of spline curves, and describe how conic sections can be given as rational spline curves.
We then describe tensor product spline surfaces and how a quadratic surface can be given as a rational tensor product spline surface.
We end by illustrating how splines can be used to interpolate or approximate discrete data and few words on tessellation and trimming of spline surfaces.
- 2.Farin, G.: Curves and surfaces for computer-aided geometric design. In: Computer Science and Scientific Computing, 4th edn., Academic Press, San Diego (1997). A practical guide, Chap. 1 by P. Bézier; Chaps. 11 and 22 by W. Boehm, with 1 IBM-PC floppy disk (3.5 inch; HD) Google Scholar
- 3.Piegl, L., Tiller, W.: The NURBS Book. Springer, New York (1987) Google Scholar
- 9.Woo, M., Neider, J., Davis, T.: OpenGL Programming Guide, 2nd edn. Addison Wesley, Reading (1998) Google Scholar