Editing Squeeze mapping (section)

==Group theory==
[[File:Hyperbolic sector squeeze mapping.svg|250px|right|thumb|A squeeze mapping moves one purple [[hyperbolic sector]] to another with the same area. <br>It also squeezes blue and green [[rectangle]]s.]]
In 1688, long before abstract [[group theory]], the squeeze mapping was described by [[Euclid Speidell]] in the terms of the day: "From a Square and an infinite company of Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same properties or  affections of any Hyperbola inscribed within a Right Angled Cone."<ref>Euclid Speidell (1688) [https://books.google.com/books?id=9l6zSrUQL0UC&q=logarithmotechnia Logarithmotechnia: the making of numbers called logarithms] from [[Google Books]]
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If {{math|''r''}} and {{math|''s''}} are positive real numbers, the [[Function composition|composition]] of their squeeze mappings is the squeeze mapping of their product. Therefore, the collection of squeeze mappings forms a [[one-parameter group]] isomorphic to the [[multiplicative group]] of [[positive real numbers]]. An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles.

From the point of view of the [[classical group]]s, the group of squeeze mappings is {{math|SO<sup>+</sup>(1,1)}}, the [[identity component]] of the [[indefinite orthogonal group]] of 2×2 real matrices preserving the [[quadratic form]] {{math|''u''<sup>2</sup> − ''v''<sup>2</sup>}}. This is equivalent to preserving the form {{math|''xy''}} via the [[change of basis]]

:<math>x=u+v,\quad y=u-v\,,</math>

and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group {{math|SO(2)}} (the connected component of the definite [[orthogonal group]]) preserving quadratic form {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} as being ''circular rotations''.

Note that the "{{math|SO<sup>+</sup>}}" notation corresponds to the fact that the reflections

:<math>u \mapsto -u,\quad v \mapsto -v</math>

are not allowed, though they preserve the form (in terms of {{math|''x''}} and {{math|''y''}} these are {{math|''x'' ↦ ''y'', ''y'' ↦ ''x''}} and {{math|''x'' ↦ −''x'', ''y'' ↦ −''y'')}}; the additional "{{math|+}}" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity component because the group {{math|O(1,1)}} has {{math|4}} [[connected component (topology)|connected component]]s, while the group {{math|O(2)}} has {{math|2}} components: {{math|SO(1,1)}} has {{math|2}} components, while {{math|SO(2)}} only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups {{math|SO ⊂ SL}} &ndash; in this case {{math|SO(1,1)&nbsp;⊂&nbsp;[[SL2(R)|SL(2)]]}} – of the subgroup of hyperbolic rotations in the [[special linear group]] of transforms preserving area and orientation (a [[volume form]]). In the language of [[Möbius transformation]]s, the squeeze transformations are the [[SL2(R)#Hyperbolic elements|hyperbolic elements]] in the [[SL2(R)#Classification of elements|classification of elements]].

A [[geometric transformation]] is called '''conformal''' when it preserves  angles. [[Hyperbolic angle]] is defined using area under ''y'' = 1/''x''. Since squeeze mappings preserve areas of transformed regions such as [[hyperbolic sector]]s, the angle measure of sectors is preserved.  Thus squeeze mappings are ''conformal'' in the sense of preserving hyperbolic angle.