Point Projection Methods Using Torus Patch Approximation And Torus/torus Intersection Algorithms

A torus is often used in the design of mechanical parts and the cutter heads ofnumerically controlled machine tools. It is needed to compute the intersection curvesor the minimum distance of a torus and a surface in CAD/CAM systems. This dis-sertation focuses on the computation of intersection and the minimum distance of twotori and point projection on surfaces based on torus patch approximation. The maincontributions are summarized as follows.A method for the collision detection and the minimum distance computation be-tween two tori is presented. This dissertation proves that the Hausdorf distance be-tween two circles in three-dimensional space can be obtained by computing theircollinear normal points, which can be calculated by solving an equation of degreeeight. With classification and comparison of the collinear normal points, the minimumdistance and the Hausdorf distance between these two circles are obtained. This dis-sertation gives the sufcient and necessary conditions of the three types of positionrelationship (i.e. inclusion, disjunction and intersection) between two tori and provesthat position relationship between two tori relates to not only the minimum distancebut also the two directed Hausdorf distances between their major circles. And then theminimum distance between two tori is calculated. The method can be carried out inreal time. Compared with existing methods, the method has the advantage that it canmake correct judgement in the case of that a torus completely includes another torusand compute the distance between them.An algorithm for torus/torus intersection is presented. The pre-image of the inter-section in the parametric space of one torus is represented by an implicit equation. Thepre-image is divided into one-valued function curve segments by characteristic points.The topological feature of these characteristic points is analyzed and the structure of thepre-image is obtained. Intersection curves satisfying required precision are generatedby a self-adaptive refinement method. Compared with the tracing method, the algo- rithm can overcome the drawbacks of straying and loop missing of the tracing method.Additionally, the points on the intersection curves obtained by the algorithm are com-puted by the analytic method, whose precision is higher than the points obtained byiteration when using the tracing method. The tracing method can only control theprecision roughly by estimating the advance step, while the algorithm can control theprecision accurately.An algorithm for point projection on surfaces is presented. This dissertation pro-poses a local surface approximation method with a torus patch. Based on it, a secondorder geometric iteration algorithm for point projection on surfaces is proposed. Ineach iteration, a second order osculating torus patch to the surface at the previous pro-jection is constructed. Then the test point is projected onto the torus patch to computethe next projection. The algorithm can apply to not only parametric surfaces but alsoimplicit surfaces, and its stability and efciency are better than the existing methods.