Superquadrics with rational and irrational symmetry

Superquadrics are important models for part level-description in computer graphics and computer vision. Their power resides in their compact characterization. To further extend the representational power of superquadrics several methods have been proposed for local and global deformations. This notwithstanding, it is very difficult, for example, to represent polygons or polyhedrons using classical superquadrics. In this paper we present a new approach to model natural and abstract shapes for computer graphics, using a Generalized Superellipse Equation, which solves the problem of symmetries. Our approach provides an elegant analytical way to fold or unfold the coordinate axis systems like a fan, thereby generalizing superquadrics and superellipses (and hyperspheres in general) to supershapes for any symmetry, rational or irrational. Very compact representations of various shapes with different symmetries are possible and this provides opportunities for CAD at the level of graphics kernels, CAD-users and their clients. For example, parts and assemblies can be represented in very small file sizes allowing to use the 3-D solid model throughout the design and manufacturing process. Our approach presents an elegant way to use 3-D models both for solid modeling and boundary representations, for rigid as well as soft models.