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Bounded-Curvature Shortest Paths through a Sequence of Points Using Convex Optimization
TLDR
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
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Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra
TLDR
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
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The Expected Number of 3D Visibility Events Is Linear
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 ourExpand
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Moving Vertices to Make Drawings Plane
TLDR
In John Tantalo's on-line game Planarity the player is given a non-plane straight-line drawing of a planar graph. Expand
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The monotonicity of f-vectors of random polytopes
TLDR
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
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Untangling a Planar Graph
TLDR
We give an algorithm that fixes at least $\sqrt{((log n)-1)/\log\log n}$ vertices when untangling a drawing of an n-vertex graph G. Expand
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Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
TLDR
We prove Helly-type theorems for line transversals to disjoint unit balls in ℝd. Expand
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Complexity analysis of random geometric structures made simpler
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 areExpand
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Smoothed complexity of convex hulls by witnesses and collectors
TLDR
We present a simple technique for analyzing the size of geometric hypergraphs defined by random point sets. Expand
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Computing Direct Shadows Cast by Convex Polyhedra
TLDR
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
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