Stereographic Projection - an alternative approach to the extended plane and i-lines

The material on these pages gives an alternative approach to
the introduction of the point at infinity and the concepts of
extended lines and i-lines.

It is not used to prove any results not proved elsewhere.

Since the World is approximately spherical, geographers have
considered the problem of representing the surface on a plane.

The easiest solution is to use a projection map.
There are many ways to do this.

We shall look at stereographic projection.

Let S be a sphere with centre O, and let N be a point on S.
let P be a plane normal to NO, not through N.
For a point P on S other than N, the
stereographic projection of P with vertex N onto P
is the point Q where NP meets P. We denote it by p(P).

In the sketch, we have taken N as the North Pole of the sphere,
so that P is parallel to the Equator.

Note that the changing P merely changes the scale of the resulting
picture since the image Q always lies on NP.

If we choose Q on P, then the line NQ cuts S-{N} exactly once.
Thus p is one-one and maps S-{N} onto P.

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}.
The inverse p -1 maps P to S-{N}, and Ñ to N.

See The Extended Plane
for an alternative approach
Circles on S

Let F be a plane which cuts S non-trivially.
Then F meets S in a circle C. You will find a proof here.
Since a circle is a plane figure, every circle on S arises in this way.

Although we shall not pursue it far, there is a geometry on S
which has as its lines the circles on S. We shall, however, show
that angles can be defined in this geometry.

If P is a point on C, then there is a tangent to C at P. This lies in F.


If C and D are circles on S meeting in P, then the
angle between C and D at P is the angle between the tangents at P.

Of course, the circles will in general meet again, at Q say.
Think of reflecting in the plane bisecting PQ and normal to PQ.
This interchanges P and Q and the tangent pairs,
so the angles at P and Q are of equal magnitude.

If we view the picture looking in towards the centre of the sphere,
we may attach a sense to angles. Since the above proof used a
reflection, we see that the angles at P and Q have opposite sense.

We now prove two theorems which together show that
this geometry on S is actually a model of inversive geometry!

Theorem 1

Let p denote stereographic projection with vertex N from
the sphere S to the plane P. Then, for any circle C on S,
p(C) is a line if N lies on C, and a circle otherwise.

Proof of Theorem 1

Theorem 2

Stereographic projection preserves angles.

Proof of Theorem 2

Main inversive page