ACKNOWLEDGMENTS 99 Acknowledgments MANY PEOPLE provided considerable help to the author as these notes were being prepared, notably Eli Goodman, Leo Guibas, Ricky Pollack, Jfirgen Richter-Gebert, David Satesin, Raimund Seidel, Bernd Sturmfels, Frances Yao, and Gfinter Ziegler. Special thanks are also due to Phyllis Winkler, who transformed more than 150 pages of scribbled manuscript into a respectable-looking scientific document. Some of the research was done during a visit to the Institute of Systems Science at the University of Singapore; the work was completed during a visit to Institut Mittag-Lemer in Djursholm, Sweden. @ 100 AXIOMS AND HULLS Bibliography [1] E. al-Aamily, A. O. Morris, and M. H. Peel, "The representations of the Weyl groups of type Bn," Journal of Algebra 68 (1981), 298-305. Cited on page 17. [2] Cecilia R. Aragon and Raimund G. Seidel, "Randomized search trees" (ex- tended abstract), 30th IEEE Symposium on Foundations of Computer Science (1989), 540-546. Cited on page 53. [3] Marshall Bern, David Eppstein, Paul Plassman, and Frances Yao, "Horizon theorems for lines and polygons," in Discrete and Computational Geometry: Papers from the DIMACS Special Year, edited by Jacob E. Goodman, Richard Pollack, and William Steiger, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 6 (1991), 45-66. Cited on page 96. [4] Marshall Bern, personal communication, January 1991. Cited on page 97. [5] Anders BjSrner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and G/inter M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics Series, Cam- bridge University Press (1992). Cited on page 97. [6] Robert G. Bland and Michel Las Vergnas, "Orientability of matroids," Journal of Combinatorial Theory B24 (1978), 94-123. Cited on pages 40, 95, 96, and 98. [7] J. Bokowski, G. Laffaille, and J. Richter-Gebert, "10 point oriented matroids and projective incidence theorems," in preparation. Cited on page 97. [8] J/irgen Bokowski, Jfirgen Richter, and Bernd Sturmfels, "Nonrealizability proofs in computational geometry," Discrete ~: Computational Geometry 5 (1990), 333-350. Cited on page 6. [9] J/irgen Bokowski and Bernd Sturmfels, "On the coordinatization of oriented matroids," Discrete & Computational Geometry 1 (1986), 293-306. Cited on page 95. [10] J/irgen Bokowski and Bernd Sturmfels; "An infinite family of minor-minimal nonrealizable 3-chirotopes," Mathematische Zeitschrift 200 (1989), 583-589. Cited on page 96. [11] Bernard Chazelle, Leonidas J. Guibas, and D. T. Lee, "The power of geometric duality," BIT 25 (1985), 76-90. Cited on page 96. [12] Kenneth L. Clarkson and Peter W. Shor, "Applications of random sampling in computational geometry, II," Discrete & Computational Geometry 4 (1989), 387-421. Cited on page 81. [13] B. Delaunay, "Neue Darstellung der geometrischen Krystallographie," Zeit- schrift f//r Kristallographie 84 (1932), 109 149; errata, 85 (1933), 332. Cited on page 69. [14] Andreas Dress, "Chirotops and oriented matroids: Diskrete Strukturen, alge- braische Methoden und Anwendungen," Bayreuther Mathematische Schriften 21 (1986), 14-68. Cited on page 95. BIBLIOGRAPHY 101 [15] Andreas Dress, Andr6 Dreiding, and Hans Haegi, "Classification of mobile molecules by category theory," in Symmetries and Properties of Non-Rigid Molecules, Proceedings of an International Symposium in Paris, France, 1-7 July 1982, edited by J. Maruani and J. Serre; Studies in Physical and Theo- retical Chemistry 23 (1983), 39-58. Cited on page 95. [16] P.H. Edelman and C. Greene, "Balanced tableaux," Advances in Mathematics 63 (1987), 42-99. Cited on page 35. [17] Herbert Edelsbrunner and Ernst Peter M/icke, "Simulation of Simplicity: A technique to cope with degenerate cases in geometric algorithms," Fourth An- nual ACM Symposium on Computational Geometry (1988), 118-133. Cited on page 59. [18] H. Edelsbrunner, J. O'Rourke, and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications," SIAM Journal on Computing 15 (1986), 341- 363. Cited on page 96. [19] H. Edelsbrunner, R. Seidel, and M. Sharir, "On the zone theorem for hyper- plane arrangements," SIAM Journal of Computing, to appear. Preprint in New Results and New Trends in Computer Science, edited by Hermann Man- rer, Lecture Notes in Computer Science 555 (1991), 108 123. Cited on page 96. [20] Robert W Floyd, personal communication, February 1964. Cited on page 29. [21] Jon Folkman and Jim Lawrence, "Oriented matroids," Journal of Combinato- rim Theory B25 (1978), 199-236. Cited on pages 40, 43, and 96. [22] Steven Fortune, "Stable maintenance of point set triangulations in two dimen- sions," 30th IEEE Symposium on Foundations of Computer Science (1989), 494-499. Cited on pages 62 and 67. [23] Fred Galvin, personal communications, November 1991 and January 1992. Cited on page 15. [24] Michael R. Garey and David S. Johnson, Computers and Intractability (San Francisco: W. H. Freeman, 1979). Cited on page 20. [25] G6rard Gonzales-Sprinberg and Guy Laffaille, "Sur les arrangements simples de huit