Arrangement of lines (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Arrangement of lines" in English language version.

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Eppstein & Hart (1999). Eppstein, D.; Hart, D. (1999), "Shortest paths in an arrangement with $k$ line orientations", Proceedings of the 10th ACM–SIAM Symposium on Discrete Algorithms (SODA '99), pp. 310–316
Grünbaum (1972); Agarwal & Sharir (2002). Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society Agarwal, P. K.; Sharir, M. (2002), "Pseudo-line arrangements: duality, algorithms, and applications", Proc. 13th ACM-SIAM Symposium on Discrete Algorithms (SODA '02), San Francisco: Society for Industrial and Applied Mathematics, pp. 800–809

ams.org

mathscinet.ams.org

Ovchinnikov (2011), p. 210. Ovchinnikov, Sergei (2011), Graphs and Cubes, Universitext, New York: Springer, doi:10.1007/978-1-4614-0797-3, ISBN 978-1-4614-0796-6, MR 3014880
Halperin & Sharir (2018), p. 724. Halperin, D.; Sharir, M. (2018), "Arrangements", in Goodman, Jacob E.; O'Rourke, Joseph; Tóth, Csaba D. (eds.), Handbook of Discrete and Computational Geometry, Discrete Mathematics and its Applications (3rd ed.), Boca Raton, Florida: CRC Press, pp. 723–762, ISBN 978-1-4987-1139-5, MR 3793131
Bern et al. (1991); an unpublished manuscript of Rom Pinchasi from 2011 claims the slightly stronger bound $\lfloor 9.5n\rfloor -3$ . Bern, M. W.; Eppstein, D.; Plassman, P. E.; Yao, F. F. (1991), "Horizon theorems for lines and polygons", in Goodman, J. E.; Pollack, R.; Steiger, W. (eds.), Discrete and Computational Geometry: Papers from the DIMACS Special Year, DIMACS Ser. Discrete Math. and Theoretical Computer Science (6 ed.), Amer. Math. Soc., pp. 45–66, MR 1143288
Bern et al. (1991). Bern, M. W.; Eppstein, D.; Plassman, P. E.; Yao, F. F. (1991), "Horizon theorems for lines and polygons", in Goodman, J. E.; Pollack, R.; Steiger, W. (eds.), Discrete and Computational Geometry: Papers from the DIMACS Special Year, DIMACS Ser. Discrete Math. and Theoretical Computer Science (6 ed.), Amer. Math. Soc., pp. 45–66, MR 1143288
Dey (1998); Tóth (2001). The problem of bounding the complexity of k-levels was first studied by Lovász (1971) and Erdős et al. (1973). Dey, T. L. (1998), "Improved bounds for planar $k$ -sets and related problems", Discrete & Computational Geometry, 19 (3): 373–382, doi:10.1007/PL00009354, MR 1608878 Tóth, G. (2001), "Point sets with many $k$ -sets", Discrete & Computational Geometry, 26 (2): 187–194, doi:10.1007/s004540010022 Lovász, L. (1971), "On the number of halving lines", Annales Universitatis Scientiarum Budapestinensis de Rolando Eőtvős Nominatae Sectio Mathematica, 14: 107–108 Erdős, P.; Lovász, L.; Simmons, A.; Straus, E. G. (1973), "Dissection graphs of planar point sets", A Survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), Amsterdam: North-Holland, pp. 139–149, MR 0363986
Ajtai et al. (1982); Leighton (1983). Ajtai, M.; Chvátal, V.; Newborn, M.; Szemerédi, E. (1982), "Crossing-free subgraphs", Theory and Practice of Combinatorics, North-Holland Mathematics Studies, vol. 60, North-Holland, pp. 9–12, MR 0806962 Leighton, F. T. (1983), Complexity Issues in VLSI, Foundations of Computing Series, Cambridge, MA: MIT Press
