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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)[reply]
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)[reply]
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)[reply]
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