Geometrical properties of a generalized cone and its 2D image

The generalized cone is a simple 3D shape that is produced by sweeping a planar cross-section along a curve. Many complex and articulated 3D objects can be represented by combining generalized cones. It has been shown that generalized cones play an important role in our visual system for perceiving the shapes of these objects and recognizing them. In this study, we analyzed the geometrical properties of generalized cones and their 2D images and found that there are invariant features in the images of the generalized cones under both 2D orthographic and perspective projections that facilitate the recovery of the 3D shapes of the cones from the images. We found that the 3D translational symmetry of generalized cones can be analyzed using tools designed for 3D mirror-symmetry.