This paper introduces a 2-dimensional triangulation scheme based on a topological triangulation that approximates a given domain X within a specified Hausdorff distance from X. The underlying space of the resulting good quality triangulation is homeomorphic to X and contains either equilateral triangles or right angled triangles with 30°]]>, 60°]]> and 90°]]> angles. For a particular range of approximation tolerance, the number of triangles in the triangulation produced by the method is O(t log² t)]]> where t is the number of triangles in an optimal triangulation where the optimum is taken over bounded aspect ratio triangulations satisfying a certain boundary condition with respect to X. The method can also produce a quadrangulation of X having similar properties. Relevant implementation issues and results are discussed.