On the stretch factor of the constrained delaunay triangulation

Given a set P of n points in the plane and a set S of non-crossing line segments whose endpoints are in P, let CDT(P, S) be the constrained Delaunay triangulation of P with respect to S. Given any two visible points p, q ∈ P, we show that there exists a path from p to q in CDT(P, S), denoted SP CDT(p, q). such that every edge in the path has length at most |pq| and the ratio |SPCDT(p, q)|/|pq| is at most 4π√3/9(≈ 2.42), thereby improving on the previously known bound of π(1+√5/2 (≈ 5.08)