Discrete & Computational Geometry

, Volume 42, Issue 1, pp 94–130

# The Voronoi Diagram of Three Lines

• Hazel Everett
• Daniel Lazard
• Sylvain Lazard
• Mohab Safey El Din
Article

## Abstract

We give a complete description of the Voronoi diagram, in ℝ3, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four unbounded branches of either a nonsingular quartic or of a nonsingular cubic and a line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. We introduce a proof technique which relies heavily upon modern tools of computer algebra and is of interest in its own right.

This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semi-algebraic tests for separating the two connected components of each two-dimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semi-algebraic tests.

## Keywords

Computational geometry Computer algebra Voronoi diagram Medial axis Quadric surface intersection

## References

1. 1.
Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. Parts I and II. Ill. J. Math. 21, 429–567 (1977)
2. 2.
Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, Chap. 5, pp. 201–290. Elsevier, Amsterdam (2000)
3. 3.
Berberich, E., Hemmer, M., Kettner, L., Schömer, E., Wolpert, N.: An exact, complete and efficient implementation for computing planar maps of quadric intersection curves. In: Proceedings of the 21st ACM Annual Symposium on Computational Geometry (SoCG’05), pp. 99–115 (2005) Google Scholar
4. 4.
Boissonnat, J.-D., Devillers, O., Pion, S., Teillaud, M., Yvinec, M.: Triangulations in CGAL. Comput. Geom. Theory Appl. 22, 5–19 (2002)
5. 5.
Boissonnat, J.-D., Wormser, C., Yvinec, M.: Curved Voronoi diagrams. In: Boissonnat, J.-D., Teillaud, M. (eds.) Effective Computational Geometry for Curves and Surfaces. Mathematics and Visualization, pp. 67–116. Springer, Berlin (2006)
6. 6.
Borcea, C., Goaoc, X., Lazard, S., Petitjean, S.: Common tangents to spheres in ℝ3. Discrete Comput. Geom. 35(2), 287–300 (2006)
7. 7.
Cheng, J., Lazard, S., Peñaranda, L., Pouget, M., Rouillier, F., Tsigaridas, E.: On the topology of planar algebraic curves. In: Proceedings of the 25th ACM Annual Symposium on Computational Geometry (SoCG’09) (2009) Google Scholar
8. 8.
Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry Theorems. World Scientific, Singapore (1994)
9. 9.
Collins, G.E., Johnson, J.R., Krandick, W.: Interval arithmetic in cylindrical algebraic decomposition. J. Symb. Comput. 34(2), 145–157 (2002)
10. 10.
Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, 2nd edn. Graduate Texts in Mathematics, vol. 185. Springer, New York (2005)
11. 11.
Culver, T.: Computing the medial axis of a polyhedron reliably and efficiently. Ph.D. thesis, University of North Carolina at Chapel Hill (2000) Google Scholar
12. 12.
Dey, T.K., Zhao, W.: Approximate medial axis as a Voronoi subcomplex. In: SMA ’02: Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications, pp. 356–366. ACM, New York (2002)
13. 13.
Dupont, L., Lazard, D., Lazard, S., Petitjean, S.: Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm. J. Symb. Comput. 43(3), 168–191 (2008)
14. 14.
Dupont, L., Lazard, D., Lazard, S., Petitjean, S.: Near-optimal parameterization of the intersection of quadrics: II. A classification of pencils. J. Symb. Comput. 43(3), 192–215 (2008)
15. 15.
Dupont, L., Lazard, D., Lazard, S., Petitjean, S.: Near-optimal parameterization of the intersection of quadrics: III. Parameterizing singular intersections. J. Symb. Comput. 43(3), 216–232 (2008)
16. 16.
Emiris, I., Tsigaridas, E., Tzoumas, G.: Predicates for the exact Voronoi diagram of ellipses under the Euclidean metric. Int. J. Comput. Geom. Appl. 18(6), 567–597 (2008) (Special Issue on SoCG’06)
17. 17.
Etzion, M., Rappoport, A.: Computing Voronoi skeletons of a 3-d polyhedron by space subdivision. Comput. Geom. Theory Appl. 21(3), 87–120 (2002)
18. 18.
Everett, H., Lazard, D., Lazard, S., Safey El Din, M.: The Voronoi diagram of three lines in ℝ3. In: Proceedings of the 23rd ACM Annual Symposium on Computational Geometry (SoCG’07), S. Korea, pp. 255–264 (2007) Google Scholar
19. 19.
Faugère, J.-C.: FGb—A software for computing Gröbner bases. http://fgbrs.lip6.fr
20. 20.
Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Handbook of Discrete and Computational Geometry, pp. 377–388. CRC, Boca Raton (1997) Google Scholar
21. 21.
Hoff, K., Culver, T., Keyser, J., Lin, M., Manocha, D.: Fast computation of generalized Voronoi diagrams using graphics hardware. Comput. Graph. 33, 277–286 (1999) (Proceedings of ACM SIGGRAPH 1999, Annual Conference Series) Google Scholar
22. 22.
Karavelas, M.I.: A robust and efficient implementation for the segment Voronoi diagram. In: International Symposium on Voronoi Diagrams in Science and Engineering, pp. 51–62 (2004) Google Scholar
23. 23.
