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

Determine whether the trisectors arising in the canonical symmetric line–line–circle family in R^3 for parameter values k<0 and k>1 are projectively homeomorphic, i.e., whether they realize the same real projective topology; if not, ascertain that these two outer chambers correspond to genuinely different curve types.

Background

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.