We describe new methods for the construction of spiral toolpaths for high-speed machining. In the simplest case, our method takes a polygon as input and a number δ > 0 and returns a spiral starting at a central point in the polygon, going around towards the boundary while morphing to the shape of the polygon. The spiral consists of linear segments and circular arcs, it is G continuous, it has n...

We study the minimum number g(m,n) (respectively, p(m, n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that dn/2e − 2 ≤ g(4, n) ≤ n 2 + o(n) and dn/4e ≤ g(n, 4) ≤ n 2 + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the min...

We study the moduli spaces of polygons in R and R, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gelfand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gelfand-Cetlin system on the Grassmannian, and with these determine the pentagon and hexagon spaces up to equivariant symplectomorphism....

[1] Data uncertainty research of rain gauge network requires generation of large numbers of Thiessen polygons. Despite its importance in hydrology, few studies on computational Thiessen polygons have been carried out, and there is little published information in the hydrological literature. This paper describes two automated approaches and the ways for their implementation in hydrological appli...

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log n) time algorithm to compute it. We also prove a number of structural properties of this ...