droites dans RP 2,'' Comptes Rendus de l'Acaddmie des Sciences, Sdrie I, 309 (1989), 341-344. Cited on page 97. [26] Jacob E. Goodman and Richard Pollack, "On the combinatorial classification of nondegenerate configurations in the plane," Journal of Combinatorial The- ory A29 (1980), 220-235. Cited on page 94. [27] Jacob E. Goodman and Richard Pollack, "Proof of Grilnbaum's conjecture on the stretchability of certain arrangements of pseudolines," Journal of Combi- natorial Theory A29 (1980), 385-390. Cited on pages 94 and 96. [28] Jacob E. Goodman and Richard Pollack, "A theorem of ordered duality," Ge- ometri~e Dedicata 12 (1982), 63-74. Cited on page 94. [29] Jacob E. Goodman and Richard Pollack, "Multidimensional sorting," SIAM Journal on Computing 12 (1983), 484-507. Cited on pages 46 and 94. 102 AXIOMS AND HULLS [30] Jacob E. Goodman and Richard Pollack, "Semispaces of configurations, cell complexes of arrangements," Journal of Combinatorial Theory A37 (1984), 257 293. Cited on pages 35 and 94. [31] Jacob E. Goodman and Richard Pollack, "Upper bounds for configurations and polytopes in Rd, '' Discrete 35 Computational Geometry 1 (1986), 219- 227. Cited on pages 40 and 96. [32] Jacob E. Goodman and Richard Pollack, "Allowable sequences and order types in discrete and computational geometry," New Trends in Discrete and Com- putational Geometry, edited by J. Pach (Springer-Verlag, 1992), to appear. Cited on page 94. [33] Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics (Reading, Mass.: Addison-Wesley, 1989). Cited on page 14. [34] Branko Griinbaum, Convex Polytopes (London Interscience, 1967). Cited on pages 94 and 98. [35] Branko Griinbaum, Arrangements and Spreads. Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, Volume 10 (Providence, RI: American Mathematical Society, 1972). Cited on pages 34 and 94. [36] Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir, "Randomized in- cremental construction of Delaunay and Voronoi diagrams," Algorithmica 7 (1992), 381 413. Abbreviated version in Automata, Languages and Program- ming, edited by M. S. Paterson, Lecture Notes in Computer Science 443 (1990), 414-431. Cited on pages 2, 3, 74, 77, and 80. [37] Leonidas Guibas, David Salesin, and Jorge Stolfi, "Constructing strongly con- vex approximate hulls with inaccurate primitives," Algorithmica, to appear. Abbreviated version in Proceedings of the International Symposium on Algo- rithms SIGAL 90, edited by T. Asano, T. Ibaraki, H. Imai, and T. Nishizeki, Lecture Notes in Computer Science 450 (1990), 261 270. Cited on page 67. [38] Leonidas Guibas and Jorge Stolfi, "Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams," ACM Transactions on Graphics 4 (1985), 74-123. Cited on pages v, 69, 72, and 97. [39] Lino Gutierrez Novoa, "On n-ordered sets and order completeness," Pacific Journal of Mathematics 15 (1965), 1337-1345. Cited on page 94. [40] Eric Richard Halsey, Zonotopal complexes on the d-cube, Ph.D. dissertation, University of Washington, Seattle, WA (1972). Cited on page 97. [41] Beat Jaggi, Peter Mani-Levitska, Bernd Sturmfels, and Neil White, "Uniform oriented matroids without the isotopy property," Discrete 35 Computational Geometry 4 (1989), 97-100. Cited on page 96. [42] J.W. Jaromczyk and G. W. Wasilkowski, "Numerical stability of a convex hull algorithm for simple polygons," University of Kentucky technical report 177-90 (1990), 18 pp. Cited on page 67. BIBLIOGRAPHY 103 [43] Arne Jonassen and Donald E. Knuth, "A trivial algorithm whose analysis isn't," Journal of Computer and System Sciences 16 (1978), 301-322. Cited on page 55. [44] Mikhail H. Klin, Serge S. Tratch, and Nikolai S. Zefirov, "2D-configurations and clique-cyclic orientations of the graphs L(Kp)," Reports in Molecular The- ory 1 (1990), 149-163. Cited on page 97. [45] Donald E. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching (Reading, MA: Addison-Wesley, 1973). Cited on pages 29 and 47. [46] Donald E. Knuth, "Two notes on notation," American Mathematical Monthly 99 (1992), 403-422. Cited on page 14. [47] Donald E. Knuth, The Stanford GraphBase, book in preparation. Cited on page 53. [48] Michel Las Vergnas, "Bases in oriented matroids," Journal of Combinatorial Theory B25 (1978), 283 289. Cited on pages 3, 40, and 95. [49] Michel Las Vergnas, "Convexity in oriented matroids," Jo,lrnal of Combinato- rial Theory B29 (1980), 231 243. Cited on page 96. [50] Alain Lascoux and Marcel-Panl Schiitzenberger, "Structure de Hopf de l'annean de cohomologie et de l'anneau de Grothendieck d'une vari6t~ de dra- peaux," Comptes Rendus des sdances de l'Acaddmie des Sciences, S~rie I, 295 (1982), 629 633. Cited on page 35. [51] Jim Lawrence, "Oriented matroids and multiply ordered sets," Linear Algebra and Its Applications 48 (1982), 1 12. Cited on pages 3 and 95. [52] F. Levi, "Die Teilung der projektiven Ebene durch Gerade oder Pseudo- gerade," Berichte fiber die Verhandlungen der ss Akademie der Wissenschaften, Leipzig, Mathematisch-physische Klasse 78 (1926), 256 267.