Configurations of Non-crossing Rays and Related Problems

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 $$R_1$$R1 and $$R_2$$R2 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 $$\begin{aligned} \overline{r}(n) = \max _{|S|=n} r(S)\quad \text { and } \quad \underline{r}(n) = \min _{|S|=n} r(S), \end{aligned}$$r¯(n)=max|S|=nr(S)andr̲(n)=min|S|=nr(S),and we prove that $$ \underline{r}(n) = \Omega ^* (2^n)$$r̲(n)=Ω∗(2n), $$ \underline{r}(n) = O^* (3.516^n)$$r̲(n)=O∗(3.516n) and that $$ \overline{r}(n) = \Theta ^* (4^n)$$r¯(n)=Θ∗(4n). We also consider the number of different ways, $$r^\gamma (S)$$rγ(S), in which a point set S can be connected to a simple curve $$\gamma $$γ using a set of non-crossing straight-line segments. We define and study $$\begin{aligned} \overline{r}^{\gamma }(n) = \max _{|S|=n} r^{\gamma }(S) \quad \text {and } \quad \underline{r}^{\gamma }(n) = \min _{|S|=n} r^{\gamma }(S), \end{aligned}$$r¯γ(n)=max|S|=nrγ(S)andr̲γ(n)=min|S|=nrγ(S),and we find these values for the following cases: When $$\gamma $$γ is a line and the points of S are in one of the halfplanes defined by $$\gamma $$γ, then $$ \underline{r}^\gamma (n) = \Theta ^* (2^n)$$r̲γ(n)=Θ∗(2n) and $$ \overline{r}^\gamma (n) = \Theta ^* (4^n)$$r¯γ(n)=Θ∗(4n). When $$\gamma $$γ is a convex curve enclosing S, then $$\overline{r}^\gamma (n) = O^* (16^n)$$r¯γ(n)=O∗(16n). If all the points of S belong to a convex closed curve $$\gamma $$γ, then $$\underline{r}^{\gamma }(n) = \overline{r}^{\gamma }(n) = \Theta ^* (5^n)$$r̲γ(n)=r¯γ(n)=Θ∗(5n).