Variational Surface Design and Surface Interrogation

The generation of technical smooth surfaces from a mesh of three-dimensional data points is an important problem in geometric modelling. In this publication we give a survey of some new techniques based on a calculus of variation approach. Apart from the pure construction of these surfaces, the analysis of their quality is equally important in the design and manufacturing process. Generalized focal surfaces are presented here as a new surface interrogation tool. 0 Introduction Curves and surfaces designed in a Computer Graphics environment have many applications, including the design of cars, air planes, shipbodies and modeling robots. The generation of "technical smooth" surfaces which are appropriate for the NC-process from a set of threedimensional data points is a key problem in the field of Computer Aided GeometricDesign. The fundamental idea for the described methods is the use of modeling tools which minimize a certain functional that can be interpretated in the sense of physics and/or geometry. In chapter 2 we deal with a variational design method for B-Spline-Surfaces. This concept is extended to NURBS-surfaces in chapter 3 . In chapter 4 we present generalized focal surfaces as a new tool for surface interrogation. 1 Fundamentals We keep this chapter short, because we assume that reader is familiar with the basic concepts of the Bezier and B-Spline techniques. The curves now known as Bezier curves and surfaces were independently developed by P. de Casteljau and by P. Bezier. The underlying mathematical theory, based on the concept of Bernstein polynomials, was first introduced by R. Forrest (see [For72]). The fundamental idea of this approach is to evaluate and manipulate the curves by a (small) number of "control C-448 H. Hagen et al. / Variational Surface Design and Surface Interrogation points”. A Bezier-curve is a segmented curve. The segments curve of degree m over the parameter interval are: of a BezierThe Bernstein polynomials are used as blending functions. This concept can be extended to NURBS (non-uniform rational B-splines). The fundamental idea of the rational Bezierand B-spline algorithms is to evaluate and manipulate the curves and surfaces by a (small) number of control points and so called weights. These weights are additional design parameters. First we give the fundamental concepts of rational curves, following [Far90]. A rational Bezier curve of degree n in is the projection of an nth degree Bezier curve in onto the hyperplane w = 1. are called (scalar) weights. We reparametrize a rational Bezier curve by changing the weights according to (see [Far90]) and get, after dividing all weights by the standard form This yields in the rational cubic case: A Bezier-surface is a segmented surface. The segments of a Bezier surface of degree m, n over the rectangular parameter domain are Instead of a control polygon a Bezier-surface(-segment) has a control polyhedron. The definition of a B-spline-surface over a rectangular parameter domain follows directly the same pattern: A rational Bezier-surface X(u,v) is definded by as the projection of a 4D-tensor product Bezier-surface. But it is common misconception to call ( 8 ) a tensor product surface itself. For more details about these basic concepts see [Far90] and [HL89]. H. Hagen et al. / Variational Surface Design and Surface Interrogation C-449 2 Variational design of B-spline surfaces In this section we leave the classical approach construct a smooth net of curves add the surface patches smoothly into the net and present a direct method to construct a technically smooth B-Spline spline surface, which uses only point data and refrains from determining a net. The construction algorithm combines a weighted least square approximation with automatic surface smoothing. The smoothing criterion is the approximate minimization of the curvature variation. This technique presented here aims at constructing tangent-plane continuous Bspline surfaces. The following mathematical model serves as variation principle: X(u, v) is the representation of the surface, is the parameter value and n , m are the number of segments in uand v-direction. are the points to be approximated and is the number of these points. The weight coefficients are valid in the interval [0,1] and fulfill the constraints 1. We apply this variation principle tu biquintic b-spline surfaces and with the knot-vectors