Triangulations of a connected subset F of parametric surfaces S(u,v) (with continuity C2 or higher) are required because a C0 approximation of such F(called a FACE) is widely required for finite element analysis, rendering, manufacturing, design, reverse engineering, etc -- The triangulation T is such an approximation, when its piecewise linear subsets are triangles (which, on the other hand, is not a compulsory condition for being C0) -- A serious obstacle for algorithms which triangulate in the parametric space u−v is that such a space may be extremely warped, and the distances in parametric space be dramatically different of the distances in R3 -- Recent publications have reported parameter -independent triangulations, which triangulate in R3 space -- However, such triangulations are not sensitive to the curvature of the S(u,v) -- The present article presents an algorithm to obtain parameter-independent, curvature-sensitive triangulations -- The invariant of the algorithm is that a vertex v of the triangulation if identified, and a quasiequilateral triangulation around v is performed on the plane P tangent to S(u,v) at v -- The size of the triangles incident to v is a function of K(v), the curvature of S(u,v) at v -- The algorithm was extensively and successfully tested, rendering short running times, with very demanding boundary representations