Pseudosphere
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In geometry, a pseudosphere of radius R is a surface of curvature −1/R2 (precisely, a complete, simply connected surface of that curvature), by analogy with the sphere of radius R, which is a surface of curvature 1/R2. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry[1].
The term is also used to refer to what is traditionally called a tractricoid: the result of revolving a tractrix about its asymptote, which is the subject of this article.
It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane.
It also denotes the entire set of points of an infinite hyperbolic space which is one of the three models of Riemannian geometry. This can be viewed as the assemblage of continuous saddle shapes to infinity. The further outward from the symmetry axis, the more increasingly ruffled the manifold becomes. [2] This makes it very hard to represent a pseudosphere in the Euclidean space of drawings. A trick mathematicians have come up with to represent it is called the Poincaré model of hyperbolic geometry. By increasingly shrinking the pseudosphere as it goes further out towards the cuspidal edge, it will fit into a circle, called the Poincaré disk; with the "edge" representing infinity. This is usually tessellated with equilateral triangles, or other polygons which become increasingly distorted towards the edges, such that some vertices are shared by more polygons than is normal under Euclidean geometry. (In normal flat space only six equilateral triangles, for instance, can share a vertex but on the Poincaré disk, some points can share eight triangles as the total of the angles in a narrow triangle of geodesic arcs is now less than 180°). Reverting the triangles back to their normal shape yields various bent sections of the pseudosphere. While smaller local sections will stretch out to saddle shapes, large sections that extend to the infinite edge, are illustrated in their expanded form by being bent until their opposite sides are joined, yielding the aforementioned "tractricoid" shape, which is also called a "Gabriel's Horn" (since it resembles a horn with the mouthpiece lying at infinity). Thus the tractricoid is really only a part of the whole pseudosphere. [3] At any point the product of two principal radii of curvature is constant. Along lines of zero normal curvature geodesic torsion is constant by virtue of Beltrami-Enneper theorem.
The name "pseudosphere" comes about because it is a two-dimensional surface of constant negative curvature just like a sphere with positive Gauss curvature. It has same formulas for area and volume (R = edge radius) 4πR2 and 4πR3/3 of the full surface in spite of the opposite Gauss curvature sign. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle.
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[edit] References
- Henderson, D. W. and Taimina, D. (2006). "Experiencing Geometry: Euclidean and Non-Euclidean with History", Aesthetics and Mathematics. Springer-Verlag.
[edit] See also
[edit] References
- ^ E. Beltrami, Saggio sulla interpretazione della geometria non euclidea, Gior. Mat. 6, 248–312 (Also Op. Mat. 1, 374-405; Ann. École Norm. Sup. 6 (1869), 251-288).
- ^ Illustration of extended manifold: "Is Space Finite", by Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey Weeks, in Scientific American, April, 1999, p.94; Reprint in Special Edition (Vol.12 no.2), 2002, p.62; [1]
- ^ Rucker, Rudy The Fourth Dimension: A guided Tour of Higher Universes; Boston: Houghton Mifflin Company, 1984, p.102-112
