A number of applications in computer-aided manufacturing, CAD, and computer-aided geometric design ask for triangulating pieces of material with defects. These tasks are known collectively as constrained triangulations. Recently, a powerful architecture called the recon gurable mesh has been proposed: in essence, a recon gurable mesh consists of a mesh-connected architecture augmented by a dynamically recon gurable bus system. The main contribution of this paper is to show that the exibility of the recon gurable mesh can be exploited for the purpose of obtaining constant-time algorithms for a number of constrained triangulation problems. These include triangulating a convex planar region containing any constant number of convex holes, triangulating a convex planar region in the presence of a collection of rectangular holes, and triangulating a set of ordered line segments. Speci cally, with a collection of O(n) such objects as input, our algorithms run in O(1) time on a recon gurable mesh of size n n. To the best of our knowledge, this is the rst time constant time solutions to constrained triangulations are reported on this architecture.