We present an unstructured triangular mesh generation algorithm that approximates a set of mutually non-intersecting simple trimmed rational B-spline surface patches within a user specified geometric tolerance. The proposed method uses numerically robust interval geometric representations/computations and also addresses the problem of topological consistency (homeomorphism) between the exact geometry and its approximation. Those are among the most important outstanding issues in geometry approximation problems. Our surface tessellation algorithm is based on the unstructured Delaunay mesh approach which leads to an efficient adaptive triangulation. A robust decision criterion is utilized to prevent possible failures in the conventional Delaunay triangulation. To satisfy the prescribed geometric tolerance, an adaptive node insertion algorithm is employed. Unstructured triangular meshes for free-form surfaces frequently involve triangles with high aspect ratio and accordingly, result in ill-conditioned meshing. Our proposed algorithm constructs 2D triangulation domains which sufficiently preserve the shape of triangles when mapped into 3D space and furthermore, the algorithm provides an efficient method that explicitly controls the aspect ratio of the triangular elements.