The swept surface of an elliptic cylinder

Abstract

In this poster, 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 [1]. 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. We reduce the problem of finding grazing points on the elliptic cylinder to that of finding points on an ellipse by slicing the elliptic cylinder with planes perpendicular to the tool axis. The method for finding the grazing points an a circular slice of a surface of revolution works because the normals to the surface along each circle pass through the axis of revolution. This allowed us to express the direction of motion of a point on the surface as the sum of the motion of a point on the axis plus a rotation around that point on the axis. This simple method for finding grazing points fails for an elliptic cylinder because the normals of a slice of the elliptic cylinder do not pass through center of ellipse. Worse, in the case of a twisted cylinder, the normals to the surface do not lie in the plane of the elliptical slice. However, these problems are readily resolved by setting up a small system of equations that we can solve for the grazing points on an elliptic slice. We begin by solving the problem for the elliptic cylinder, after which we will show how to generalize the solution to the twisted elliptic cylinder.