Ordinary trigonometry studies triangles in the Euclidean plane R2. There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations, definitions using functional equations. Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space. A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and n-simplices.
Trigonometry
- In spherical trigonometry, triangles on the surface of a sphere are studied. The spherical triangle identities are written in terms of the ordinary trigonometric functions but differ from the plane triangle identities.
- Hyperbolic trigonometry:
- Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions.
- Hyperbolic functions in Euclidean geometry: The unit-circle is parameterized by (cos t, sin t) whereas the equilateral hyperbola is parameterized by the points (cosh t, sinh t).
- Gyrotrigonometry: A form of trigonometry used in the gyrovector space approach to hyperbolic geometry, with applications to special relativity and quantum computation.
- Rational trigonometry – a reformulation of trigonometry in terms of spread and quadrance rather than angle and length.
- Trigonometry for taxicab geometry[1]
- Spacetime trigonometries[2]
- Fuzzy qualitative trigonometry[3]
- Operator trigonometry[4]
- Lattice trigonometry[5]
- Trigonometry on symmetric spaces[6][7][8]
Higher-dimensions
- Polar sine
- Trigonometry of a tetrahedron[9]
- A law of sines for tetrahedra
- Simplexes with an "orthogonal corner" - Pythagorean theorems for n-simplexes
- De Gua's theorem - a Pythagorean theorem for a tetrahedron with a cube corner
Trigonometric functions
- Polar sine
- Trigonometry of a tetrahedron[9]
- A law of sines for tetrahedra
- Simplexes with an "orthogonal corner" - Pythagorean theorems for n-simplexes
- De Gua's theorem - a Pythagorean theorem for a tetrahedron with a cube corner
Other
Polar/Trigonometric forms of hypercomplex numbers[11][12]
Polygonometry - trigonometric identities for multiple distinct angles[13]
See also
The Pythagorean theorem in non-Euclidean geometry
References
Thompson, K.; Dray, T. (2000), "Taxicab angles and trigonometry" (PDF), Pi Mu Epsilon Journal, 11 (2): 87–96, arXiv:1101.2917, Bibcode:2011arXiv1101.2917T
Herranz, Francisco J.; Ortega, Ramón; Santander, Mariano (2000), "Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry", Journal of Physics A, 33 (24): 4525–4551, arXiv:math-ph/9910041, Bibcode:2000JPhA...33.4525H, doi:10.1088/0305-4470/33/24/309, MR 1768742
Liu, Honghai; Coghill, George M. (2005), "Fuzzy Qualitative Trigonometry", 2005 IEEE International Conference on Systems, Man and Cybernetics (PDF), 2, pp. 1291–1296, archived from the original (PDF) on 2011-07-25
Gustafson, K. E. (1999), "A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin", Вычислительные технологии, 4 (3): 73–83
Karpenkov, Oleg (2008), "Elementary notions of lattice trigonometry", Mathematica Scandinavica, 102 (2): 161–205, arXiv:math/0604129, doi:10.7146/math.scand.a-15058, MR 2437186
Aslaksen, Helmer; Huynh, Hsueh-Ling (1997), "Laws of trigonometry in symmetric spaces", Geometry from the Pacific Rim (Singapore, 1994), Berlin: de Gruyter, pp. 23–36, CiteSeerX 10.1.1.160.1580, MR 1468236
Leuzinger, Enrico (1992), "On the trigonometry of symmetric spaces", Commentarii Mathematici Helvetici, 67 (2): 252–286, doi:10.1007/BF02566499, MR 1161284
Masala, G. (1999), "Regular triangles and isoclinic triangles in the Grassmann manifolds G2(RN)", Rendiconti del Seminario Matematico Università e Politecnico di Torino., 57 (2): 91–104, MR 1974445
Richardson, G. (1902-03-01). "The Trigonometry of the Tetrahedron" (PDF). The Mathematical Gazette. 2 (32): 149–158. doi:10.2307/3603090. JSTOR 3603090.
West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003), Physics of fractal operators, Institute for Nonlinear Science, New York: Springer-Verlag, p. 101, doi:10.1007/978-0-387-21746-8, ISBN 0-387-95554-2, MR 1988873
Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers", Mathematics Magazine, 77 (2): 118–129, doi:10.1080/0025570X.2004.11953236, JSTOR 3219099, MR 1573734
Yamaleev, Robert M. (2005), "Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics" (PDF), Advances in Applied Clifford Algebras, 15 (1): 123–150, doi:10.1007/s00006-005-0007-y, MR 2236628, archived from the original (PDF) on 2011-07-22
Antippa, Adel F. (2003), "The combinatorial structure of trigonometry" (PDF), International Journal of Mathematics and Mathematical Sciences, 2003 (8): 475–500, doi:10.1155/S0161171203106230, MR 1967890
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