Goodman & Pollack (1993), p. 109 Archived 2023-01-01 at the Wayback Machine: "The natural setting for arrangements of lines is the real projective plane" Goodman, Jacob E.; Pollack, Richard (1993), "Allowable sequences and order types in discrete and computational geometry", in Pach, János (ed.), New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 10, Berlin: Springer, pp. 103–134, doi:10.1007/978-3-642-58043-7_6, MR 1228041
Polster (1998), p. 223. Polster, Burkard (1998), A geometrical picture book, Universitext, Springer-Verlag, New York, doi:10.1007/978-1-4419-8526-2, ISBN 0-387-98437-2, MR 1640615
Goodman & Pollack (1993), p. 110. Goodman, Jacob E.; Pollack, Richard (1993), "Allowable sequences and order types in discrete and computational geometry", in Pach, János (ed.), New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 10, Berlin: Springer, pp. 103–134, doi:10.1007/978-3-642-58043-7_6, MR 1228041
This is the earliest proof cited by Borwein & Moser (1990), but they write that the same proof was likely given "much earlier by others". Borwein, P.; Moser, W. O. J. (1990), "A survey of Sylvester's problem and its generalizations", Aequationes Mathematicae, 40 (1): 111–135, doi:10.1007/BF02112289, MR 1069788, S2CID 122052678
Melchior (1940); Grünbaum (2009). Melchior, E. (1940), "Über Vielseite der projektiven Ebene", Deutsche Mathematik, 5: 461–475 Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
Grünbaum (1972); Grünbaum (2009). Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
Grünbaum (2009). Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
Eppstein (2006). Eppstein, D. (2006), "Cubic partial cubes from simplicial arrangements", Electronic Journal of Combinatorics, 13 (1, R79): 1–14, arXiv:math.CO/0510263, doi:10.37236/1105, MR 2255421, S2CID 8608953, archived from the original on 2012-02-14, retrieved 2008-12-14
Moreno & Prieto-Martínez (2021). Moreno, José Pedro; Prieto-Martínez, Luis Felipe (2021), "El problema de los triángulos de Kobon" [The Kobon triangles problem], La Gaceta de la Real Sociedad Matemática Española (in Spanish), 24 (1): 111–130, hdl:10486/705416, MR 4225268
Abramenko & Brown (2008). Abramenko, Peter; Brown, Kenneth S. (2008), Buildings: Theory and Applications, Graduate Texts in Mathematics, vol. 248, New York: Springer, doi:10.1007/978-0-387-78835-7, ISBN 978-0-387-78834-0, MR 2439729; see Example 10.14, pp. 519–520
Cole et al. (1989). Cole, Richard; Salowe, Jeffrey S.; Steiger, W. L.; Szemerédi, Endre (1989), "An optimal-time algorithm for slope selection", SIAM Journal on Computing, 18 (4): 792–810, doi:10.1137/0218055, MR 1004799
Goodman et al. (1994). Goodman, Jacob E.; Pollack, Richard; Wenger, Rephael; Zamfirescu, Tudor (1994), "Every arrangement extends to a spread", Combinatorica, 14 (3): 301–306, doi:10.1007/BF01212978, MR 1305899, S2CID 42055590
Dress, Koolen & Moulton (2002). Dress, A.; Koolen, J. H.; Moulton, V. (2002), "On line arrangements in the hyperbolic plane", European Journal of Combinatorics, 23 (5): 549–557, doi:10.1006/eujc.2002.0582, MR 1931939
Martin (1996), pp. 41, 338. Martin, George E. (1996), The foundations of geometry and the non-Euclidean plane, Undergraduate Texts in Mathematics, Springer-Verlag, ISBN 0-387-90694-0, MR 1410263

