Chapter 6 - B-spline surfaces

Abstract

The natural extension of the Bézier surface is the Cartesian product B-spline surface. This chapter analyzes nonrational B-spline surfaces. Nonrational B-spline surfaces are useful for describing sculptured surfaces, such as automobile bodies, aircraft, ships, or any surface where the fairness or smoothness of the surface is a design requirement. Nonrational B-spline surfaces are not quite as flexible as nonuniform rational B-spline surfaces (NURBS), which have additional degrees of freedom. Nonrational B-spline surfaces can only approximate the conic surfaces, whereas NURBS can reproduce them exactly. A nonrational Bézier surface is a special case of the nonrational B-spline surface. The shape and character of a B-spline surface is significantly influenced by knot vectors. Periodic B-spline surfaces are easily generated using periodic knot vectors to obtain periodic basis functions. The development of appropriate techniques for determining and visualizing the fairness or smoothness of surfaces is a fundamental concern in computer-aided design.