Talk:Hyperbolic orthogonality

Tale of two hyperbolas

The initial statement of this concept presumes the analytic hyperbola

$\ x^{2}-y^{2}=1$ to set the context.

A more general approach in terms of conjugate hyperbola and conjugate diameters is stated later. When this approach is invoked for the algebraic hyperbola $\ xy=1$ then hyperbolic-orthogonality corresponds to additive inverse slopes. The article requires editing to give the correct slope interpretation regardless of initial hyperbola taken, the analytic or the algebraic.Rgdboer (talk) 21:35, 11 August 2009 (UTC)

Reflection in asymptote of given hyperbola now taken as primary. See the new Felsager link for visuals and related concepts. Graphic contributions could enhance this article.Rgdboer (talk) 22:00, 12 August 2009 (UTC)

Archaic?

The term "hyperbolic orthogonality" appears to have been subsumed in the more general term "orthogonality", and I suspect that it is only used in limited (mainly historical) contexts. For example, one does not single out the timelike element of an orthogonal basis of a Lorentzian vector space as being "hyperbolic orthogonal" to the others. The most recent two references do not use the term, and the remainder are all are over a century old. It would be useful if this article made the standing and usage of the term "hyperbolic orthogonality" clear, or perhaps this article should be merged into a "History" section of Orthogonality. —Quondum 17:32, 27 July 2016 (UTC)

Is relativity archaic? Is Simultaneity a relative concept? No and Yes. Minkowski used the term normal, and his lead has not been followed. Orthogonal has been adopted by authors, and the modifier Hyperbolic makes the distinction:

Suppose (x,y) and (u,v) are in ℝ² and we presume hyperbola (B). Then

$(x,y)\perp (u,v)\equiv xu-yv=0\equiv Re((x+iy)(u+iv))=0,$ and

$(x,y)\ h.o.(u,v)\equiv xu+yv=0\equiv Re((x+jy)(u+jv))=0,$

where the arithmetics of complex numbers and split-complex numbers are used.

While Orthogonality has no History section, there is a link to bring readers here when they deem that important. Relativity is only possible by using some concepts of hyperbolic geometry such as hyperbolic angle (rapidity) and the mathematics of this article that provides the basis of modern simultaneity. Note the redirect Hyperbolic number in this regard. Rgdboer (talk) 22:33, 27 July 2016 (UTC)

I in no way implied that relativity is archaic; I only suggested that the terminology has changed. For example, one refers to an "orthogonal basis" of a Lorentzian vector space, not to "a basis in which the space-like elements are mutually orthogonal and the time-like element is hyperbolic-orthogonal to the space-like elements". The examples given above do not generalize naturally to an arbitrary number of dimensions other than through quadratic or bilinear forms, which deal with all cases without distinguishing them. I will simply leave my comments here for others more familiar with modern usage to consider. —Quondum 23:18, 27 July 2016 (UTC)

Alternate introduction

In a space with an isotropic quadratic form, two points $(x_{1},y_{1}),\ \ (x_{2},y_{2})$ are hyperbolic orthogonal with respect to the origin when $x_{1}x_{2}-y_{1}y_{2}=0,$ where the quadratic form reduces to $x^{2}-y^{2}$ in the plane spanned by the two points.

Geometrically, given a rectangular hyperbola and its conjugate hyperbola, diameters of the hyperbolas are hyperbolically orthogonal when they are conjugate diameters, which occurs when they are reflections of one another across an asymptote.

The model of relativity of simultaneity expressed by Herman Minkowski in 1908 uses hyperbolic orthogonality in this sense: In a spacetime plane a worldline has a tangent vector. A line hyperbolically orthogonal to this tangent represents simultaneous events with respect to the worldline.

This introduction specifies the type of quadratic form found in pseudo-Euclidean space such as Minkowski space. It also uses the classical concept of conjugate diameters.

Rgdboer (talk) 20:48, 17 October 2024 (UTC)