Typical methods for the triangulation of parametric surfaces use a sampling of the parameter space, and the wrong choice of parameterization can spoil a triangulation or even cause the algorithm to fail. We present a new method that uses a local tessellation primitive for almost-uniformly sampling and triangulating a surface, so that its parameterization becomes irrelevant. If sampling density or triangle shape has to be adaptive, the uniform mesh can be used either as an initial coarse mesh for a refinement process, or as a fine mesh to be reduced.