What is the concept of the extended complex plane/Riemann Sphere?

[jimmycricket]

In a report I am writing I want to define the extended complex plane/Riemann Sphere and I would like to check if I grasp the concept properly:
Consider the Euclidean space $\mathbb{R}^3$ where the $x-y$ plane represents $\mathbb{C}$. Consider the sphere with south pole $(0,0,0)$ and north pole $(0,0,\infty)$. For any point in the $x-y$ plane there exists a unique point where the straight line from this point to the north pole crosses the sphere. Hence the complex plane $\mathbb{C}$ can be mapped bijectively onto this sphere.

I know this isn't rigorous but as a worded explanation of the concept does this capture the crux of the matter.

Jim

 

[fresh_42]

The north pole is at infinity, so strictly speaking not in the complex plane anymore; extended yes. Here is a picture and some formulas:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/