The Number of Congruent Simplices in a Point Set

Abstract

We derive improved bounds on the number of fc-dimensional simplices spanned by a set of n points in R'^ that are congruent to a given fc-simplex, for fc < d -1. Let f^ '{n) be the maximum number of fc-simplices spanned by a set of n points in R"* that are congruent to a given fc-simplex. We prove that /j (n) = 0("5/3 .20(a^(n)))_ y(4)(") ^ 0{n^+n, A^\n) = 0(n^/3), and /^^'(n) = 0(TI^/**+^). We also derive a recurrence to bound fjf'^ (n) for arbitrary values of k and d, and use it to derive the bound fjfhn) = 0{n^/'^) for d < 7 and A; < d -2. Following Erdos and Purdy, we conjecture that this bound holds for laxger values of d as well, and for fc < d -2.

Similar simplices in a d-dimensional point set

“…As already noted, d = 5 is the last interesting case for triangles, since, already for the congruent case, g 2,6 (n) = Θ(n 3 ) [4].…”

4-points congruent sets for robust pairwise surface registration

“…In such a scenario, we can extract all the congruent 4-points sets from point sets of size n in O(n 2 ). However, with no approximation (δ = 0), the number of reported subsets is upper bounded by O(n 5/3 ) [Agarwal and Sharir 2002]. Later we extend our algorithm to handle affine transformations.…”

“…When the overlap margin is not arbitrarily small, a randomized verification takes O(k3) where k3 is the number of actual 4-points bases from Q that are approximately congruent to B, assuming that P does not have many copies in Q. Worst case bounds for such ki-s is a topic of interest in combinatorial geometry, and such bounds exist for both the exact (δ = 0) and approximate cases [Indyk et al 1999,Agarwal andSharir 2002]. However, for point sets with samples approximately evenly distributed on the scanned surface, all ki-s are O(n).…”

Problems and Results on Geometric Patterns

“…For higher dimensions, Lenz's construction or, in the odd-dimensional cases, a combination of Lenz's construction with the best known three-dimensional point set (Erdős et al, 1989;Ábrego and Fernández-Merchant, 2002), are most likely to be optimal. The only results in this direction, given in Agarwal and Sharir (2002), are for d ≤ 7 and do not quite attain this bound.…”

“…This justifies the choice of the word "pattern" for the resulting equivalence classes. Indeed, the algorithmic aspects of these problems have also been studied in the context of geometric pattern matching (Akutsu et al, 1998;Brass, 2000;Agarwal and Sharir, 2002;Brass, 2002). A typical algorithmic question is the following.…”