English: Homothetic transformation or inversion to make two decentralized conics concentric.

Given two conics ΔP and ΔQ, which are homothetic but not axial and not concentric with each other, the goal is to make them concentric using projective geometry. To do so, one conic, for example, ΔP representing the outer conic, is fixed, and the pole I of the two coincident polars with respect to the two conics ΔP and ΔQ is determined. The point I and its inverse I^ (with respect to ΔP and ΔQ) act as conjugate poles with respect to the same conics. In this way, we can make the decentralized conics concentric by establishing I as the center of homothety (for the conics similar to ΔQ) to scale ΔQ until its center coincides with that of ΔP. This procedure is faster and does not require the inversion of ΔP and ΔQ with respect to another inversion conic Σ with center I.

https://www.researchgate.net/profile/Hasan-Isawi