Editing Hyperbolic orthogonality (section)

==Geometry==
Two lines are '''hyperbolic orthogonal''' when they are [[reflection (mathematics)|reflections]] of each other over the asymptote of a given [[hyperbola]].
Two particular hyperbolas are frequently used in the plane:
{{ordered list | list-style-type = upper-alpha
| 1 = ''xy'' = 1  with ''y'' = 0 as asymptote.

When reflected in the x-axis, a line ''y'' = ''mx'' becomes ''y'' = −''mx''.

In this case the lines are hyperbolic orthogonal if their [[slope]]s are [[additive inverse]]s.

| 2 = ''x''<sup>2</sup> − ''y''<sup>2</sup> = 1  with ''y'' = ''x'' as asymptote.

For lines ''y'' = ''mx'' with &minus;1 < ''m'' < 1, when ''x'' = 1/''m'', then ''y'' = 1.

The point (1/''m'' , 1) on the line is reflected across ''y'' = ''x'' to (1, 1/''m'').

Therefore the reflected line has slope 1/m and the slopes of hyperbolic orthogonal lines are [[multiplicative inverse|reciprocal]]s of each other.}}

The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote ''A'', a pair of lines (''a'', ''b'') are hyperbolic orthogonal if there is a pair (''c'', ''d'') such that <math>a \rVert c ,\  b \rVert d </math>, and ''c'' is the reflection of ''d'' across ''A''.

Similar to the perpendularity of a circle radius to the [[tangent]], a radius to a hyperbola is hyperbolic orthogonal to a tangent to the hyperbola.<ref name=L&W/><ref>Bjørn Felsager (2004), [https://www.dynamicgeometry.com/Documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf Through the Looking Glass – A glimpse of Euclid’s twin geometry, the Minkowski geometry] {{Webarchive|url=https://web.archive.org/web/20110716173907/http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf |date=2011-07-16 }}, ICME-10 Copenhagen; pages 6 & 7.</ref>

A [[bilinear form]] is used to describe orthogonality in analytic geometry, with two elements orthogonal when their bilinear form vanishes. In the plane of [[complex number]]s <math>z_1 =u + iv, \quad z_2 = x + iy</math>, the bilinear form is <math>xu + yv</math>, while in the plane of [[hyperbolic number]]s <math>w_1 =  u + jv,\quad w_2 =  x +jy,</math> the bilinear form is <math>xu - yv .</math>
:The vectors ''z''<sub>1</sub> and ''z''<sub>2</sub> in the complex number plane, and ''w''<sub>1</sub> and ''w''<sub>2</sub> in the hyperbolic number plane are said to be respectively ''Euclidean orthogonal'' or ''hyperbolic orthogonal'' if their respective inner products [bilinear forms] are zero.<ref>Sobczyk, G.(1995) [https://garretstar.com/secciones/publications/docs/HYP2.PDF Hyperbolic Number Plane], also published in ''College Mathematics Journal'' 26:268–80.</ref>

The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other. Then
:<math>z_1 z_2^* + z_1^* z_2 = 0</math> entails perpendicularity in the complex plane, while
:<math>w_1 w_2^* + w_1^* w_2 = 0</math> implies the ''w'''s are hyperbolic orthogonal.

The notion of hyperbolic orthogonality arose in [[analytic geometry]] in consideration of [[conjugate diameters]] of ellipses and hyperbolas.<ref>Barry Spain (1957) [http://catalog.hathitrust.org/Record/000660610 Analytical Conics], ellipse §33, page 38 and hyperbola §41, page 49, from [[Hathi Trust]]</ref> If ''g'' and ''g''′ represent the slopes of the conjugate diameters, then <math>g g' = - \frac{b^2}{a^2}</math> in the case of an ellipse and <math>g g' =  \frac{b^2}{a^2}</math> in the case of a hyperbola. When ''a'' = ''b'' the ellipse is a circle and the conjugate diameters are perpendicular while the hyperbola is rectangular and the conjugate diameters are hyperbolic-orthogonal.

In the terminology of [[projective geometry]], the operation of taking the hyperbolic orthogonal line is an [[involution (mathematics)|involution]]. Suppose the slope of a vertical line is denoted ∞ so that all lines have a slope in the [[projectively extended real line]]. Then whichever hyperbola (A) or (B) is used, the operation is an example of a [[involution (mathematics)#Projective geometry|hyperbolic involution]] where the asymptote is invariant. Hyperbolically orthogonal lines lie in different sectors of the plane, determined by the asymptotes of the hyperbola, thus the relation of hyperbolic orthogonality is a [[heterogeneous relation]] on sets of lines in the plane.