On a proper acute triangulation of a polyhedral surface

Let Σ be a polyhedral surface in R 3 with n edges. Let L be the length of the longest edge in Σ , δ be the minimum value of the geodesic distance from a vertex to an edge that is not incident to the vertex, and θ be the measure of the smallest face angle in Σ . We prove that Σ can be triangulated into at most C L n / ( δ θ ) planar and rectilinear acute triangles, where C is an absolute constant.