This page is a sub-page of our page on Surfaces.
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Related KMR-surfaces:
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Other relevant sources of information:
• Pseudospherical Surfaces
• Dini’s Surface
• Minding Surfaces
• Ferdinand Minding
• Amsler’s Surface
/////// Quoting Stillwell (2016, section 5.11, p. 184):
Non-Euclidean Geometry
In this book I have made the judgement that non-Euclidean geometry is more advanced than Euclidean. There is ample historical reason to support this call, since non-Euclidean geometry was discovered more than 2000 years after Euclid. The “points” and “lines” of non-Euclidean geometry can be modeled by Euclidean concepts, so they are not advanced in themselves, but the concept of non-Euclidean distance surely is.
One way to see this is to map a portion of the non-Euclidean plane onto a piece of surface \, S \, in \, \mathbb{R}^3 \, in such a way that distance is preserved. Then ask: how simple is \, S \, ? Well, the simplest possible \, S \, is the trumpet-shaped surface shown in figure 5.33 and known as the pseudosphere.
Henry Segerman on the pseudosphere (YouTube):
The pseudosphere is obtained by rotating the tractrix curve with equation
\, x \, = \, \ln {\dfrac{1 + \sqrt{1 - y^2}}{y}} - \sqrt{1 - y^2} \,
about the \, x -axis.
The formula is complicated enough, but the conceptual complication is much greater. It is possible to compare only small pieces of the non-Euclidean plane with small pieces of a surface in \, \mathbb{R}^3 , because a complete non-Euclidean plane does not “fit” smoothly in \, \mathbb{R}^3 . This was proved by Hilbert (1901). The pseudosphere, for example, represents just a thin wedge of the non-Euclidean plane, the edges of which are two non-Euclidean lines that approach each other at infinity. These two edges are joined together to form the tapering tube shown in figure 5.33.
In contrast, Euclidean geometry is modeled by the simplest possible surface in \, \mathbb{R}^3 \,
– the plane!
[Footnote 7: It may be thought unfair to the hyperbolic plane to force it into the Euclidean straightjacket of \, \mathbb{R}^3 . Might not the Euclidean plane look equally bad if forced to live in non-Euclidean space? Actually, this is not the case. Beltrami showed that the Euclidean plane fits beautifully into non-Euclidean space, where it is a “sphere with center at infinity.”]
/////// End of Quote from Stillwell (2016)
Tractrix and Catenary – Involute and Evolute of each other
The catenary is the evolute of the tractrix, and the tractrix is an involute of the catenary:
/// Connect with the Beltrami-Klein model
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/// Connect with Horocycles in Hyperbolic Geometry
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