A Möbius transformation of the plane is a rational function of the form $$f(z)=\frac{az+b}{cz+d}$$ of one complex variable $z$. Here the coefficients $a,b,c,d$ are complex numbers satisfying $ad-bc\neq 0$.
Geometrically, a Möbius transformation can be obtained by stereographic projection of the complex plane onto an admissible sphere in $\mathbb R^3$, followed by a rigid motion of the sphere in $\mathbb R^3$ which maps it to another admissible sphere, followed by stereographic projection back to the plane.
The following simulation shows a square transformed by a Möbius transformation, on the complex plane, and its corresponding stereographic projection in $\mathbb R^3.$
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The University of Queensland