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We show that conformal transformations on the generalized Minkowski space$\mathbb{R}^{p,q}$ map hyperboloids and affine hyperplanes into hyperboloidsand affine hyperplanes. We also show that this action on hyperboloids andaffine hyperplanes is transitive when $p$ or $q$ is $0$, and that this actionhas exactly three orbits if $p, q \ne 0$. Then we extend these results tohyperboloids and affine planes of arbitrary dimension. These propertiesgeneralize the well-known properties of M\"{o}bius (or fractional linear)transformations on the complex plane $\mathbb{C}$.

Geometric Properties of Conformal Transformations on $\mathbb{R}^{p,q}$