Limit theorems for generalized perimeters of random inscribed polygons. I

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.

With Random Parametric Equation of Circle Inscribed Polygon Generation Algorithm and Analysis

For getting the parametric equation of random polygon, by the basic extension factor to stretch the round, there generates polygon of meeting the required conditions. Such algorithm overcomes the disadvantage in the generation of polygon from round under control of a single parameter and the complication of mathematical background. Firstly, a detailed introduction is given to the basic extension factor and the selection of relevant coefficient, and then the errors of such algorithm are analyzed. And the experimental result shows that the mathematical background of such method is relatively easy, the parameters are easy to control and constitute, and the generation of arbitrary polygon is available with high precision. Such the parametric equation of random polygon and method can be used in relevant fields such as geometric modeling, CAD/CAM, and computer graphics.