Hyperbolic geometry From Wikipedia, the free encyclopedia Lines through a given point P and asymptotic to line R. A triangle immersed in a saddle-shape plane (ahyperbolic paraboloid ), as well as two diverging ultraparallel lines. In mathematics , hyperbolic geometry (also called Lobachevskian geometry or Bolyai - Lobachevskian geometry) is a non-Euclidean geometry , meaning that the parallel postulate ofEuclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line R and point P not on R, there is exactly one line through P that does not intersect R; i.e., that is parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid (assuming that those other postulates are in fact consistent). Because there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies among writers. In this article,
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Hyperbolic geometryFrom Wikipedia, the free encyclopedia
Lines through a given point P and asymptotic to line R.
A triangle immersed in a saddle-shape plane (ahyperbolic paraboloid), as well as two diverging ultraparallel lines.
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai-Lobachevskian
geometry) is a non-Euclidean geometry, meaning that the parallel postulate ofEuclidean geometry is replaced.
The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for
any given line R and point P not on R, there is exactly one line through P that does not intersect R; i.e., that is
parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, so
the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms
of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid
(assuming that those other postulates are in fact consistent).
Because 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 calledasymptotic and lines sharing a
common perpendicular are called ultraparallel; the simple wordparallel may apply to both.
(2012) Notes on hyperbolic geometry, in: Strasbourg
Master class on Geometry, pp. 1–182, IRMA Lectures in
Mathematics and Theoretical Physics, Vol. 18, Zürich:
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SBN ISBN 978-3-03719-105-7, DOI 10.4171/105.
COXETER, H. S. M. , (1942) Non-Euclidean geometry,
University of Toronto Press, Toronto
Fenchel , Werner (1989). Elementary geometry in
hyperbolic space. De Gruyter Studies in
mathematics. 11. Berlin-New York: Walter de Gruyter &
Co..
Fenchel , Werner; Nielsen, Jakob; edited by Asmus L.
Schmidt (2003). Discontinuous groups of isometries in
the hyperbolic plane. De Gruyter Studies in
mathematics. 29. Berlin: Walter de Gruyter & Co..
LOBACHEVSKY, NIKOLAI I., (2010) Pangeometry, Edited
and translated by Athanase Papadopoulos, Heritage of
European Mathematics, Vol. 4. Zürich: European
Mathematical Society (EMS). xii, 310~p, ISBN 978-3-
03719-087-6/hbk
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 of hyperbolic geometry.
History of Mathematics. 10. Providence, R.I.: American
Mathematical Society. ISBN 978-0-8218-0529-
9. MR 1402697
SAMUELS, DAVID., (March 2006) Knit Theory Discover
Magazine, volume 27, Number 3.
JAMES W. ANDERSON, Hyperbolic Geometry, Springer
2005, ISBN 1-85233-934-9
JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON,
and WALTER R. PARRY (1997) Hyperbolic Geometry,
MSRI Publications, volume 31.
[edit]External links
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.
Hazewinkel, Michiel, ed. (2001), "Lobachevskii
geometry", Encyclopedia of
Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W. , "Gauss-Bolyai-Lobachevsky Space"
from MathWorld.
Weisstein, Eric W. , "Hyperbolic Geometry"
from MathWorld.
More on hyperbolic geometry, including movies and
equations for conversion between the different
models University of Illinois at Urbana-Champaign
Hyperbolic Voronoi diagrams made easy, Frank Nielsen
Stothers, Wilson (2000). Hyperbolic
geometry. University of Glasgow, interactive
instructional website.
Universal Hyperbolic Geometry II: A pictorial overview
Universal Hyperbolic Geometry III: First Steps in
Projective Triangle Geometry
3: What is Non-Euclidean Geometry
1.1 Euclidean Geometry: The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. His book, called "The Elements", is a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book.
Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land.
1.2 Non-Euclidean Geometry: non-Euclidean geometry is any geometry that is different from Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. The essential difference between
Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.
1.3 Spherical Geometry: Spherical geometry is a plane geometry on the surface of a sphere. In a plane geometry, the basic concepts are points and lines. In spherical geometry, points are defined in the usual way, but lines are defined such that the shortest distance between two points lies along them. Therefore, lines in spherical geometry are great circles. A great circle is the largest circle that can be drawn on a sphere. The longitude lines and the equator are great circles of the Earth. Latitude lines, except for the equator, are not great circles. Great circles are lines that divide a sphere into two equal hemispheres.
Spherical geometry is used by pilots and ship captains as they navigate around the globe. Working in spherical geometry has some non-intuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are south of Florida - why is flyingnorth to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a great circle). Another odd property of spherical geometry is that the sum of the angles of a triangle is always greater then 180°. Small triangles, like those drawn on a football field, have very, very close to 180°. Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have significantly more than 180°.
1.4 Hyperbolic Geometry: hyperbolic geometry is the geometry of which the NonEuclid software is a model. Hyperbolic geometry is a "curved" space, and plays an important role in Einstein's General theory of Relativity. hyperbolic geometry is also has many applications within the field of Topology.
Hyperbolic geometry shares many proofs and theorems with Euclidean geometry, and provides a novel and beautiful prospective from which to view those theorems. Hyperbolic geometry also has many differences from Euclidean geometry. The following sections discuss and explore hyperbolic geometry in some detail.
Hyperbolic Geometry
Figure 1
In the Fun Fact on Spherical Geometry, we saw an example of a space which is curved in such a way that the sum of angles in a triangle is greater than 180 degrees, where the sides of the triangle are "intrinsically" straight lines, or geodesics.
Is it also possible to have a space that "curves" in such a way that the sum of angles in a triangle is less than 180 degrees?
Yes! For instance, consider a saddle-shaped surface. A triangle that extends over the saddle of this surface (whose edges are geodesics) will have this property.
Another space with this property is something called the hyperbolic plane. This can be modeled by disc in which is "curved" in such a strange way that a bug on this disc would think that the "straight" lines are the pieces of circles or straight lines (viewed in planar geometry) that intersect the disc boundary at right angles. Any 3-sided figure using such lines will have angles in the corners that sum to less than 180 degrees!
Presentation Suggestions:Convince students of the triangle assertion by drawing a saddle-shaped surface and a triangle on it. Alternatively, you could show that the angles of a square do not add to 360 degrees. Follow by showing drawing the hyperbolic disc and explaining what the "straight lines" are. You can also construct and bring to class an approximate physical model of a hyperbolic plane; the references discuss ways to construct them.
The Math Behind the Fact:These spaces are examples of spaces with a kind of non-Euclidean geometry called hyperbolic geometry. Unlike planar geometry, the parallel postulate does not hold in hyperbolic geometry. Two lines are said to be parallel if they do not intersect. In Euclidean geometry, given a line L there is exactly one line through any given point P that is parallel to L (the parallel postulate). However in hyperbolic geometry,
there are infinitely many lines parallel to L passing through P.
Mathematicians sometimes work with strange geometries by defining them in terms of a Riemannian metric, which gives a local notion of how to measure "distance" and "angles" on an arbitrary set. You can learn more about such metrics by taking a first course on real analysis, then following with an advanced course in differential geometry.
How to Cite this Page: Su, Francis E., et al. "Hyperbolic Geometry." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.
Hyperbolic Geometry
The McDonnell Planetarium at the St. Louis Science Center.