Combinatorial complexity bounds for arrangements of curves and spheres

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 bounding m cells in an arrangement of n lines is O(m2/~n 2/~ + n), and that it is O(m2/an2/3~(n) + n) for n unit-circles, where p(n) (and later/~(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 to O(m3/Sn*/513(n) + n). The same bounds (without the B(n)-terms) hold for the maximum sum of degrees of m 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 of m vertices in an arrangement of n spheres in three dimensions is * O(m4/Tng/713(m, n) + n2), in general, and O(m3/4na/413(m, n) + n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances among m points in three dimensions is O (m3/2fl(rn)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.The techniques for studying this problem, both here and in [t9] and [20], can be viewed as divide-and-conquer attacks on this problem. In the approach taken in this paper, which we call the primal approach, the points are partitioned into groups according to an underlying subdivision of space into cells, to be specified shortly.We call this subdivision the funneling subdivision and its cells funnels, to distinguish 106

The Number of Congruent Simplices in a Point Set

“…SO assume that w is at distance 77 from at least four points of AT. In case (i), w must lie on a 2-sphere (pc that is concentric with and orthogonal to -yA, and thus lies in the same 3-space containing ipB • We have thus reduced the problem to the following one: We have two concentric spheres, (p, <p', in three dimensions and two finite point sets Q, Q', with Q Cf and Q' dp', and we wish to bound the number of pairs of points in Q X Q' that are at distance C from each other. We claim that the number of such pairs is 0{\Q\^'^\Q'\^'^ + \Q\ + \Q'\)-This is proved exactly as in the analysis in [6] of the number of repeated distances in a planar point set, and as in the proof of Theorem 2.1. In other words, the number of triangles under consideration is…”

“…That is, u lies on the sphere of radius | centered at v. Conversely, any such point i; gives rise to a circle 7"" € G that is contained in (r". The asserted bound is now an immediate consequence of the bound on the number of incidences between points and unit spheres in R^, as given in [6]. a…”

“…The analysis is similar to the one used in [6] to obtain the same bound for the planar case. First, the pointcircle incidence graph does not contain 1^3,2 as a subgraph (with 3 points and 2 circles), so the incidence graph can have at most 0{nt^^^ +1) edges.…”

“…Repeating this analysis for each pair p,q at distance ^, we obtain a (multi)set 6 of congruent circles, one for each such pair of points, and the number of triangles under consideration is equal to the number of incidences between the circles of 6 and the points of P. It is easily checked that at most two pairs of points p, q can give rise to the same circle in C, so we may assume that all circles in 6 are distinct. Since each circle in 6 is generated by a pair of points of P at distance ^ apart, we have, by the results of [6], |e| = 0{n'''''l3{n)), where /3(n) = 2®(°'("» is as above. For each u € P, let o-" denote the sphere of radius 77 centered at u.…”

“…[6,16]; in fact, the proof in [16] translates practically verbatim to the case of congruent circles on a sphere).…”