Global comparison between outer chambers (k<0 vs k>1) for the canonical line–line–circle trisector

The paper studies the canonical symmetric family of trisectors formed by two skew lines and one circle in R^3, parameterized by (k, R, t) with R>0 and t≠0. The authors compute the certified topological transition set and prove that topology can change only at k=0 and k=1.

They establish affine smoothness throughout the admissible region and analyze the unique projective point at infinity. Using slope-coordinate analysis, they certify local topological constancy within each chamber of the parameter complement, which, after accounting for the symmetry t↦−t, yields three regimes: k<0, 0<k\<1, and k\>1.

While the chamberwise analysis shows stability within each region, the paper explicitly leaves open whether the two outer chambers (k<0 and k>1) exhibit the same real projective topology or represent distinct curve types, motivating a global comparison across these chambers.