IV.5 - Fast Generation of Ellipsoids

Abstract

This chapter describes the fast generation of ellipsoids. Among the many methods for generating a polygonal approximation to an ellipse in standard position, the one shown in the chapter is simple and common. The idea is to subdivide the angles in plane around the center of the unit circle equally, forming a polygonal approximation to the unit circle that is then scaled by a in the x-axis, and by b in the y-axis. Vertices get denser around the sharper-axis direction of the ellipse rather than around the smoother-axis one, which is a merit of the method. When the degree of approximation n is given and the approximation is to yield a symmetric polygon of 4·n vertices, the whole 4n vertices on the unit circle need not actually be computed. The idea for 2D ellipse generation is directly generalized to 3D ellipsoid generation. Vertices on the surface of the unit sphere are generated by subdividing the angles in space around the center of the unit sphere regularly, forming a polyhedral approximation to the unit sphere, which is then scaled appropriately in each of the x, y, and z-axes.