Floating Point Geometric Algorithms for Topologically Correct Scientific Visualization

The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations. Some emerging solutions based upon topological considerations will be presented. A novel geometric seeding algorithm for Newton’s method was used in experiments to determine feasible support for these visualization applications. 1 Computing the pipe surface radius Parametric curves have been shown to have a particular neighborhood whose boundary is non-self-intersecting [4]. It has also been shown that specified movements of the curve within this neighborhood preserve the topology of the curve [8, 7], as is desired in visualization. This neighborhood is defined by a single value, which is the radius of a pipe surface, where that radius depends on curvature and the minimum length over those line segments which are normal to the curve at both endpoints of the line segment [4]. The focus of this paper is efficient and accurate floating point techniques to compute that radius. ∗Department of Computer Science & Engineering, University of Connecticut, Storrs, CT 06269-2155, [email protected]. 1 Dagstuhl Seminar Proceedings 06021 Reliable Implementation of Real Number Algorithms: Theory and Practice http://drops.dagstuhl.de/opus/volltexte/2006/717 Definition 1.1 For a non-self-intersecting, parametric curve c, where c : [0, 1] → R, and for distinct values s, t ∈ [0, 1], the line segment [c(s), c(t)] is doubly normal if it is normal to c at both of the points c(s) and c(t). Definition 1.2 The global separation is the minimum over the lengths of all doubly normal segments. (For compact curves, this minimum has been shown in be positive [5].) An example cubic B-splines curve is given in Figure 1, with 1. control points: (0.0 0.0 0.0) (-1.0 1.0 0.0) (4.5 5.5 2.0) (5.0 -1.0 8.5) (-1.5 2.5 -4.5) (4.5 6.0 8.5) (3.5 -3.5 0.0) (0.0 0.0 0.0) and 2. knot vector: {0 0 0 0 0.2 0.4 0.6 0.8 1 1 1 1} For this curve, there exist many doubly normal segments, as shown in Figure 1. The problem is how to efficiently find all these doubly normal segments, Figure 1: Many doubly normal segments exist on this curve. and then find the pair which represents the global separation distance. A pair of distinct points at s and t on a parametric curve will be endpoints of a doubly normal segment if they satisfy the two equations [4]: [c(s) − c(t)] · c′(s) = 0 (1)