On a proper acute triangulation of a polyhedral surface

Let ΣΣ be a polyhedral surface in R3R3 with nn edges. Let LL be the length of the longest edge in ΣΣ, δδ be the minimum value of the geodesic distance from a vertex to an edge that is not incident to the vertex, and θθ be the measure of the smallest face angle in ΣΣ. We prove that ΣΣ can be triangulated into at most CLn/(δθ)CLn/(δθ) planar and rectilinear acute triangles, where CC is an absolute constant.