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In
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
, the angle of parallelism \Pi(a) , is the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point. Let ''a'' be the length of this perpendicular segment, and \Pi(a) be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel, : \lim_ \Pi(a) = \tfrac\pi\quad\text\quad\lim_ \Pi(a) = 0. There are five equivalent expressions that relate '' \Pi(a)'' and ''a'': : \sin\Pi(a) = \operatorname a = \frac =\frac \ , : \cos\Pi(a) = \tanh a = \frac \ , : \tan\Pi(a) = \operatorname a = \frac = \frac \ , : \tan \left( \tfrac\Pi(a) \right) = e^, : \Pi(a) = \tfrac\pi - \operatorname(a), where sinh, cosh, tanh, sech and csch are
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s and gd is the
Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwe ...
.


Construction

János Bolyai János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consis ...
discovered a construction which gives the asymptotic parallel ''s'' to a line ''r'' passing through a point ''A'' not on ''r''. Drop a perpendicular from ''A'' onto ''B'' on ''r''. Choose any point ''C'' on ''r'' different from ''B''. Erect a perpendicular ''t'' to ''r'' at ''C''. Drop a perpendicular from ''A'' onto ''D'' on ''t''. Then length ''DA'' is longer than ''CB'', but shorter than ''CA''. Draw a circle around ''C'' with radius equal to ''DA''. It will intersect the segment ''AB'' at a point ''E''. Then the angle ''BEC'' is independent of the length ''BC'', depending only on ''AB''; it is the angle of parallelism. Construct ''s'' through ''A'' at angle ''BEC'' from ''AB''. : \sin BEC = \frac = \frac = \frac = \frac = \frac = \frac = \frac = \frac \,. See Trigonometry of right triangles for the formulas used here.


History

The angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, ...
. This publication became widely known in English after the Texas professor
G. B. Halsted George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and ...
produced a translation in 1891. (''Geometrical Researches on the Theory of Parallels'') The following passages define this pivotal concept in hyperbolic geometry: :''The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism) which we will here designate by Π(p) for AD = p''.
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, ...
(1840)
G. B. Halsted George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and ...
translator (1891
Geometrical Researches on the Theory of Parallels
link from
Google Books Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google Inc. that searches the full text of books and magazines that Google has scanned, converted to text using optical ...


Demonstration

In the
Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincar� ...
of the hyperbolic plane (see
Hyperbolic motion In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geom ...
s), one can establish the relation of ''Φ'' to ''a'' with
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. Let ''Q'' be the semicircle with diameter on the ''x''-axis that passes through the points (1,0) and (0,''y''), where ''y'' > 1. Since ''Q'' is tangent to the unit semicircle centered at the origin, the two semicircles represent ''parallel hyperbolic lines''. The ''y''-axis crosses both semicircles, making a right angle with the unit semicircle and a variable angle ''Φ'' with ''Q''. The angle at the center of ''Q'' subtended by the radius to (0, ''y'') is also ''Φ'' because the two angles have sides that are perpendicular, left side to left side, and right side to right side. The semicircle ''Q'' has its center at (''x'', 0), ''x'' < 0, so its radius is 1 − ''x''. Thus, the radius squared of ''Q'' is : x^2 + y^2 = (1 - x)^2, hence : x = \tfrac(1 - y^2). The
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
of the
Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincar� ...
of hyperbolic geometry parametrizes distance on the ray with
logarithmic measure In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used ...
. Let log ''y'' = ''a'', so ''y'' = ''ea'' where ''e'' is the base of the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
. Then the relation between ''φ'' and ''a'' can be deduced from the triangle , for example: : \tan\phi = \frac = \frac = \frac = \frac.


References

* Marvin J. Greenberg (1974) ''Euclidean and Non-Euclidean Geometries'', pp. 211–3, W.H. Freeman & Company. *
Robin Hartshorne __NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
(1997) ''Companion to Euclid'' pp. 319, 325,
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
, {{ISBN, 0821807978. *
Jeremy Gray Jeremy John Gray (born 25 April 1947) is an English mathematician primarily interested in the history of mathematics. Biography Gray studied mathematics at Oxford University from 1966 to 1969, and then at Warwick University, obtaining his Ph.D ...
(1989) ''Ideas of Space: Euclidean, Non-Euclidean, and Relativistic'', 2nd edition,
Clarendon Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
, Oxford (See pages 113 to 118). *
Béla Kerékjártó Béla Kerékjártó (1 October 1898, in Budapest – 26 June 1946, in Gyöngyös) was a Hungarian mathematician who wrote numerous articles on topology. Kerékjártó earned his Ph.D. degree from the University of Budapest in 1920. He taught a ...
(1966) ''Les Fondements de la Géométry'', Tome Deux, §97.6 Angle de parallélisme de la géométry hyperbolique, pp. 411,2, Akademiai Kiado, Budapest. Hyperbolic geometry Functions and mappings Angle