Gabriel-constrained Parametric Surface Triangulation

Abstract

The Boundary Representation of a 3D manifold contains FACES (connected subsets of a parametric surface S : R2−R3) -- In many science and engineering applications it is cumbersome and algebraically difficult to deal with the polynomial set and constraints (LOOPs) representing the FACE -- Because of this reason, a Piecewise Linear (PL) approximation of the FACE is needed, which is usually represented in terms of triangles (i.e. 2-simplices) -- Solving the problem of FACE triangulation requires producing quality triangles which are: (i) independent of the arguments of S, (ii) sensitive to the local curvatures, and (iii) compliant with the boundaries of the FACE and (iv) topologically compatible with the triangles of the neighboring FACEs -- In the existing literature there are no guarantees for the point (iii) -- This article contributes to the topic of triangulations conforming to the boundaries of the FACE by applying the concept of parameter independent Gabriel complex, which improves the correctness of the triangulation regarding aspects (iii) and (iv) -- In addition, the article applies the geometric concept of tangent ball to a surface at a point to address points (i) and (ii) -- Additional research is needed in algorithms that (i) take advantage of the concepts presented in the heuristic algorithm proposed and (ii) can be proved correct