Cost analysis of the longest-side (triangle bisection) refinement algorithm for triangulations

The triangulation refinement problem, as formulated in the adaptive finite element setting (also useful in the rendering of complex scenes), is discussed. This can be formulated as follows: given a valid, non-degenerate triangulation of a polygonal region, construct a locally refined triangulation, with triangles of prescribed size in a refinement region R, and such that the smallest (or the largest) angle is bounded. To cope with this problem, longest-side refinement algorithms guarantee the construction of good quality irregular triangulations. This is due in part to their natural refinement propagation strategy farther than the (refinement) area of interest R. In this paper we prove that, asymptotically, the number N of points inserted in R to obtain triangles of prescribed size, is optimal. Furthermore, in spite of the unavoidable propagation outside the refinement region R, the time cost of the algorithm is linear in N, independent of the size of the triangulation. Specifically, the number of points inserted outside R is of order O(n log(2) n) where N = O(n(2)). We prove the latter result for circular and rectangular refinement regions, which allows us to conclude that this is true for general convex refinement regions. We also include empirical evidence, both in two and three dimensions, which is in complete agreement with the theory even for small values for N.