For any convex quadrilateral, the sum of the lengths of the diagonals is greater than the corresponding sum of a pair of opposite sides, and all four of its interior angles cannot be simultaneously acute. In this article, we use these two properties to estimate the number of unit distance edges in convex n-gons and we: (i) exhibit three large groups of cycles formed by unit distance edges that ...

In this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are better-suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to...

Clipping a triangle or a convex quadrilateral to a plane is a common operation in computer graphics. This clipping is implemented by fixed-function units within the graphics pipeline under most rasterization APIs. It is increasingly interesting to perform clipping in programmable stages as well. For example, to clip bounding volumes generated in the Geometry unit to the near plane, or to clip a...

Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge e, which is the set of all edges reachable from e via flips. We establish the...

In this paper, the block method for convex polygon triangulation is presented. The method is based on the usage of the previously generated triangulations for polygon with smaller number of vertices. In the beginning, we establish some relations between the triangulations of the polygons having the consecutive number of vertices. Using these relations, we decompose the triangulation problem int...

Let ES(n) be the smallest integer such that any set of ES(n) points in the plane in general position contains n points in convex position. In their seminal 1935 paper, Erdős and Szekeres showed that ES(n) ≤ ( 2n−4 n−2 ) + 1 = 4. In 1960, they showed that ES(n) ≥ 2 + 1 and conjectured this to be optimal. In this paper, we nearly settle the Erdős-Szekeres conjecture by showing that ES(n) = 2.

In [12], the widely known linear time algorithm for computing the maximum area triangle in a convex polygon was found incorrect, and whether it can still be solved in linear time was proposed as an open problem. We resolved it affirmatively by presenting a new linear time algorithm. Our algorithm is simple and easy to implement, and it computes all the locally maximal triangles. 1998 ACM Subjec...

Visibility methods are often based on the existence of large convex occluders. We present an algorithm that for a given simple non-convex polygon P finds an approximate inner-cover by large convex polygons. The algorithm is based on an initial partitioning of P into a set C of disjoint convex polygons which are an exact tessellation of P . The algorithm then builds a set of large convex polygon...