An algorithm for constructing constrained Delaunay triangulation (CDT) of a planar straight-line graph (PSLG) is presented. Although the uniform grid method can reduce the time cost of visibility determinations, the time needed to construct the CDT is still long. The algorithm proposed in this paper decreases the number of edges involved in the computation of visibility by replacing traditional visibility with Delaunay visibility. With Delaunay visibility introduced, all strongly Delaunay edges are excluded from the computation of visibility. Furthermore, a sufficient condition for DT (CDT whose triangles are all Delaunay) existence is presented to decrease the times of visibility determinations. The mesh generator is robust and exhibits a linear time complexity for randomly generated PSLGs.