Editing Non-Euclidean geometry (section)

==Planar algebras==
In [[analytic geometry]] a [[plane (geometry)|plane]] is described with [[Cartesian coordinate]]s: 

:<math>C = \{ (x,y) : x,y \isin \mathbb{R} \}</math>

The [[point (geometry)|point]]s are sometimes identified with generalized complex numbers {{math|1=''z'' = ''x'' + ''y'' ε}} where ε<sup>2</sup> ∈ { –1, 0, 1}.

The Euclidean plane corresponds to the case {{math|1=ε<sup>2</sup> = &minus;1}}, an [[imaginary unit]]. Since the modulus of {{mvar|z}} is given by
:<math>z z^\ast = (x + y \epsilon) (x - y \epsilon) = x^2 + y^2 ,</math> this quantity is the square of the [[Euclidean distance]] between {{mvar|z}} and the origin.
For instance, {{math|1={''z'' {{!}} ''z z''* = 1} }} is the [[unit circle]].

For planar algebra, non-Euclidean geometry arises in the other cases.
When {{math|1=ε<sup>2</sup> = +1}}, a [[hyperbolic unit]]. Then {{mvar|z}} is a [[split-complex number]] and conventionally {{math|'''j'''}} replaces epsilon. Then
:<math>z z^\ast = (x + y\mathbf{j}) (x - y\mathbf{j}) = x^2 - y^2 \!</math>
and {{math|1={''z'' {{!}} ''z z''* = 1} }} is the [[unit hyperbola]].

When {{math|1=ε<sup>2</sup> = 0}}, then {{mvar|z}} is a [[dual number]].<ref>[[Isaak Yaglom]] (1968) ''Complex Numbers in Geometry'', translated by E. Primrose from 1963 Russian original, appendix "Non-Euclidean geometries in the plane and complex numbers", pp 195–219, [[Academic Press]], N.Y.</ref>

This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of [[slope]] in the dual number plane and [[hyperbolic angle]] in the split-complex plane correspond to [[angle]] in Euclidean geometry. Indeed, they each arise in [[polar decomposition#Alternative planar decompositions|polar decomposition]] of a complex number {{mvar|z}}.<ref>[[Richard C. Tolman]] (2004) ''Theory of Relativity of Motion'', page 194, §180 Non-Euclidean angle, §181 Kinematical interpretation of angle in terms of velocity</ref>