Random Convex Polygon Construction Algorithm

Abstract

In this paper, we propose an algorithm for constructing arbitrary convex polygons with a random arrangement of vertices. Earlier, we have already described an algorithm for constructing arbitrary polygons with a random arrangement of vertices. The construction method is based on the sequential addition of new vertices and is a modification of the algorithm described Earlier. For a randomly selected edge of the polygon, a random point Pnew is taken – a candidate for a new additional vertex. If after adding Pnew the polygon remains convex, then instead of a randomly selected existing edge Ek = [Pk , Pk+1] between the vertices Pk and Pk+1 two new edges are added $E_{new}^1 = \left[ {{P_k},{P_{{\text{new }}}}} \right]\quad {\text{and}}\quad E_{new}^2 = \left[ {{P_{{\text{new }}}},{P_{k + 1}}} \right]$. The procedure is repeated until the specified number of vertices is obtained. If it is not possible to find a new additional vertex for all edges of the polygon the algorithm stops. When choosing an admissible point Pnew, the convex zone CZk is constructed for the edge Ek - this is a polygon all points of which can become a new additional vertex without breaking the convexity of the polygon. A random point from CZk is selected as Pnew.