archive.org

Aharoni et al. (1999). Aharoni, Y.; Halperin, D.; Hanniel, I.; Har-Peled, S.; Linhart, C. (1999), "On-line zone construction in arrangements of lines in the plane", in Vitter, Jeffrey S.; Zaroliagis, Christos D. (eds.), Algorithm Engineering: 3rd International Workshop, WAE'99, London, UK, July 19–21, 1999, Proceedings, Lecture Notes in Computer Science, vol. 1668, Springer-Verlag, pp. 139–153, CiteSeerX 10.1.1.35.7681, doi:10.1007/3-540-48318-7_13, ISBN 978-3-540-66427-7

arxiv.org

Eppstein (2006). Eppstein, D. (2006), "Cubic partial cubes from simplicial arrangements", Electronic Journal of Combinatorics, 13 (1, R79): 1–14, arXiv:math.CO/0510263, doi:10.37236/1105, MR 2255421, S2CID 8608953, archived from the original on 2012-02-14, retrieved 2008-12-14

books.google.com

Halperin & Sharir (2018), p. 724. Halperin, D.; Sharir, M. (2018), "Arrangements", in Goodman, Jacob E.; O'Rourke, Joseph; Tóth, Csaba D. (eds.), Handbook of Discrete and Computational Geometry, Discrete Mathematics and its Applications (3rd ed.), Boca Raton, Florida: CRC Press, pp. 723–762, ISBN 978-1-4987-1139-5, MR 3793131
Goodman & Pollack (1993), p. 109 Archived 2023-01-01 at the Wayback Machine: "The natural setting for arrangements of lines is the real projective plane" Goodman, Jacob E.; Pollack, Richard (1993), "Allowable sequences and order types in discrete and computational geometry", in Pach, János (ed.), New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 10, Berlin: Springer, pp. 103–134, doi:10.1007/978-3-642-58043-7_6, MR 1228041

combinatorics.org

Eppstein (2006). Eppstein, D. (2006), "Cubic partial cubes from simplicial arrangements", Electronic Journal of Combinatorics, 13 (1, R79): 1–14, arXiv:math.CO/0510263, doi:10.37236/1105, MR 2255421, S2CID 8608953, archived from the original on 2012-02-14, retrieved 2008-12-14

depaul.edu

ovid.cs.depaul.edu

Shor (1991); Schaefer (2010). Shor, P. W. (1991), "Stretchability of pseudolines is NP-hard", in Gritzmann, P.; Sturmfels, B. (eds.), Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, Providence, R.I.: American Mathematical Society, pp. 531–554 Schaefer, Marcus (2010), "Complexity of some geometric and topological problems" (PDF), Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers, Lecture Notes in Computer Science, vol. 5849, Springer-Verlag, pp. 334–344, doi:10.1007/978-3-642-11805-0_32, ISBN 978-3-642-11804-3, archived (PDF) from the original on 2021-06-26, retrieved 2011-02-25

