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 \(R_1\) and \(R_2\) 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 $$\begin{aligned} \overline{r}(n) = \max _{|S|=n} r(S)\quad \text { and } \quad \underline{r}(n) = \min _{|S|=n} r(S), \end{aligned}$$and we prove that \( \underline{r}(n) = \Omega ^* (2^n)\), \( \underline{r}(n) = O^* (3.516^n)\) and that \( \overline{r}(n) = \Theta ^* (4^n)\). We also consider the number of different ways, \(r^\gamma (S)\), in which a point set S can be connected to a simple curve \(\gamma \) using a set of non-crossing straight-line segments. We define and study $$\begin{aligned} \overline{r}^{\gamma }(n) = \max _{|S|=n} r^{\gamma }(S) \quad \text {and } \quad \underline{r}^{\gamma }(n) = \min _{|S|=n} r^{\gamma }(S), \end{aligned}$$and we find these values for the following cases: When \(\gamma \) is a line and the points of S are in one of the halfplanes defined by \(\gamma \), then \( \underline{r}^\gamma (n) = \Theta ^* (2^n)\) and \( \overline{r}^\gamma (n) = \Theta ^* (4^n)\). When \(\gamma \) is a convex curve enclosing S, then \(\overline{r}^\gamma (n) = O^* (16^n)\). If all the points of S belong to a convex closed curve \(\gamma \), then \(\underline{r}^{\gamma }(n) = \overline{r}^{\gamma }(n) = \Theta ^* (5^n)\).