The swept surface of an elliptic cylinder

Abstract

In this paper, we present a method for computing a piecewise linear approximation to the surface swept by a moving, rotating elliptic cylinder. Our method is a generalization of the imprint point method we developed for computing points on a surface of revolution. The method is based on on identifying grazing points on the surface of revolution at a sequence of positions, and for each position connecting the grazing points with a piecewise linear curve. A collection of grazing curves is joined to approximate the swept surface and stitched into a solid model. Previously this method has been tested on cylinders, toruses, and cones.This work extends the imprint point method to compute the swept surface generated by a general closed surface. The method is demonstrated by translating and rotating an elliptic cylinder along a spline curve. The application of this method is in interference detection of moving machinery such as robots. It is anticipated that a robot would be modeled as a collection of components enclosed in elliptic cylinders, cones, toruses and spheres. Our method can be used to generate the swept volume of these enclosing objects, and used for interference detection and avoidance.1 Introduction Swept surface are frequently needed in many engineering applications. Swept volumes are important for determining interference free trajectories for robots and for generation of gouge free tool paths. The existing methods rely on SEDE [1] or envelop theory [2,3]. Implementation of these techniques results in simultaneous solution non-linear equations which can be tedious to generalize and time consuming to compute.A new method was presented for the creation of swept surfaces swept by surfaces of revolution. This method is similar to envelop theory, but uses some surface properties to speed up the calculation. The idea is based on the observation that when a solid cylinder translates it sweeps a surfaces made up of straight lines. To find points on the straight lines, the cylinder can be imagined to be made of a stack of disks and each disk translates in the same direction. Then the grazing point on one disk on the surface at a particular tool position can be identified by taking a cross product of the normal to the disk with the direction of motion. This basic idea can be extended to a cylinder executing simultaneous translation and rotation, although in this case the grazing curves are not straight lines but curves on the cylinder. The cross product method works for tori and spheres; a more general method can be applied to find grazing curves on an arbitrary surface of revolution, which we review in the next section.In this paper, we further extend the method to determine the surfaces swept by straight and twisted elliptic cylinders (Figures 4 and 5). The concept can be generalized to convex closed surfaces, but examples in this paper will be limited to the elliptic case.In the next section, we will review grazing curves and our earlier work on computing the surface swept by an APT cutter (i.e., a surf...