Results on nonsingular, cyclide transition surfaces Original Research Article

In this paper, we generate nonsingular, transition surfaces between natural quadrics using possibly singular cyclides. Both Dupin cyclides (ring, spindle, and horned) and parabolic cyclides (singular and nonsingular) are considered. Necessary and sufficient conditions for the existence of cyclide transition surfaces and easily implementable tests of these conditions are given in each quadric/quadric case. The relationship of the existence of cyclide transition surfaces to the common inscribed sphere condition is examined. Also, we show that except in one special case, blends and joins cannot simultaneously exist between two quadrics.