Configurations of non-crossing rays and related problems

Abstract

Let S be a set of n points in the plane and let R be a set of n pairwise non-crossing rays, each with an apex at a different point of S. Two sets of non-crossing rays R1R1 and R2R2 are considered to be different if the cyclic permutations they induce at infinity are different. In this paper, we study the number r(S) of different configurations of non-crossing rays that can be obtained from a given point set S. We define the extremal values r¯¯(n)=max|S|=nr(S) and r–(n)=min|S|=nr(S), r¯(n)=max|S|=nr(S) and r_(n)=min|S|=nr(S), and we prove that r–(n)=O*(2n)r_(n)=O*(2n) , r–(n)=O*(3.516n)r_(n)=O*(3.516n) and that r¯¯(n)=T*(4n)r¯(n)=T*(4n) . We also consider the number of different ways, r¿(S)r¿(S) , in which a point set S can be connected to a simple curve ¿¿ using a set of non-crossing straight-line segments. We define and study r¯¯¿(n)=max|S|=nr¿(S)and r–¿(n)=min|S|=nr¿(S), r¯¿(n)=max|S|=nr¿(S)and r_¿(n)=min|S|=nr¿(S), and we find these values for the following cases: When ¿¿ is a line and the points of S are in one of the halfplanes defined by ¿¿ , then r–¿(n)=T*(2n)r_¿(n)=T*(2n) and r¯¯¿(n)=T*(4n)r¯¿(n)=T*(4n) . When ¿¿ is a convex curve enclosing S, then r¯¯¿(n)=O*(16n)r¯¿(n)=O*(16n) . If all the points of S belong to a convex closed curve ¿¿ , then r–¿(n)=r¯¯¿(n)=T*(5n)r_¿(n)=r¯¿(n)=T*(5n) .