Configurations of Non-crossing Rays and Related Problems

3 Let S be a set of n points in the plane and let R be a set of n pairwise non-crossing rays, 4 each with an apex at a different point of S. Two sets of non-crossing rays R1 and R2 are 5 considered to be different if the cyclic permutations they induce at infinity are different. 6 In this paper, we study the number r(S) of different configurations of non-crossing rays 7 that can be obtained from a given point set S. We define the extremal values 8 r(n) = max |S|=n r(S) and r(n) = min |S|=n r(S), and we prove that r(n) = Ω∗(2n), r(n) = O∗(3.516n) and that r(n) = Θ∗(4n). 9 We also consider the number of different ways, r(S), in which a point set S can be 10 connected to a simple curve γ using a set of non-crossing straight-line segments. We define 11 and study 12 r(n) = max |S|=n r(S) and r(n) = min |S|=n r(S), and we find these values for the following cases: When γ is a line and the points of S are 13 in one of the halfplanes defined by γ, then r(n) = Θ∗(2n) and r(n) = Θ∗(4n). When γ 14 is a convex curve, then r(n) = O∗(16n). If all the points are on a convex curve γ, then 15 r(n) = r(n) = Θ∗(5n). 16