Let us consider Fig. 1. In Fig. 1, there are four regions, as indicated (R4 9 epresents the entire space not occupied by the graph). Suppose we are given all of he edges of the graph and their geometrical orientation. Can we find all of the 'egions automatically? A similar problem was discussed in [Nie 82]. In their )roblem, we are given a polygon which is not simple and we are asked to produce...

This paper considers transformations of a planar polygon P according to two types of operations. A vertex pop (or a pop) reflects a vertex vi, i ∈ {1, . . . , n}, across the line through the two adjacent vertices vi−1 and vi+1 (where index arithmetic is modulo n). A popturn rotates vi in the plane by 180◦ about the midpoint of the line segment vi−1vi+1. Pops and popturns are polygon reconfigura...

let e be the ellipse with major and minor radii a and b respectively, and pbe its perimeter, then p = lim 4 tan(p/n)(a + b + 2) σ a2 cos2 (2k-2)pi/n+ sin2 (2k-2)pi/n; where n = 2m. so without considering the limit, it gives a reasonable approxi-mation for p, it means that we can choose n large enough such that the amountof error be less than any given small number. on the other hand, the formul...

Recent progress in acquiring shape from range data permits the acquisition of seamless million-polygon meshes from physical models. While dense polygon meshes are an adequate representation for some applications, many users prefer smooth surface representations for reasons of compactness, control, manufacturability, or appearance. In this thesis, we present algorithms and an end-to-end software...

The plane sweep technique is one of the best known paradigms for the design of geometric algorithms [2]. Here an imaginary line sweeps over the plane while computing the property of interest at the moment the sweep line passes the required information needed to compute that property. There are also three-dimensional problems that are solved by space sweep, where a plane sweeps the space. This p...

We solve the problem of designing a simple device that uses rotary motion to drill a hole with a cross-section that is a regular polygon with an odd number of sides: the main idea is to use a polygonal trammel and a family of rotors. By using different rotors, one can produce a hole with a cross-section that is in any proportion of the trammel size from zero to exactly one. The key geometric id...

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type A (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed Σ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed Σ. Our formula is give...