Related papers: Finding Closed Quasigeodesics on Convex Polyhedra

In this paper, we consider enumeration of geodesics on a polyhedron, where a geodesic means locally-shortest path between two points. Particularly, we consider the following preprocessing problem: given a point $s$ on a polyhedral surface…

Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff…

Given a polygon $P$, for two points $s$ and $t$ contained in the polygon, their \emph{geodesic distance} is the length of the shortest $st$-path within $P$. A \emph{geodesic disk} of radius $r$ centered at a point $v \in P$ is the set of…

We consider triangle faced convex polyhedra inscribed in the unit sphere $S^2$ in ${\Bbb{R}}^3$. One way of measuring their deviation from regular polyhedra with triangular faces is to consider the quotient of the lengths of the longest and…

It is a widely observed phenomenon in computer graphics that the size of the silhouette of a polyhedron is much smaller than the size of the whole polyhedron. This paper provides, for the first time, theoretical evidence supporting this for…

A convex polygon Q is circumscribed about a convex polygon P if every vertex of P lies on at least one side of Q. We present an algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time.…

We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}^3$, every point in $\mathbb R^3$ that does not see any vertex of $\mathcal P$ must see eight or more edges of $\mathcal P$, and this bound is tight. More generally, this remains…

In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some…

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in…

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In…

The minimum convex cover problem seeks to cover a polygon $P$ with the fewest convex polygons that lie within $P$. This problem is $\exists\mathbb R$-complete, and the best previously known algorithm, due to Eidenbenz and Widmayer (2001),…

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $L/k$, where $L$ is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show…

Let $K\subseteq{\mathbb R}^n$ be a convex semialgebraic set. The semidefinite extension degree ${\mathrm{sxdeg}}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of…

Let G be a graph that may be drawn in the plane in such a way that all internal faces are centrally symmetric convex polygons. We show how to find a drawing of this type that maximizes the angular resolution of the drawing, the minimum…

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.

Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,…

We prove that, if a closed geodesic $\Gamma$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $\Gamma$ has an lower bound $2\log(5+2\sqrt6)$, and the lower bound is sharp, attained on a…

In this paper, we prove that if a quasi-Fuchsian 3-manifold $M$ contains a simple closed geodesic with complex length $\Lscr=l+i\theta$ such that $\theta/l\gg{}1$, then it contains at least two minimal surfaces which are incompressible in…

Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $\Omega$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input…

Given a polygonal curve P, a pointset S, and an \epsilon > 0, we study the problem of finding a polygonal curve Q whose vertices are from S and has a Frechet distance less or equal to \epsilon to curve P. In this problem, Q must visit every…