Barycentric coordinates were introduced by Möbius in 1827 as an alternative to Cartesian coordinates. They describe points relative to the vertices of a simplex and are commonly used to express the linear interpolant of data given at these vertices. Generalized barycentric coordinates and kernels extend this idea from simplices to polyhedra and smooth domains. In this paper, we focus on Wachspr...

A convex hull is a polygon which encloses all given set of points.Euler number or Euler characteristic of an image has proven to be an important feature in many image analyses and visual inspection applications. This paper presents an algorithm for fast computing the convex hull of a planar scattered point set, which pre-strike an initial convex hull boundary, remove internal points in the boun...

The relative convex hull, or the minimum-perimeter polygon (MPP) of a simple polygon A, contained in a second polygon B, is a unique polygon in the set of nested polygons between A and B. The computation of the minimum-length polygon (MLP), as a special case for isothetic polygons A and B, is useful for various applications in image analysis and robotics. The paper discusses the first recursive...

A simple polyhedron is weakly-monotonic in direction ~ d provided that the intersection of the polyhedron and any plane with normal ~ d is simply-connected (i.e. empty, a point, a line-segment or a simple polygon). Furthermore, if the intersection is a convex set, then the polyhedron is said to be weakly-monotonic in the convex sense. Toussaint [10] introduced these types of polyhedra as genera...

Separating axis theorem is used in a large number of game development, it is mainly used to detect whether two convex polygon product a collision. When the number of sides of a convex polygon is less, separating axis theorem can quickly determine whether it products a collision. However, When the number of edges is increased, the polygon will become more complex and the greater costs will be.A ...

Given a polygon P , a ipturn involves re ecting a pocket p of P through the midpoint of the lid of p. In 1973, Joss and Shannon (published in Gr unbaum (1995)) showed that any polygon on n vertices will become convex after a sequence of at most (n 1)! ipturns. They conjectured that this bound was not tight, and that n=4 ipturns would always be suÆcient. In this work, we show that any polygon o...

We consider the problem of guarding curvilinear art galleries. A closed arc a joining two points, p and q, in the plane is called a convex arc if the curve obtained by joining a with the line segment pq encloses a convex set. A piece-wise convex polygon P with vertices v0, . . . , vn−1 is the region bounded by a set {a0, . . . , an−1} of n convex arcs with pairwise disjoint interiors such that ...

Shamos [1] recently showed that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be us...

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout w...