doi.org

Ovchinnikov (2011), p. 210. Ovchinnikov, Sergei (2011), Graphs and Cubes, Universitext, New York: Springer, doi:10.1007/978-1-4614-0797-3, ISBN 978-1-4614-0796-6, MR 3014880
Steiner (1826); Agarwal & Sharir (2000). Steiner, J. (1826), "Einige Gesetze über die Theilung der Ebene und des Raumes", J. Reine Angew. Math., 1: 349–364, doi:10.1515/crll.1826.1.349, S2CID 120477563 Agarwal, P. K.; Sharir, M. (2000), "Arrangements and their applications" (PDF), in Sack, J.-R.; Urrutia, J. (eds.), Handbook of Computational Geometry, Elsevier, pp. 49–119, archived (PDF) from the original on 2021-04-11, retrieved 2018-11-11
Chazelle, Guibas & Lee (1985), Edelsbrunner (1987), Edelsbrunner, O'Rourke & Seidel (1986). Chazelle, B.; Guibas, L. J.; Lee, D. T. (1985), "The power of geometric duality", BIT Numerical Mathematics, 25 (1): 76–90, doi:10.1007/BF01934990, S2CID 122411548 Edelsbrunner, H. (1987), Algorithms in Combinatorial Geometry, EATCS Monographs in Theoretical Computer Science, Springer-Verlag, ISBN 978-3-540-13722-1 Edelsbrunner, H.; O'Rourke, J.; Seidel, R. (1986), "Constructing arrangements of lines and hyperplanes with applications", SIAM Journal on Computing, 15 (2): 341–363, doi:10.1137/0215024
Aronov, Matoušek & Sharir (1994). Aronov, B.; Matoušek, J.; Sharir, M. (1994), "On the sum of squares of cell complexities in hyperplane arrangements", Journal of Combinatorial Theory, Series A, 65 (2): 311–321, doi:10.1016/0097-3165(94)90027-2
Dey (1998); Tóth (2001). The problem of bounding the complexity of k-levels was first studied by Lovász (1971) and Erdős et al. (1973). Dey, T. L. (1998), "Improved bounds for planar $k$ -sets and related problems", Discrete & Computational Geometry, 19 (3): 373–382, doi:10.1007/PL00009354, MR 1608878 Tóth, G. (2001), "Point sets with many $k$ -sets", Discrete & Computational Geometry, 26 (2): 187–194, doi:10.1007/s004540010022 Lovász, L. (1971), "On the number of halving lines", Annales Universitatis Scientiarum Budapestinensis de Rolando Eőtvős Nominatae Sectio Mathematica, 14: 107–108 Erdős, P.; Lovász, L.; Simmons, A.; Straus, E. G. (1973), "Dissection graphs of planar point sets", A Survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), Amsterdam: North-Holland, pp. 139–149, MR 0363986
Alon & Győri (1986). Alon, N.; Győri, E. (1986), "The number of small semi-spaces of a finite set of points in the plane", Journal of Combinatorial Theory, Series A, 41: 154–157, doi:10.1016/0097-3165(86)90122-6
Balogh et al. (2004); see also Matoušek (1991). Balogh, J.; Regev, O.; Smyth, C.; Steiger, W.; Szegedy, M. (2004), "Long monotone paths in line arrangements", Discrete & Computational Geometry, 32 (2): 167–176, doi:10.1007/s00454-004-1119-1 Matoušek, J. (1991), "Lower bounds on the length of monotone paths in arrangements", Discrete & Computational Geometry, 6 (1): 129–134, doi:10.1007/BF02574679
Canham (1969); Clarkson et al. (1990). Canham, R. J. (1969), "A theorem on arrangements of lines in the plane", Israel Journal of Mathematics, 7 (4): 393–397, doi:10.1007/BF02788872, S2CID 123541779 Clarkson, K.; Edelsbrunner, H.; Guibas, L. J.; Sharir, M.; Welzl, E. (1990), "Combinatorial complexity bounds for arrangements of curves and spheres", Discrete & Computational Geometry, 5 (1): 99–160, doi:10.1007/BF02187783
Székely (1997). Székely, L. A. (1997), "Crossing numbers and hard Erdős problems in discrete geometry" (PDF), Combinatorics, Probability and Computing, 6 (3): 353–358, doi:10.1017/S0963548397002976, S2CID 36602807, archived (PDF) from the original on 2017-08-08, retrieved 2019-09-23
Goodman & Pollack (1993), p. 109 Archived 2023-01-01 at the Wayback Machine: "The natural setting for arrangements of lines is the real projective plane" Goodman, Jacob E.; Pollack, Richard (1993), "Allowable sequences and order types in discrete and computational geometry", in Pach, János (ed.), New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 10, Berlin: Springer, pp. 103–134, doi:10.1007/978-3-642-58043-7_6, MR 1228041
