We show that when consecutive waypoints are a distance of at least four apart, this question reduces to a family of convex optimization problems over polyhedra in $\mathbb{R}^n$.Expand

We prove that the set of lines tangent to four possibly intersecting convex polyhedra in $\mathbb{R}^3$ with a total of $n$ edges consists of $\Theta(n^2k^2)$ connected components in the worst case.Expand

In this paper, we show that, amongst $n$ uniformly distributed unit balls in $\mathbb{R}^3$, the expected number of maximal nonoccluded line segments tangent to four balls is linear. Using our… Expand

We show that for planar convex sets, the number of facets of the convex hull of $n$ points chosen uniformly and independently in a convex body is asymptotically increasing.Expand

Average-case analysis of data-structures or algorithms is commonly used in computational geometry when the, more classical, worst-case analysis is deemed overly pessimistic. Since these analyses are… Expand

We present an exact method to compute the boundaries between umbra, penumbra and full-light regions cast on a plane by a set of disjoint convex polyhedra, some of which are light sources.Expand