For the continuous case I believe what you show are confocal quadrics.
For the discrete version though, according to the paper by Simon Schleicher et al that your first figure comes from, to get the planar quad mesh discretization there with the touching incircles, they used the quasiisothermic mesh layout approach of Sechelmann et al, as described in this paper.
The touching incircle energy described in that paper is actually something I added as part of the latest Kangaroo release.
Here’s an example of applying it to an ellipsoid:
isothermic_ellipsoid.gh (17.4 KB)
I’ve been quite liking the result of applying this energy on more general quad meshes too:
edit - this paper from Bobenko et al is also relevant https://arxiv.org/abs/1511.01777