Editing Length contraction (section)

=== Geometrical considerations ===

[[File:Slabs.svg|thumb|250px|Cuboids in Euclidean and Minkowski spacetime]]

Additional geometrical considerations show that length contraction can be regarded as a ''trigonometric'' phenomenon, with analogy to  parallel slices through a [[cuboid]] before and after a ''rotation'' in '''E'''<sup>3</sup> (see left half figure at the right).  This is the Euclidean analog of ''boosting'' a cuboid in '''E'''<sup>1,2</sup>.  In the latter case, however, we can interpret the boosted cuboid as the ''world slab'' of a moving plate.

''Image'': Left: a ''rotated cuboid'' in three-dimensional euclidean space '''E'''<sup>3</sup>. The cross section is ''longer'' in the direction of the rotation than it was before the rotation. Right: the ''world slab'' of a moving thin plate in Minkowski spacetime (with one spatial dimension suppressed) '''E'''<sup>1,2</sup>, which is a ''boosted cuboid''. The cross section is ''thinner'' in the direction of the boost than it was before the boost. In both cases, the transverse directions are unaffected and the three planes meeting at each corner of the cuboids are ''mutually orthogonal'' (in the sense of '''E'''<sup>1,2</sup> at right, and in the sense of '''E'''<sup>3</sup> at left).

In special relativity, [[Poincaré group|Poincaré transformations]] are a class of [[affine transformation]]s which can be characterized as the transformations between alternative [[Cartesian coordinates|Cartesian coordinate charts]] on [[Minkowski spacetime]] corresponding to alternative states of [[inertial frame|inertial motion]] (and different choices of an [[origin (mathematics)|origin]]).  Lorentz transformations are Poincaré transformations which are [[linear transformation]]s (preserve the origin). Lorentz transformations play the same role in Minkowski geometry (the [[Lorentz group]] forms the ''isotropy group'' of the self-isometries of the spacetime) which are played by [[rotation]]s in euclidean geometry.  Indeed, special relativity largely comes down to studying a kind of noneuclidean [[trigonometry]] in Minkowski spacetime, as suggested by the following table:

{| class="wikitable" style="text-align:center"
|+ Three plane trigonometries
|-
! Trigonometry 
! Circular !! Parabolic !! Hyperbolic
|-
! Kleinian Geometry 
| Euclidean plane      || Galilean plane        || Minkowski plane
|-
! Symbol 
| '''E'''<sup>2</sup>  || '''E'''<sup>0,1</sup> || '''E'''<sup>1,1</sup>
|-
! Quadratic form 
| Positive definite || Degenerate || Non-degenerate but indefinite
|- 
! Isometry group 
|  '''E'''(2) ||  '''E'''(0,1) ||  '''E'''(1,1)
|-
! Isotropy group 
| '''SO'''(2) || '''SO'''(0,1) || '''SO'''(1,1)
|-
! Type of isotropy
| Rotations || Shears || Boosts  
|- 
! Algebra over R 
| [[Complex number]]s || [[Dual number]]s || [[Split-complex number]]s 
|-
! ε<sup>2</sup> 
| −1 || 0 || 1
|-
! Spacetime interpretation 
| None || Newtonian spacetime || Minkowski spacetime
|-
! Slope 
| tan φ = m || tanp φ = u  || tanh φ = v
|-
! "cosine" 
| cos φ = (1 + m<sup>2</sup>)<sup>−1/2</sup> || cosp φ = 1 || cosh φ = (1 − v<sup>2</sup>)<sup>−1/2</sup>
|-
! "sine" 
| sin φ = m (1 + m<sup>2</sup>)<sup>−1/2</sup> || sinp φ = u || sinh φ = v (1 − v<sup>2</sup>)<sup>−1/2</sup> 
|- 
! "secant" 
| sec φ = (1 + m<sup>2</sup>)<sup>1/2</sup> || secp φ = 1 || sech φ = (1 − v<sup>2</sup>)<sup>1/2</sup>
|-
! "cosecant" 
| csc φ = m<sup>−1</sup> (1 + m<sup>2</sup>)<sup>1/2</sup> || cscp φ = u<sup>−1</sup> || csch φ = v<sup>−1</sup> (1 − v<sup>2</sup>)<sup>1/2</sup>
|}