Not sure about the question, please help correct it to convey properly or I could even delete if not meaningful.
The common intersecting plane is through the north pole of top unit sphere. By Möbius transformation we have stereographic mappings between small blue circles of top sphere through its north pole and straight lines in the horizontal complex plane via the Möbius transformation
$$ f(z)=\frac{az+b}{cz+d}$$
discussed for example in Stereographic Projn/Moebius Transfrmn. The geometrical connection appears to be through inversions/bipolar coordinates.
If now a second larger sphere is brought into mutual contact with its north pole, the complex plane and the south pole of the top unit sphere... can we have another complex function to describe the second set of small red circle mappings with the common tangent plane?
If so, could it belong to the same Möbius group?