Arrangements

Effective Computational Geometry for Curves and Surfaces

Volumn , Issue , 2006, Pages 1-66

  Fogel, Efi ^a   Halperin, Dan ^a   Kettner, Lutz ^b   Teillaud, Monique ^c   Wein, Ron ^a   Wolpert, Nicola ^b  

^a TEL AVIV UNIVERSITY   (Israel)

^b MAX PLANCK INSTITUTE FOR INFORMATICS   (Germany)

^c INRIA   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84892282471     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-3-540-33259-6_1     Document Type: Chapter

References (23)

1. http://www-sop.inria.fr/prisme/ECG/
2. The original algorithm of Bentley and Ottmann only detects and reports the intersection points between the input segments. However, it can be easily augmented to compute the arrangement.
3. The work of Emiris et al. [145], which is based on different methods, reducing the comparison of algebraic numbers to computing signs of polynomial expressions, and thus does not require the manipulation of such number types, is not described in this section. We refer the reader to Chap. 3.
4. This is the number of points on C with a vertical tangent-see below. In degenerate cases, this is the number of singular points on C.
5. 2 + vy + w) = 0, so it is not difficult to distinguish between a vertical tangency point and a singular point.
6. For conic arcs we used a slightly different technique, computing the roots of the quadratic polynomial (1.2). However, using resultant calculus we obtain an equivalent polynomial (having the same roots).
7. For intersection points that involve a singular point we refer the reader to [139].
8. 2) time. Moreover, this was also verified by many experimental results.
9. The arrangement package of Cgal lets users choose between the two alternatives: (i) supplying an additional predicate, or (ii) resorting to the basic sweep-line methods (see [167] for more details). While the first option is usually more efficient, implementing the additional predicate may be a major endeavor in some cases; see for example Sect. 1.3.1.
10. Cgal prescribes the suffix 2 for all data structures of planar objects as a convention.
11. See [242, 285] for the theoretical background behind the implementation of the root operator.
12. For a detailed discussion of what constitute well-behaved surfaces or surface patches in the context of arrangements, see [15]. These are, for example, a collection of algebraic surface patches of bounded degree each bounded by at most some constant number of algebraic curves of bounded degree and each decomposed into a constant number of xy-monotone patches.
13. M. Goodrich, Personal communication
14. Although we have only discussed arrangements in Euclidean space so far in the chapter, arrangements are naturally defined on curved surfaces as well. For instance, a very useful type of arrangements is defined on the surface of a sphere-see Sect. 1.7 for an application of such arrangements.
15. In Cgal though, the implementation of algorithms for two-dimensional arrangements do not assume general position and handle degeneracies.
16. The exact size of Δ depends on the specific application of the perturbed arrangement.
17. It would have also been suitable to call it a separation bound, but we use resolution bound to avoid confusion with separation bounds of exact algebraic computing.
18. A variant of this problem has been studied under the name of red-blue segment intersection for the linear case, but we do not know of extensions for curves. For example, the algorithm in [248] (and also others) makes explicit use of the fact that the edges intersect at most once.
19. In the NC-machining literature, a configuration is often referred to as a contact location (CL) point, as the tool tip is typically in contact with the workpiece, and this is the only type of contact that we allow.
20. Given a set of planar curves, we can regard each x-monotone curve as the graph of a continuous univariate function defined on an interval of the x-axis, such that the lower envelope of the set is the point-wise minimum of these functions.
21. The silhouette is often referred to as the envelope. To avoid confusion with lower envelopes of finite sets of curves, we stick with the term silhouette.
22. The so-called smooth molecular surfaces are possibly self intersecting; see, e.g., [303].
23. The Voronoi diagram of polygons is a collection of line segments and parabolic arcs-a parabolic arc is the locus of points equidistant from a polygon vertex and an edge of another polygon.