Do blending and offsetting commute for Dupin cyclides?

A common method for constructing blending Dupin cyclides for two cones having a common inscribed sphere of radius r > 0 involves three steps: (1) computing the (−r)-offsets of the cones so that they share a common vertex, (2) constructing a blending cyclide for the offset cones, and (3) computing the r-offset of the cyclide. Unfortunately, this process does not always work properly. Worse, for some halfcones cases, none of the blending cyclides can be constructed this way. This paper studies this problem and presents two major contributions. First, it is shown that the offset construction is correct for the case of = −r, where is the signed offset value; otherwise, a procedure must be followed for properly selecting a pair of principal circles of the blending cyclide. Second, based on Shene’s construction in “Blending two cones with Dupin cyclides”, CAGD, Vol. 15 (1998), pp. 643– 673, a new algorithm is available for constructing all possible blending cyclides for two half-cones. This paper also examines Allen and Dutta’s theory of pure blends, which uses the offset construction. To help overcome the difficulties of Allen and Dutta’s method, this paper suggests a new algorithm for constructing all possible pure blends. Thus, Shene’s diagonal construction is better and more reliable than the offset construction.