Stereographic Projection  an alternative approach to the extended plane and ilines
The material on these pages gives an alternative approach to the introduction of the point at infinity and the concepts of extended lines and ilines. It is not used to prove any results not proved elsewhere.
Since the World is approximately spherical, geographers have
The easiest solution is to use a projection map. We shall look at stereographic projection.
Definition
In the sketch, we have taken N as the North Pole of the sphere,
Note that the changing P merely changes the scale of the resulting
If we choose Q on P, then the line NQ cuts S{N} exactly once.


The map p is so far defined only on S {N}. To extend it to all of S, we introduce the extended plane P^{+}=P+{Ñ}, where Ñ is called the point at infinity. We say that p(N) = Ñ.
Now p gives a bijection from S to P+{Ñ}.

See The Extended Plane for an alternative approach 
Circles on S
Let F be a plane which cuts S nontrivially.
Although we shall not pursue it far, there is a geometry on S If P is a point on C, then there is a tangent to C at P. This lies in F.


Definition
If C and D are circles on S meeting in P, then the
Of course, the circles will in general meet again, at Q say.
If we view the picture looking in towards the centre of the sphere,
We now prove two theorems which together show that


Theorem 1
Let p denote stereographic projection with vertex N
from Theorem 2 Stereographic projection preserves angles.

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