A New Torus Bounding for Line-Torus Intersection Vaclav Skala Department of Computer Science and Engineering Faculty of Applied Sciences, University of West Bohemia CZ 306 14 Plzen, Czech Republic http://www.VaclavSkala.eu Abstract—Intersection algorithms are very important in computation of geometrical problems. An intersection of a line with linear or quadratic surfaces is well done, however a line intersection with other surfaces is more complex and time consuming. In this case the object is usually closed into a simple bounding volume to speed up the cases when the given line cannot intersect the given object. In this paper a new formulation of the line-torus intersection problem is given and new specification of the bounding volume for a torus is given. The presented approach is based on an idea of a line intersection with an envelope of rotating sphere that forms a torus. Due to this approach new bounding volume can be formulated which is more effective as it enables to detect cases when the line passes the “hole” of a torus. Keywords—torus intersection; bounding volume; intersection; line clipping; CAD systems I. INTRODUCTION Intersection algorithms play a significant role in all geometric problems and CAD/CAM systems. Intersection algorithms are well documented for linear cases, e.g. line-plane or line-triangle etc., and also for some specific non-linear surfaces like line-sphere intersection etc. However, there are other objects like bicubic parametric patches, torus etc. In this case computation of intersection points is more complex and usually complex formula or iterative formula are to be used. Intersection of a line and closed surface can be considered as generalized well known clipping problem. Intersection of a line or ray with a surface is the key problem solved in all ray- tracing techniques. Due to the computational complexity a bounding volumes are used to detect cases when a line cannot intersect the given object. In this paper we present torus-line intersection problem, which leads to a quartic equation in principle, and show other possible formulations of line-torus intersection problem. These reformulations lead to a formulation of a new problem – generalized line clipping by an envelope (convex or non- convex) of parametric closed surfaces. II. TORUS LINE INTERSECTION A. Traditional approach Torus-line intersection is actually a solution of a line in E 3 usually given in a parametric form as                 (1) and a torus, which is generally a surface of the 4 th order and can be given as :                     (2) Note that the z axis is rotational. This torus equation can be reformulated as                 (3) where                (4) Fig. 1. Torus Fig. 2. Bounding Volume r x t min p A p C p B p D t max R 1 R y Proceedings of the 2013 International Conference on Applied Mathematics and Computational Methods in Engineering 225 AMCME 2013 Conference on Applied Mathematics and Computational Methods in Engineering, pp.225-230, ISBN 978-1-61804-200-2, Rhodos, Greece, 2013