Editing Split-complex number (section)

==Geometry==
<!-- This section is linked from [[Lorentz transformation]] -->
[[Image:Drini-conjugatehyperbolas.svg|thumb|
{{legend-line|solid blue|Unit hyperbola: {{math|1=‖''z''‖ = 1}}}}
{{legend-line|solid green|Conjugate hyperbola: {{math|1=‖''z''‖ = −1}}}}
{{legend-line|solid red|Asymptotes: {{math|1=‖''z''‖ = 0}}}}]]

A two-dimensional real [[vector space]] with the Minkowski inner product is called {{math|(1 + 1)}}-dimensional [[Minkowski space]], often denoted {{tmath|\R^{1,1}.}} Just as much of the [[geometry]] of the Euclidean plane {{tmath|\R^2}} can be described with complex numbers, the geometry of the Minkowski plane {{tmath|\R^{1,1} }} can be described with split-complex numbers.

The set of points

<math display=block>\left\{ z : \lVert z \rVert^2 = a^2 \right\}</math>

is a [[hyperbola]] for every nonzero {{mvar|a}} in {{tmath|\R.}} The hyperbola consists of a right and left branch passing through {{math|(''a'', 0)}} and {{math|(−''a'', 0)}}. The case {{math|1=''a'' = 1}} is called the [[unit hyperbola]]. The [[conjugate hyperbola]] is given by

<math display=block>\left\{ z : \lVert z \rVert^2 = -a^2 \right\}</math>

with an upper and lower branch passing through {{math|(0, ''a'')}} and {{math|(0, −''a'')}}. The hyperbola and conjugate hyperbola are separated by two diagonal [[asymptote]]s which form the set of null elements:

<math display=block>\left\{ z : \lVert z \rVert = 0 \right\}.</math>

These two lines (sometimes called the [[null cone]]) are [[perpendicular]] in {{tmath|\R^2}} and have slopes ±1.

Split-complex numbers {{mvar|z}} and {{mvar|w}} are said to be [[hyperbolic-orthogonal]] if {{math|1=⟨''z'', ''w''⟩ = 0}}. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the [[Minkowski space#Causal structure|simultaneous hyperplane]] concept in spacetime.

The analogue of [[Euler's formula]] for the split-complex numbers is

<math display=block>\exp(j\theta) = \cosh(\theta) + j\sinh(\theta).</math>

This formula can be derived from a [[power series]] expansion using the fact that [[hyperbolic cosine|cosh]] has only even powers while that for [[hyperbolic sine|sinh]] has odd powers.<ref>James Cockle (1848) [https://www.biodiversitylibrary.org/item/20157#page/452/mode/1up On a New Imaginary in Algebra], ''Philosophical Magazine'' 33:438</ref> For all real values of the [[hyperbolic angle]] {{mvar|θ}} the split-complex number {{math|1=''λ'' = exp(''jθ'')}} has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as {{mvar|λ}} have been called [[versor#Hyperbolic versor|hyperbolic versors]].

Since {{mvar|λ}} has modulus 1, multiplying any split-complex number {{mvar|z}} by {{mvar|λ}} preserves the modulus of {{mvar|z}} and represents a ''hyperbolic rotation'' (also called a [[Lorentz boost]] or a [[squeeze mapping]]). Multiplying by {{mvar|λ}} preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.

The set of all transformations of the split-complex plane which preserve the modulus (or equivalently, the inner product) forms a [[group (mathematics)|group]] called the [[generalized orthogonal group]] {{math|O(1, 1)}}. This group consists of the hyperbolic rotations, which form a [[subgroup]] denoted {{math|SO{{sup|+}}(1, 1)}}, combined with four [[discrete mathematics|discrete]] [[Reflection (mathematics)|reflection]]s given by

<math display=block>z \mapsto \pm z</math> and <math>z \mapsto \pm z^*.</math>

The exponential map

<math display=block>\exp\colon (\R, +) \to \mathrm{SO}^{+}(1, 1)</math>

sending {{mvar|θ}} to rotation by {{math|exp(''jθ'')}} is a [[group isomorphism]] since the usual exponential formula applies:

<math display=block>e^{j(\theta + \phi)} = e^{j\theta}e^{j\phi}.</math>

If a split-complex number {{mvar|z}} does not lie on one of the diagonals, then {{mvar|z}} has a [[polar decomposition#Alternative planar decompositions|polar decomposition]].