Polster (1998), p. 223. Polster, Burkard (1998), A geometrical picture book, Universitext, Springer-Verlag, New York, doi:10.1007/978-1-4419-8526-2, ISBN 0-387-98437-2, MR 1640615
Goodman & Pollack (1993), p. 110. Goodman, Jacob E.; Pollack, Richard (1993), "Allowable sequences and order types in discrete and computational geometry", in Pach, János (ed.), New Trends in Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 10, Berlin: Springer, pp. 103–134, doi:10.1007/978-3-642-58043-7_6, MR 1228041
This is the earliest proof cited by Borwein & Moser (1990), but they write that the same proof was likely given "much earlier by others". Borwein, P.; Moser, W. O. J. (1990), "A survey of Sylvester's problem and its generalizations", Aequationes Mathematicae, 40 (1): 111–135, doi:10.1007/BF02112289, MR 1069788, S2CID 122052678
Melchior (1940); Grünbaum (2009). Melchior, E. (1940), "Über Vielseite der projektiven Ebene", Deutsche Mathematik, 5: 461–475 Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
Grünbaum (1972); Grünbaum (2009). Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
Grünbaum (2009). Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
Crowe & McKee (1968); Dirac (1951); Kelly & Moser (1958); Grünbaum (1972), page 18. Crowe, D. W.; McKee, T. A. (1968), "Sylvester's problem on collinear points", Mathematics Magazine, 41 (1): 30–34, doi:10.2307/2687957, JSTOR 2687957 Dirac, G. (1951), "Collinearity properties of sets of points", Quarterly Journal of Mathematics, 2 (1): 221–227, Bibcode:1951QJMat...2..221D, doi:10.1093/qmath/2.1.221 Kelly, L. M.; Moser, W. O. J. (1958), "On the number of ordinary lines determined by $n$ points", Canadian Journal of Mathematics, 10: 210–219, doi:10.4153/CJM-1958-024-6 Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society
Eppstein (2006). Eppstein, D. (2006), "Cubic partial cubes from simplicial arrangements", Electronic Journal of Combinatorics, 13 (1, R79): 1–14, arXiv:math.CO/0510263, doi:10.37236/1105, MR 2255421, S2CID 8608953, archived from the original on 2012-02-14, retrieved 2008-12-14
Grünbaum (1972); Levi (1926); Roudneff (1988). Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society Levi, F. (1926), "Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade", Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig, 78: 256–267 Roudneff, J.-P. (1988), "Arrangements of lines with a minimum number of triangles are simple", Discrete & Computational Geometry, 3 (1): 97–102, doi:10.1007/BF02187900
Füredi & Palásti (1984); Grünbaum (1972). Füredi, Z.; Palásti, I. (1984), "Arrangements of lines with a large number of triangles" (PDF), Proceedings of the American Mathematical Society, 92 (4): 561–566, doi:10.2307/2045427, JSTOR 2045427, archived (PDF) from the original on 2016-03-03, retrieved 2008-12-15 Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society
Purdy (1979); Purdy (1980); Strommer (1977). Purdy, G. B. (1979), "Triangles in arrangements of lines", Discrete Mathematics, 25 (2): 157–163, doi:10.1016/0012-365X(79)90018-9 Purdy, G. B. (1980), "Triangles in arrangements of lines, II", Proceedings of the American Mathematical Society, 79: 77–81, doi:10.1090/S0002-9939-1980-0560588-4 Strommer, T. O. (1977), "Triangles in arrangements of lines", Journal of Combinatorial Theory, Series A, 23 (3): 314–320, doi:10.1016/0097-3165(77)90022-X
Abramenko & Brown (2008). Abramenko, Peter; Brown, Kenneth S. (2008), Buildings: Theory and Applications, Graduate Texts in Mathematics, vol. 248, New York: Springer, doi:10.1007/978-0-387-78835-7, ISBN 978-0-387-78834-0, MR 2439729; see Example 10.14, pp. 519–520
Edelsbrunner & Guibas (1989). Edelsbrunner, H.; Guibas, L. J. (1989), "Topologically sweeping an arrangement", Journal of Computer and System Sciences, 38 (1): 165–194, doi:10.1016/0022-0000(89)90038-X
