Motivated by problems from computer graphics and robotics---namely, ray tracing and assembly planning---we investigate the combinatorial structure of arrangements of segments on a line and of arcs on a circle. We show that there are, respectively, $1\times 3\times5\times\hbox{\mathversion{normal}$\cdots$}\times(2n{-}1)$ and $(2n) !$/$n!$ such arrangements;
that the probability for the $i$-th endpoint of a random arrangement to be an initial endpoint is $(2n{-}i)$/$(2n{-}1)$ or $\half$, respectively; and that the average number of segments or arcs the $i$-th endpoint is contained in are $(i{-}1)(2n{-}i)$/$(2n{-}1)$ or $(n{-}1)$/$2$, respectively. The constructions used to prove these results provide sampling schemes for generating random inputs that can be used to test programs manipulating arrangements.
¶ We also point out how arrangements are classically related to Catalan
numbers and the ballot problem.