Isotropic Möbius Geometry and i-M Circles on Singular Isotropic Cyclides

The Möbius geometry of \({\mathbb R}^3\) has an isotropic counterpart in \(\mathbb {R}^{3}_{++0}\). We describe the isotropic Möbius model of surfaces in \(\mathbb {R}^{3}_{++0}\) and show how the degree of a surface changes under i-M inversions while the number of families of i-M circles remain constant. This gives us a generalization of the classification of families of lines and i-M circles on quadratic surfaces in \(\mathbb {R}^{3}_{++0}\) to isotropic cyclides with real singularities, containing up to 4 such families.