Fortune & Milenkovic (1991); Greene & Yao (1986); Milenkovic (1989). Fortune, S.; Milenkovic, V. (1991), "Numerical stability of algorithms for line arrangements", Proc. 7th ACM Symposium on Computational Geometry (SoCG '91), pp. 334–341, CiteSeerX 10.1.1.56.2404, doi:10.1145/109648.109685, ISBN 978-0897914260, S2CID 2861855 Greene, D.; Yao, F. F. (1986), "Finite-resolution computational geometry", Proceedings of the 27th IEEE Symposium on Foundations of Computer Science (FOCS '86), pp. 143–152, doi:10.1109/SFCS.1986.19, ISBN 978-0-8186-0740-0, S2CID 2624319 Milenkovic, V. (1989), "Double precision geometry: a general technique for calculating line and segment intersections using rounded arithmetic", Proceedings of the 30th IEEE Symposium on Foundations of Computer Science (FOCS '89), pp. 500–505, doi:10.1109/SFCS.1989.63525, ISBN 978-0-8186-1982-3, S2CID 18564700
Aharoni et al. (1999). Aharoni, Y.; Halperin, D.; Hanniel, I.; Har-Peled, S.; Linhart, C. (1999), "On-line zone construction in arrangements of lines in the plane", in Vitter, Jeffrey S.; Zaroliagis, Christos D. (eds.), Algorithm Engineering: 3rd International Workshop, WAE'99, London, UK, July 19–21, 1999, Proceedings, Lecture Notes in Computer Science, vol. 1668, Springer-Verlag, pp. 139–153, CiteSeerX 10.1.1.35.7681, doi:10.1007/3-540-48318-7_13, ISBN 978-3-540-66427-7
Agarwal et al. (1998); Chan (1999); Cole, Sharir & Yap (1987); Edelsbrunner & Welzl (1986). Agarwal, P. K.; de Berg, M.; Matoušek, J.; Schwarzkopf, O. (1998), "Constructing levels in arrangements and higher order Voronoi diagrams", SIAM Journal on Computing, 27 (3): 654–667, CiteSeerX 10.1.1.51.5064, doi:10.1137/S0097539795281840 Chan, T. (1999), Remarks on $k$ -level algorithms in the plane, archived from the original on 2010-11-04 Cole, R.; Sharir, M.; Yap, C.-K. (1987), "On $k$ -hulls and related problems", SIAM Journal on Computing, 16 (1): 61–77, doi:10.1137/0216005 Edelsbrunner, H.; Welzl, E. (1986), "Constructing belts in two-dimensional arrangements with applications", SIAM Journal on Computing, 15 (1): 271–284, doi:10.1137/0215019
Agarwal (1990); Agarwal, Matoušek & Sharir (1998); Edelsbrunner, Guibas & Sharir (1990). Agarwal, P. K. (1990), "Partitioning arrangements of lines II: Applications", Discrete & Computational Geometry, 5 (1): 533–573, doi:10.1007/BF02187809 Agarwal, P. K.; Matoušek, J.; Sharir, M. (1998), "Computing many faces in arrangements of lines and segments", SIAM Journal on Computing, 27 (2): 491–505, doi:10.1137/S009753979426616X, hdl:1874/17088 Edelsbrunner, H.; Guibas, L. J.; Sharir, M. (1990), "The complexity and construction of many faces in arrangements of lines and of segments", Discrete & Computational Geometry, 5 (1): 161–196, doi:10.1007/BF02187784
Cole et al. (1989). Cole, Richard; Salowe, Jeffrey S.; Steiger, W. L.; Szemerédi, Endre (1989), "An optimal-time algorithm for slope selection", SIAM Journal on Computing, 18 (4): 792–810, doi:10.1137/0218055, MR 1004799
Shor (1991); Schaefer (2010). Shor, P. W. (1991), "Stretchability of pseudolines is NP-hard", in Gritzmann, P.; Sturmfels, B. (eds.), Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, Providence, R.I.: American Mathematical Society, pp. 531–554 Schaefer, Marcus (2010), "Complexity of some geometric and topological problems" (PDF), Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers, Lecture Notes in Computer Science, vol. 5849, Springer-Verlag, pp. 334–344, doi:10.1007/978-3-642-11805-0_32, ISBN 978-3-642-11804-3, archived (PDF) from the original on 2021-06-26, retrieved 2011-02-25
Goodman et al. (1994). Goodman, Jacob E.; Pollack, Richard; Wenger, Rephael; Zamfirescu, Tudor (1994), "Every arrangement extends to a spread", Combinatorica, 14 (3): 301–306, doi:10.1007/BF01212978, MR 1305899, S2CID 42055590
Dress, Koolen & Moulton (2002). Dress, A.; Koolen, J. H.; Moulton, V. (2002), "On line arrangements in the hyperbolic plane", European Journal of Combinatorics, 23 (5): 549–557, doi:10.1006/eujc.2002.0582, MR 1931939

