We consider an optimization problem of inscribing a unit rectangle in a convex polygon. An axis-aligned unit rectangle is an axis-aligned rectangle whose horizontal sides are of length 1. A unit rectangle of orientation θ is a copy of an axis-...

This paper uses the technology of weighted triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Given a collection of Euclidean simplices glued together along their boundary, a geometric structure on the ...

We compute the shortest sequence of local connectivity modifications that transform a genus 0 quad mesh to a polycube. The modification operations are (dual) loop preserving and thus, we are restricted to quad meshes where loops don't ...

We consider the following Euclidean 2-center problem. Given n disks in the plane, find two smallest congruent disks such that every input disk intersects at least one of the two congruent disks. We present a deterministic algorithm for ...

We investigate the problem of partitioning a rectilinear polygon P with n vertices and no holes into rectangles using disjoint line segments drawn inside P under two optimality criteria. In the minimum ink partition, the total length ...

We show that every outerplanar graph can be linearly embedded in the plane such that the number of distinct distances between pairs of adjacent vertices is at most thirteen and there is no intersection between the image of a vertex and ...

A t-spanner is a subgraph of a graph G in which the length of the shortest path between two vertices never exceeds t times the length of the shortest path between them in G. A geometric graph is one whose vertices are points and whose ...

Given a simple polygon P on n vertices and a set D of m pairwise intersecting geodesic disks in P, we show that five points in P are always sufficient to pierce all the disks in D. The points can be computed in O ( ( n + m ) log ⁡ n r )...

We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an O ( n log ⁡ n )-time algorithm for the two-...

In the acrophobic guard watchtower problem for a polyhedral terrain, a square axis-aligned platform is placed on the top of a tower whose bottom end-point lies on the surface of the terrain. As in the standard watchtower problem, the ...

We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid – the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of ...

A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets ...

We study a family of line segment visibility problems, related to classical art gallery problems, which are motivated by monitoring requirements in commercial data centers. Given a collection of non-overlapping line segments in the ...

• We study the number of red lines required to pierce the intersections of a finite family of blue lines.

Let L be a set of n non-concurrent blue lines and let R be a set of m red lines in the real projective plane. In this note, using elementary geometric arguments, we show that if L ∩ R = ∅ and there is a line from R through every ...

We compute shortest paths connecting two axis-aligned rectilinear simple polygons in the domain consisting of axis-aligned rectilinear obstacles in the plane. The bounding boxes, one defined for each polygon and one defined for each ...

We study the Tukey layers and convex layers of a planar point set, which consists of n points independently and uniformly sampled from a convex polygon with k vertices. We show that the expected number of vertices on the first t Tukey ...

A convex polygon Q is inscribed in a convex polygon P if every side of P contains at least one vertex of Q. We present algorithms for finding a minimum area and a minimum perimeter convex polygon inscribed in any given convex n-gon in ...

A rectilinear polygon is a simple polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of ...

Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or ...

Let S be a set of n weighted points in the plane and let R be a query range in the plane. In the range closest pair problem, we want to report the closest pair in the set R ∩ S. In the range minimum weight problem, we want to report ...