Klein's Geometry in Group

This is the "New Geometry" introduced by Klein 200 years ago in his Erlangan Program (his PhD Thesis).

Rigid Motion is defined by 3 components: Translation, Rotation and Reflection.

If fixed at one point (origin), there is no translation, only Rotation ρ(θ) and Reflection r(θ) are possible around that fixed point.

We can prove r(θ) and ρ(θ) form a Group O2, namely Orthogonal Group with this property:

$latex A^{T}. A = A. A^{T} = I $

where A can be any of the 2 matrices represented by ρ(θ)

or r(θ),

$latex A^{T} $ is the transpose of A (columns => rows, rows => columns).

1. Rotation

ρ(θ)=

(cos θ  -sin θ)

(sin θ   cos θ)

2. Reflection

r(θ) =

(cos θ   sin θ)

(sin θ   -cos θ)

when θ =0,

r0 =

(1 0)

(0 -1)

=> r(θ) = ρ(θ).r0

Change of Reference Axis:

 Make a shift from fixed origin A to another fixed original A' by a translation t(α), the first Orthogonal Group O at A and the second Orthogonal Group O' at A' are related by:

O' = t(α).O.t^-1(α)
ρ'(θ) = t(α).ρ(θ).t^-1(α)
r'(θ) = t(α).r(θ).t^-1(α)

Note: this looks analogous to Conjugate groups (Normal Subgroups).

Einstein Relativity interpreted by Rigid Motion (M4).

If first origin A is the Earth, second origin A' is the spaceship traveling at speed of light, ie t(α) = c

O' = t(α).O.t^-1(α) ; O & O' ∈ O4

<=> O'.t(α) = t(α).O