Geodesic Cutting and Grinding curve

Abstract

An algorithm to compute a geodesic path over an arbitrary topological mesh was presented. Given two points over an arbitrary topological mesh, an initial approximation of discrete geodesic between two points is obtained by using Fast Marching Method. Initial approximation is iteratively corrected by getting the intersection point of two space straight lines. Thus a discrete geodesic between two points is obtained. The method avoids classing mesh vertex and computing equation of a plane where mesh vertex is so that it is practicable for computing a discrete geodesic over an arbitrary topological mesh. Furthermore, an algorithm of geodesic Cutting and Grinding curve is given over an arbitrary topological mesh. Geodesic free curves over arbitrary topological mesh were constructed by using the algorithm, such as geodesic B-spline curve of degree two. Some important properties were proved for geodesic Cutting and Grinding curve, such as convex hull, local adjustment and convexity preserving properties. Under Visual C++6.0, some examples of geodesic curves on convex and concave triangulated surface, geodesic curves on quadrangular subdivided surface and geodesic B-spline curve of degree two on triangulated arbitrary topological surface were given by OpenGL. Examples show that two algorithms are correct, stable, fast, easy in implementation. There is a good effect of simulation for all examples.