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/