Talk:Hyperbolic coordinates

Hyperboloid model

The following contribution deserves reconsideration:

In special relativity, hyperbolic geometry is used as a non-Euclidean geometry of spacetime, in particular in the hyperboloid model presented by Herman Minkowski in his ground-breaking paper "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" of 1908.<ref Scott Walter (1999). "The non-Euclidean style of Minkowskian relativity". Chapter 4 in: Jeremy J. Gray (ed.), The Symbolic Universe: Geometry and Physics 1890-1930, pp. 91–127. Oxford University Press. {{ISBN|0-19-850088-2}}. ref>

The hyperbolic geometry of velocities of Minkowski is 3-dimensional. The radial coordinate of proper time is not included in the hyperbolic geometry. As Scott Walter says on page 8, "Minkowski never again referred to a manifold as both four-dimensional and non-Euclidean." Since the quadrant model has only one parameter of hyperbolic angle, the hyperboloid model is irrelevant to Q.

As for the Minkowski paper linked by wikisource, there are two notable errors on page 59 in the second part of equation (1): the minus sign is wrong and the following x₄ should be a x₃. The passage shows Minkowski using circular functions where hyperbolic are required, so it detracts from the hyperbolic message here.Rgdboer (talk) 02:09, 9 June 2010 (UTC)

Orthogonality

At the moment, the article is included in the box on orthogonal coordinate systems, and the article on orthogonal coordinates uses “coordinate surfaces all meet at right angles” as the defining property. The illustration at the beginning of this article here however does not exhibit orthogonality between coordinate lines. It might be that this orthogonality is only with respect to some non-trivial metric, but if so, that fact should imho. be discussed in the article. --Martin von Gagern (talk) 21:28, 27 July 2014 (UTC)

The curves with one coordinate constant do not meet at right angles. Therefore this article has been removed from the Template. Thank you for noting this error. Yes, one might refer to hyperbolic orthogonality. The references there may help the discussion you suggest.Rgdboer (talk) 21:55, 27 July 2014 (UTC)

350 years

This year we pass this mark in time since the work of Gregoire de Saint-Vincent noted in the History section. Accordingly, the OR tag has been removed. — Rgdboer (talk) 02:55, 10 April 2017 (UTC)