Riemann sphere $$\hat{\mathbb{C} }

is the one-point compactification of the complex plane $$\mathbb{C}, obtained by identifying the limits of all infinitely extending rays from the origin as one single ``point at infinity.''

An atlas for the Riemann sphere is given by two charts:

$$\hat{\mathbb{C} }\backslash\{\infty\}\rightarrow\mathbb{C}:z\mapsto z and $$\hat{\mathbb{C} }\backslash\{0\}\rightarrow\mathbb{C}:z\mapsto \frac{1}{z}

An atlas for the Riemann sphere is given by two charts:

$$\hat{\mathbb{C} }\backslash\{\infty\}\rightarrow\mathbb{C}:z\mapsto z and $$\hat{\mathbb{C} }\backslash\{0\}\rightarrow\mathbb{C}:z\mapsto \frac{1}{z}

Möbius transform(s) / resource(s)