Surfaces of revolution belong to an important class of geometric models with simpler shape characteristics and useful geometric properties. In CAD and geometric modeling systems, solid models are often composed of planes and surfaces of revolution. Thus, it is important to develop efficient algorithms that compute the intersection of planes and surfaces of revolution and the intersection between two surfaces of revolution. Despite their importance, there seems to be a lack of reported work in this area....[ Read more ]

Surfaces of revolution belong to an important class of geometric models with simpler shape characteristics and useful geometric properties. In CAD and geometric modeling systems, solid models are often composed of planes and surfaces of revolution. Thus, it is important to develop efficient algorithms that compute the intersection of planes and surfaces of revolution and the intersection between two surfaces of revolution. Despite their importance, there seems to be a lack of reported work in this area.

In this thesis, we propose two efficient algorithms to compute the intersection set: (1) planar sections of a surface of revolution and (2) the intersection between two surfaces of revolution. These algorithms first subdivide the i-th surface of revolution into a series of consecutive coaxial truncated cones T_ij or revolute quadrics according to a user supplied error bound. Thus, the problem is reduced to finding the intersections between truncated cones and planes or revolute quadrics and planes.

Our planar intersection algorithm detects intervals of consecutive truncated cones or revolute quadrics that contain non-empty intersections using a bounding volume tree. Then, it finds planar sections of each resulting truncated cone or revolute quadric that would intersect the plane. Finally joining them together into complete intersection curves. Our representations of complete intersection curves (piecewise conics, piecewise quadratic rational Bezier arcs, NURBS curves) facilitates 2D operations on the intersection plane and generates 3D points quickly on demand. We compare our representation with the exact planar intersection representation we derived.

Our surface-surface intersection algorithm detects intervals of consecutive truncated cones that contain non-empty intersections using binary search. Then, for each truncated cone T_1i within the valid intersection interval of the first revolution surface, we find the intersection curves S_ij between T_1i and all truncated cones T_2j within the valid intersection interval of the other revolution surface. Finally, we join all S_ij together into a complete curve representation which generates 3D points quickly on demand.