The Klein View of Geometry

The inversive group

In the inversive geometry pages, we introduced inversive geometry.
This is defined on the set C⁺ = C_u{∞}. We defined i-lines and inversion
in i-lines. Here, we determine the group I(2) generated by such inversions.
This is the inversive group. Its elements are inversive transformations.
We know that each inversion preseves the size of angles, but reverses their sense.
Thus I(2) has a subgroup I⁺(2) consisting of those which preserve the sense of angles.
These are known as direct transformations. The others are indirect.
An i-line is either a circle or an extended line (a euclidean line plus the point ∞).
In the case of an extended line L, inversion with respect to L is simply reflection
in the euclidean line, extended by mapping ∞ to ∞. By Theorem E1, E(2) is
generated by the reflections. The extended versions generate a subgroup E(2)
of I(2) which is isomorphic to E(2).
In particular, I(2) contains extended versions of

reflection in the real axis: r₀(z)=z*,
the translations : t_c(z)=z+c,
We can also extend elements of S(2) to C⁺ by defining s(∞)=∞ for each sεS(2)
These extended versions form a group S(2) isomorphic to S(2). This is also a
subgroup of I(2), as we now show. The result is of interest in its own right.
The ∞ Theorem
S(2) is the subgroup of I(2) consisting of elements which fix ∞.
Proof of the ∞ theorem
However, not all elements of I(2) belong to S(2).
For example, i_r, inversion in the circle |z|=r is given by

i_r(z)=r²/z*, when z ≠ 0, ∞
i_r(0)=∞,
i_r(∞)=0.
Note that the elements k_r = i_ror₀ have

k_r(z) = r²/z, z ≠ ∞,
k_r(∞) = ∞.
They belong to I⁺(2) as composites of two inversions.

We observe that the elements t_c and k_r of I⁺(2) can be written as
Mobius transformations, i.e. in the form m(z) = (αz+β)/(γz+δ),
where α, β, γ, δ are in C, with αδ - βγ ≠ 0.
The set of all such transformations is the Mobius Group
Find out about the Mobius Group

Theorem I1
I⁺(2) = M(2).
Proof of Theorem I1

This leads immediately to the corresponding theorem for the entire inversive group:
Theorem I2
If tεI(2) is direct, then t(z) = m(z) for some m in M(2).
If tεI(2) is indirect, then t(z) = m(z*) for some m in M(2).
Proof
If t is direct, we have the result by Theorem I1.
If t is indirect, then tor₀ is direct, so by Theorem I1, tor₀ = mεM(2).
Then t = mor₀^-1 = mor₀, as required.
With our knowledge of the groups, it is now possible to prove further results on
the geometry, in particular, the Fundamental Theorem of Inversive Geometry.

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In the inversive geometry pages, we introduced inversive geometry. This is defined on the set C⁺ = C_u{∞}. We defined i-lines and inversion in i-lines. Here, we determine the group I(2) generated by such inversions. This is the inversive group. Its elements are inversive transformations. We know that each inversion preseves the size of angles, but reverses their sense. Thus I(2) has a subgroup I⁺(2) consisting of those which preserve the sense of angles. These are known as direct transformations. The others are indirect. An i-line is either a circle or an extended line (a euclidean line plus the point ∞). In the case of an extended line L, inversion with respect to L is simply reflection in the euclidean line, extended by mapping ∞ to ∞. By Theorem E1, E(2) is generated by the reflections. The extended versions generate a subgroup E(2) of I(2) which is isomorphic to E(2). In particular, I(2) contains extended versions of reflection in the real axis: r₀(z)=z, the translations : t_c(z)=z+c, We can also extend elements of S(2) to C⁺ by defining s(∞)=∞ for each sεS(2) These extended versions form a group S(2) isomorphic to S(2). This is also a subgroup of I(2), as we now show. The result is of interest in its own right. The ∞ Theorem* S(2) is the subgroup of I(2) consisting of elements which fix ∞. Proof of the ∞ theorem However, not all elements of I(2) belong to S(2). For example, i_r, inversion in the circle \|z\|=r is given by i_r(z)=r²/z*, when z ≠ 0, ∞ i_r(0)=∞, i_r(∞)=0. Note that the elements k_r = i_ror₀ have k_r(z) = r²/z, z ≠ ∞, k_r(∞) = ∞. They belong to I⁺(2) as composites of two inversions.
We observe that the elements t_c and k_r of I⁺(2) can be written as Mobius transformations, i.e. in the form m(z) = (αz+β)/(γz+δ), where α, β, γ, δ are in C, with αδ - βγ ≠ 0. The set of all such transformations is the Mobius Group Find out about the Mobius Group
Theorem I1 I⁺(2) = M(2). Proof of Theorem I1
This leads immediately to the corresponding theorem for the entire inversive group: Theorem I2 If tεI(2) is direct, then t(z) = m(z) for some m in M(2). If tεI(2) is indirect, then t(z) = m(z) for some m in M(2). Proof* If t is direct, we have the result by Theorem I1. If t is indirect, then tor₀ is direct, so by Theorem I1, tor₀ = mεM(2). Then t = mor₀^-1 = mor₀, as required. With our knowledge of the groups, it is now possible to prove further results on the geometry, in particular, the Fundamental Theorem of Inversive Geometry.