Discrete & Computational Geometry

, Volume 5, Issue 2, pp 99–160

# Combinatorial complexity bounds for arrangements of curves and spheres

• Kenneth L. Clarkson
• Herbert Edelsbrunner
• Leonidas J. Guibas
• Micha Sharir
• Emo Welzl
Article

DOI: 10.1007/BF02187783

Clarkson, K.L., Edelsbrunner, H., Guibas, L.J. et al. Discrete Comput Geom (1990) 5: 99. doi:10.1007/BF02187783

## Abstract

We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n2/3 +n), and that it isO(m2/3n2/3β(n) +n) forn unit-circles, whereβ(n) (and laterβ(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m3/5n4/5β(n) +n). The same bounds (without theβ(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m4/7n9/7β(m, n) +n2), in general, andO(m3/4n3/4β(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m3/2β(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

© Springer-Verlag New York Inc. 1990

## Authors and Affiliations

• Kenneth L. Clarkson
• 1
• Herbert Edelsbrunner
• 2
• Leonidas J. Guibas
• 3
• 4
• Micha Sharir
• 5
• 6
• Emo Welzl
• 7
1. 1.AT&T Bell LaboratoriesMurray HillUSA
2. 2.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA
3. 3.DEC Systems Research CenterPalo AltoUSA
4. 4.Computer Science DepartmentStanford UniversityUSA
5. 5.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
6. 6.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
7. 7.Fachbereich MathematikFreie Universität BerlinBerlin 33Germany