Möbius Group

The equation

$${x_1}^2+{x_2}^2+\ldots+{x_n}^2-2x_0x_\infty=0$$

represents an $n$-D Hypersphere $\Bbb{S}^n$ as a quadratic hypersurface in an $(n+1)$-D real projective space $\Bbb{P}^{n+1}$, where $x_a$ are homogeneous coordinates in $\Bbb{P}^{n+1}$. Then the Group $M(n)$ of projective transformations which leave $\Bbb{S}^n$ invariant is called the Möbius group.

References

Iyanaga, S. and Kawada, Y. (Eds.). ``Möbius Geometry.'' §78A in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 265-266, 1980.