Lines through a given point P and asymptotic to line l.

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

Since there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies among writers. In this article, the two limiting lines are called asymptotic and lines sharing a common perpendicular are called ultraparallel; the simple word parallel may apply to both.

Non-intersecting lines

An interesting property of hyperbolic geometry follows from the occurrence of more than one parallel line through a point P: there are two classes of non-intersecting lines. Let B be the point on l such that the line PB is perpendicular to l. Consider the line x through P such that x does not intersect l, and the angle ? between PB and x counterclockwise from PB is as small as possible; i.e., any smaller angle will force the line to intersect l. This is called an asymptotic line in hyperbolic geometry. Symmetrically, the line y that forms the same angle ? between PB and itself but clockwise from PB will also be asymptotic. x and y are the only two lines asymptotic to l through P. All other lines through P not intersecting l, with angles greater than ? with PB, are called ultraparallel (or disjointly parallel) to l. Notice that since there are an infinite number of possible angles between ? and 90 degrees, and each one will determine two lines through P and disjointly parallel to l, there exist an infinite number of ultraparallel lines.

Thus we have this modified form of the parallel postulate: In hyperbolic geometry, given any line l, and point P not on l, there are exactly two lines through P which are asymptotic to l, and infinitely many lines through P ultraparallel to l.

The differences between these types of lines can also be looked at in the following way: the distance between asymptotic lines shrinks toward zero in one direction and grows without bound in the other; the distance between ultraparallel lines (eventually) increases in both directions. The ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each of a given pair of ultraparallel lines.

In Euclidean geometry, the angle of parallelism is a constant; that is, any distance \lVert BP \rVert between parallel lines yields an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with the \Pi(p) function. This function, described by Nikolai Ivanovich Lobachevsky, produces a unique angle of parallelism for each distance p = \lVert BP \rVert. As the distance gets shorter, \Pi(p) approaches 90°, whereas with increasing distance \Pi(p) approaches 0°. Thus, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. Indeed, on small scales compared to \frac{1}{\sqrt{-K}}, where K\! is the (constant) Gaussian curvature of the plane, an observer would have a hard time determining whether he is in the Euclidean or the hyperbolic plane.

History

A number of geometers made attempts to prove the parallel postulate by assuming its negation and trying to derive a contradiction, including Proclus, Ibn al-Haytham (Alhacen), Omar Khayyám,^[1] Nasir al-Din al-Tusi, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Lambert, and Legendre.http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html Their attempts failed, but their efforts gave birth to hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.^[2]

In the nineteenth century, hyperbolic geometry was extensively explored by János Bolyai and Nikolai Ivanovich Lobachevsky, after whom it sometimes is named. Lobachevsky published in 1830, while Bolyai independently discovered it and published in 1832. Karl Friedrich Gauss also studied hyperbolic geometry, describing in a 1824 letter to Taurinus that he had constructed it, but did not publish his work. In 1868, Eugenio Beltrami provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was.

The term "hyperbolic geometry" was introduced by Felix Klein in 1871^[3].

For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor.

Models of the hyperbolic plane

There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disc model, the Poincaré half-plane model, and the Lorentz model, or hyperboloid model. These models define a real hyperbolic space which satisfies the axioms of a hyperbolic geometry. Despite the naming, the two disc models and the half-plane model were introduced as models of hyperbolic space by Beltrami, not by Poincaré or Klein.

Poincaré disc model of great rhombitruncated {3,7} tiling

Lines through a given point and asymptotic to a given line, illustrated in the Poincaré disc model

1. The Klein model, also known as the projective disc model and Beltrami-Klein model, uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines.
   • This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted.
   • The distance in this model is the cross-ratio, which was introduced by Arthur Cayley in projective geometry.
2. The Poincaré disc model, also known as the conformal disc model, also employs the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle.
3. The Poincaré half-plane model takes one-half of the Euclidean plane, as determined by a Euclidean line B, to be the hyperbolic plane (B itself is not included).
   • Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B.
   • Both Poincaré models preserve hyperbolic angles, and are thereby conformal. All isometries within these models are therefore Möbius transformations.
4. The Lorentz model or hyperboloid model employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds (see below) says that Wilhelm Killing and Karl Weierstrass used this model from 1872.
   • This model has direct application to special relativity, as Minkowski 3-space is a model for spacetime, suppressing one spatial dimension. One can take the hyperboloid to represent the events that various moving observers, radiating outward in a spatial plane from a single point, will reach in a fixed proper time. The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.

Visualizing hyperbolic geometry

A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring

For example, in Circle Limit III every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In Circle Limit IV, for example, one can see that the number of within a distance of n from the center rises exponentially. The demons have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

There are several ways to physically realize a hyperbolic plane (or approximation thereof). A particularly well-known paper model based on the pseudosphere is due to William Thurston. The art of crochet has been used to demonstrate hyperbolic planes with the first being made by Daina Taimina.^[4] In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball".

See also

• Elliptic geometry
• Hyperbolic structure
• Hyperboloid structure
• Hjelmslev transformation
• Hyperbolic 3-manifold

• Kleinian group
• Poincaré metric
• Khayyam-Saccheri quadrilateral
• Spherical geometry
• Systolic geometry

External links

• Visions of Infinity: Tiling a hyperbolic floor inspires both mathematics and art Science News: Dec. 23, 2000; Vol. 158, No. 26/27, p. 408
• Java freeware for creating sketches in both the Poincaré Disk and the Upper Half-Plane Models of Hyperbolic Geometry University of New Mexico
• "The Hyperbolic Geometry Song" A short music video about the basics of Hyperbolic Geometry available at Youtube.
• More on hyperbolic geometry, including movies and equations for conversion between the different models University of Illinois at Urbana-Champaign
• , interactive instructional website.

References

Literature

• Coxeter, H. S. M. (1942) Non-Euclidean geometry, University of Toronto Press, Toronto
• Milnor, John W. (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9-24.
• Reynolds, William F. (1993) Hyperbolic Geometry on a Hyperboloid, American Mathematical Monthly 100:442-455.
• Stillwell, John. (1996) Sources in Hyperbolic Geometry, volume 10 in AMS/LMS series History of Mathematics.
• Samuels, David. (March 2006) Knit Theory Discover Magazine, volume 27, Number 3.
• James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1852339349

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