Keyser, J., Krishnan, S., Manocha, D.: Efficient and accurate B-Rep generation of low degree sculptured solids using exact arithmetic: I Representations, II Computation. Comput. Aided Geom. Des. 16(9), 841–859, 861–882 (1999)
24. 24.
Koltun, V., Sharir, M.: Three dimensional Euclidean Voronoi diagrams of lines with a fixed number of orientations. SIAM J. Comput. 32(3), 616–642 (2003)
25. 25.
Kurdyka, K., Orro, P., Simon, S.: Semialgebraic Sard theorem for generalized critical values. J. Differ. Geom. 56, 67–92 (2000)
26. 26.
LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006) (Also available at http://planning.cs.uiuc.edu/)
27. 27.
Lax, P.: On the discriminant of real symmetric matrices. Commun. Pure Appl. Math. 51(11–12), 1387–1396 (1998)
28. 28.
Lazard, D.: On the representation of rigid-body motions and its application to generalized platform manipulators. In: Angeles, J., Hommel, G., Kovàcs, P. (eds.) Computational Kinematics. Solid Mechanics and Its Applications, vol. 28, pp. 175–182. Kluwer, Dordrecht (1993) Google Scholar
29. 29.
Lazard, D.: On the specification for solvers of polynomial systems. In: 5th Asian Symposium on Computers Mathematics—ASCM 2001. Lecture Notes Series in Computing, vol. 9, pp. 66–75. World Scientific, Singapore (2001) Google Scholar
30. 30.
Levin, J.: A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces. Commun. ACM 19(10), 555–563 (1976)
31. 31.
Maple System: Waterloo Maple Software. http://www.maplesoft.com
32. 32.
Milenkovic, V.J.: Robust construction of the Voronoi diagram of a polyhedron. In: Proceedings of the 5th Canadian Conference on Computational Geometry (CCCG’93), pp. 473–478 (1993) Google Scholar
33. 33.
Mourrain, B., Técourt, J.-P., Teillaud, M.: On the computation of an arrangement of quadrics in 3D. Comput. Geom. Theory Appl. 30(2), 145–164 (2005) (Special issue, 19th European Workshop on Computational Geometry)
34. 34.
Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM 55(2), 1–29 (2008)
35. 35.
Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations—Concepts and Applications of Voronoi Diagrams, 2nd edn. Wiley, New York (2000)
36. 36.
Parillo, P., Papachristodoulou, A., Prajna, S., Seiler, P.: SOSTOOLS—A MATLAB toolbox for sums of squares optimization programs. http://www.cds.caltech.edu/sostools/
37. 37.
Robertson, N., Sanders, D.P., Seymour, P.D., Thomas, R.: The four colour theorem. J. Comb. Theory Ser. B 70, 2–44 (1997)
38. 38.
Rouillier, F., Roy, M.-F., Safey El Din, M.: Finding at least one point in each connected component of a real algebraic set defined by a single equation. J. Complex. 16, 716–750 (2000)
39. 39.
Safey El Din, M.: Testing sign conditions on a multivariate polynomial and applications. Math. Comput. Sci. 1(1), 177–207 (2007)
40. 40.
Safey El Din, M.: RAG’Lib—A library for real algebraic geometry. http://www-calfor.lip6.fr/~safey/RAGLib/
41. 41.
Safey El Din, M., Schost, E.: Polar varieties and computation of at least one point in each connected component of a smooth real algebraic set. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC’03), pp. 224–231. ACM, Philadelphia (2003) Google Scholar
42. 42.
Safey El Din, M., Schost, E.: Properness defects of projections and computation of one point in each connected component of a real algebraic set. Discrete Comput. Geom. 32(3), 417–430 (2004)
43. 43.
Schömer, E., Wolpert, N.: An exact and efficient approach for computing a cell in an arrangement of quadrics. Comput. Geom. Theory Appl. 33(1–2), 65–97 (2006) (Special Issue on Robust Geometric Algorithms and Their Implementations)
44. 44.
Schwartz, J.T., Sharir, M.: On the “piano movers” problem: V. The case of a rod moving in three-dimensional space amidst polyhedral obstacles. Commun. Pure Appl. Math. 37, 815–848 (1984)
45. 45.
Schwarzkopf, O., Sharir, M.: Vertical decomposition of a single cell in a three-dimensional arrangement of surfaces and its applications. Discrete Comput. Geom. 18, 269–288 (1997)
46. 46.
Segre, C.: Studio sulle quadriche in uno spazio lineare ad un numero qualunque di dimensioni. Mem. della R. Acc. delle Scienze di Torino 36(2), 3–86 (1883) Google Scholar
47. 47.
Sharir, M.: Almost tight upper bounds for lower envelopes in higher dimensions. Discrete Comput. Geom. 12, 327–345 (1994)
48. 48.
Teichmann, M., Teller, S.: Polygonal approximation of Voronoi diagrams of a set of triangles in three dimensions. Technical report 766, Laboratory of Computer Science, MIT (1997) Google Scholar
49. 49.
Viro (Drobotukhina), J., Viro, O.: Configurations of skew lines. Leningrad Math. J. 1(4), 1027–1050 (1990) (Revised version in English: arXiv:math/0611374)

## Authors and Affiliations

• Hazel Everett
• 1
• Daniel Lazard
• 2
• Sylvain Lazard
• 1
Email author
• Mohab Safey El Din
• 2
1. 1.LORIA, INRIA LorraineUniversity Nancy 2NancyFrance
2. 2.LIP6, INRIA RocquencourtUniversity Pierre et Marie CurieParisFrance