Cyclides in computer aided geometric design

Abstract

The Dupin cyclides provide a generalisation of all the surfaces conventionally used in solid modelling —the plane, cylinder, cone, sphere and torus. They offer the potential for a significant extension of the geometric capabilities of solid modelling systems. This paper presented a rational biquadratic representation for cyclide surface patches, and also gives details of an important parallel offset property of these surfaces. The main results of the paper are however concerned with the use of cyclides for blending in solid modelling. It is shown that they can be used to give exact G 1-continuous blends between pairs of other cyclide surfaces in several situations which frequently occur in engineering design. The basic principle is extended by using a new theorem due to Sabin and by employing piecewise cyclide blending surfaces to give still greater generality. All the blends created are subject to the restriction that their boundaries must be circles lying on the surfaces being blended. Such blends occur surprisingly often in the design of real objects