The intersection of two ringed surfaces

Abstract

Presents an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep /spl cup//sub u/C/sup u/ generated by a moving circle. Given two ringed surfaces /spl cup//sub u/C/sub 1//sup u/ and /spl cup//sub v/C/sub 2//sup v/, we formulate the condition C/sub 1//sup u//spl cap/C/sub 2//sup v//spl ne/O (i.e. that the intersection of the two circles C/sub 1//sup u/ and C/sub 2//sup v/ is non-empty) as a bivariate equation /spl lambda/(u,v)= 0 of relatively low degree. Except for some redundant solutions and degenerate cases, there is a rational map from each solution of /spl lambda/(u,v)=0 to the intersection point C/sub 1//sup u//spl cap/C/sub 2//sup v/. Thus, it is trivial to construct the intersection curve once we have computed the zero-set of /spl lambda/(u,v)=0. We also analyze some exceptional cases and consider how to construct the corresponding intersection curves.