Triangulating a convex polygon with fewer number of non-standard bars

Abstract

For a given convex polygon with inner angle no less than 2 3 π and boundary edge bounded by [ l , α l ] for 1 ≤ α ≤ 1.4 , where l is a given standard bar’s length, we investigate the problem of triangulating the polygon using some Steiner points such that (i) the length of each edge in triangulation is bounded by [ β l , 2 l ] , where β is a given constant and meets 0 < β ≤ 1 2 , and (ii) the number of non-standard bars in the triangulation is minimum. This problem is motivated by practical applications and has not been studied previously. In this paper, we present a heuristic to solve the above problem, which is based on the heuristic to generate a triangular mesh with less number of non-standard bars and shorter maximal edge length, and a process to make the length of each edge lower bounded. Our procedure is simple and easily implemented for this problem, and we prove that it has good performance guaranteed.