Useful geometric properties of the generalized cone

Abstract

The authors present results on geometric properties of the generalized cone, in an effort to utilize it for a shape description system. They first derive the relationship between the generalized cone description and the surface description given by differential geometry. Then they derive expressions for the Gaussian and mean curvatures of a generalized cone, in general, and obtain expressions for some special cases like the torus, the solid of revolution etc. They study the planarity property of the contour generators of a generalized cone, in particular, one with a planar axis. They find that homogeneous generalized cones with planar axes and circular cross sections or constant-size cross sections have planar contour generators in an orthographic side view. An example of such a generalized cone is the torus. However, the contour generators are not planar in a general view. They also study symmetry properties of some generalized cones and find, in particular, that in orthographic projection the contour of the solid of revolution is symmetric about the projection of its axis from any point of view. >