The determination of surface intersection using subdivision and Polyhedron Intersection Methods

The need for the determination of surface intersection exists in many real-world applications. Such determination is based on a recurrent operation so that its computation should be fast, reliable and suitable for the surfaces involved. Two methods are applied and two new methods are proposed in this work for determining the intersection of two Bezier surfaces: Subdivision, Marching Methods, Polyhedron Intersection, and Hybrid between subdivision and poly-hedron intersection. Combining the two methods: subdivision and polyhedron intersection, a hybrid method is obtained. It integrates the subdivision method with the intersection of two triangulated polyhedra. Thus, the final result is much more precise than that for plane/plane intersection. Consequently, it is closer to the exact result. Unfortunately, none of the three methods can produce the true intersection curves. Further refinement of the intermediate result needs to be performed. Two marching methods, using Tangential or Circular steps, are used to calculate exact intersection points. A major difference between the two techniques is the step size. In the method using tangential steps, the step size has to be fixed and predefined while that of the circular step is dynamic and automatically changed. The selection of which techniques to be used depends on the problem itself: if the result is crooked or winding, the circular step is recommended, otherwise it is sufficient to use the tangential step.