duke.edu

users.cs.duke.edu

Steiner (1826); Agarwal & Sharir (2000). Steiner, J. (1826), "Einige Gesetze über die Theilung der Ebene und des Raumes", J. Reine Angew. Math., 1: 349–364, doi:10.1515/crll.1826.1.349, S2CID 120477563 Agarwal, P. K.; Sharir, M. (2000), "Arrangements and their applications" (PDF), in Sack, J.-R.; Urrutia, J. (eds.), Handbook of Computational Geometry, Elsevier, pp. 49–119, archived (PDF) from the original on 2021-04-11, retrieved 2018-11-11

handle.net

hdl.handle.net

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Grünbaum (1972); Grünbaum (2009). Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
Grünbaum (2009). Grünbaum, Branko (2009), "A catalogue of simplicial arrangements in the real projective plane", Ars Mathematica Contemporanea, 2 (1): 1–25, doi:10.26493/1855-3974.88.e12, hdl:1773/2269, MR 2485643
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Agarwal (1990); Agarwal, Matoušek & Sharir (1998); Edelsbrunner, Guibas & Sharir (1990). Agarwal, P. K. (1990), "Partitioning arrangements of lines II: Applications", Discrete & Computational Geometry, 5 (1): 533–573, doi:10.1007/BF02187809 Agarwal, P. K.; Matoušek, J.; Sharir, M. (1998), "Computing many faces in arrangements of lines and segments", SIAM Journal on Computing, 27 (2): 491–505, doi:10.1137/S009753979426616X, hdl:1874/17088 Edelsbrunner, H.; Guibas, L. J.; Sharir, M. (1990), "The complexity and construction of many faces in arrangements of lines and of segments", Discrete & Computational Geometry, 5 (1): 161–196, doi:10.1007/BF02187784

harvard.edu

ui.adsabs.harvard.edu

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jstor.org

Crowe & McKee (1968); Dirac (1951); Kelly & Moser (1958); Grünbaum (1972), page 18. Crowe, D. W.; McKee, T. A. (1968), "Sylvester's problem on collinear points", Mathematics Magazine, 41 (1): 30–34, doi:10.2307/2687957, JSTOR 2687957 Dirac, G. (1951), "Collinearity properties of sets of points", Quarterly Journal of Mathematics, 2 (1): 221–227, Bibcode:1951QJMat...2..221D, doi:10.1093/qmath/2.1.221 Kelly, L. M.; Moser, W. O. J. (1958), "On the number of ordinary lines determined by $n$ points", Canadian Journal of Mathematics, 10: 210–219, doi:10.4153/CJM-1958-024-6 Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society
Füredi & Palásti (1984); Grünbaum (1972). Füredi, Z.; Palásti, I. (1984), "Arrangements of lines with a large number of triangles" (PDF), Proceedings of the American Mathematical Society, 92 (4): 561–566, doi:10.2307/2045427, JSTOR 2045427, archived (PDF) from the original on 2016-03-03, retrieved 2008-12-15 Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society

oeis.org

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psu.edu

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semanticscholar.org

api.semanticscholar.org

Steiner (1826); Agarwal & Sharir (2000). Steiner, J. (1826), "Einige Gesetze über die Theilung der Ebene und des Raumes", J. Reine Angew. Math., 1: 349–364, doi:10.1515/crll.1826.1.349, S2CID 120477563 Agarwal, P. K.; Sharir, M. (2000), "Arrangements and their applications" (PDF), in Sack, J.-R.; Urrutia, J. (eds.), Handbook of Computational Geometry, Elsevier, pp. 49–119, archived (PDF) from the original on 2021-04-11, retrieved 2018-11-11
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tau.ac.il

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tue.nl

alexandria.tue.nl

de Bruijn (1981). de Bruijn, N. G. (1981), "Algebraic theory of Penrose's non-periodic tilings of the plane" (PDF), Indagationes Mathematicae, 43: 38–66, archived (PDF) from the original on 2021-05-07, retrieved 2008-12-14

uiuc.edu

math.uiuc.edu

Füredi & Palásti (1984); Grünbaum (1972). Füredi, Z.; Palásti, I. (1984), "Arrangements of lines with a large number of triangles" (PDF), Proceedings of the American Mathematical Society, 92 (4): 561–566, doi:10.2307/2045427, JSTOR 2045427, archived (PDF) from the original on 2016-03-03, retrieved 2008-12-15 Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society

compgeom.cs.uiuc.edu

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uwaterloo.ca

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web.archive.org

Steiner (1826); Agarwal & Sharir (2000). Steiner, J. (1826), "Einige Gesetze über die Theilung der Ebene und des Raumes", J. Reine Angew. Math., 1: 349–364, doi:10.1515/crll.1826.1.349, S2CID 120477563 Agarwal, P. K.; Sharir, M. (2000), "Arrangements and their applications" (PDF), in Sack, J.-R.; Urrutia, J. (eds.), Handbook of Computational Geometry, Elsevier, pp. 49–119, archived (PDF) from the original on 2021-04-11, retrieved 2018-11-11
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Füredi & Palásti (1984); Grünbaum (1972). Füredi, Z.; Palásti, I. (1984), "Arrangements of lines with a large number of triangles" (PDF), Proceedings of the American Mathematical Society, 92 (4): 561–566, doi:10.2307/2045427, JSTOR 2045427, archived (PDF) from the original on 2016-03-03, retrieved 2008-12-15 Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society
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