Basic lines, axes and geometric modeling on implicit algebraic surfaces

We introduce some new concepts such as basic lines and divide the blending problems into two classes, Class I blending and Class II blending. We further develop the results [Li, A software system on blending of pipe surfaces, Master Thesis, Jilin University, June, 2000; Wu et al., Blending of implicit algebraic surfaces, Proceedings of the ASCM 1995, Beijing, China, August 18–20, 1995, pp. 125–131; Wu and Han, Newton form formulae for the n-way blending of quadratic surfaces, Proceedings of IJCC Workshop on Digital Engineering, August 21–22, 2003; Wu and Zhou, On blending of several quadratic algebraic surfaces, Comput. Aided Geom. Design 17(9) (2000) 759–766; Wu, Zhou and Feng, Blending two quadratic algebraic surfaces with cubic surfaces, Proceedings of ASCM, 1996, Kobe, Japan, August 20–22, 1996, pp. 73–79] and analyze the geometric condition for the Class I blending problem by using characteristic roots. We obtain the lowest degree of Class II blending according to the different position of the axes of the primary surfaces. And we derive the efficient parameterization method for a kind of cubic blending surfaces on the basis of the results obtained by Berry and Patterson [Implicitization and parameterization of nonsingular cubic surfaces, Comput. Aided Geom. Design 19 (2001) 723–738].