# Generalized trigonometry

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.
• Rational trigonometry – a reformulation of trigonometry in terms of spread and quadrance rather than angle and length.
• Spacetime trigonometries[3]
• Fuzzy qualitative trigonometry[4]
• Operator trigonometry[5]
• Lattice trigonometry[6]

## References

1. ^ Wildberger, N. J. (2009), Universal Hyperbolic Geometry I: Trigonometry, arXiv:0909.1377
2. ^ Thompson, K.; Dray, T. (2000), "Taxicab angles and trigonometry", Pi Mu Epsilon Journal 11 (2): 87–96, arXiv:1101.2917
3. ^ 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, doi:10.1088/0305-4470/33/24/309, MR 1768742
4. ^ Liu, Honghai; Coghill, George M. (2005), "Fuzzy Qualitative Trigonometry", 2005 IEEE International Conference on Systems, Man and Cybernetics 2, pp. 1291–1296
5. ^ Gustafson, K. E. (1999), "A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin", Вычислительные технологии 4 (3): 73–83
6. ^ Karpenkov, Oleg (2008), "Elementary notions of lattice trigonometry", Mathematica Scandinavica 102 (2): 161–205, arXiv:math/0604129, MR 2437186
7. ^ 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, MR 1468236, CiteSeerX: 10.1.1.160.1580
8. ^ Leuzinger, Enrico (1992), "On the trigonometry of symmetric spaces", Commentarii Mathematici Helvetici 67 (2): 252–286, doi:10.1007/BF02566499, MR 1161284
9. ^ 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
10. ^ West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003), Physics of fractal operators, Institute for Nonlinear Science, New York: Springer-Verlag, p. 101, ISBN 0-387-95554-2, MR 1988873
11. ^ Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers", Mathematics Magazine 77 (2): 118–129, JSTOR 3219099, MR 1573734
12. ^
13. ^ Antippa, Adel F. (2003), "The combinatorial structure of trigonometry", International Journal of Mathematics and Mathematical Sciences (8): 475–500, doi:10.1155/S0161171203106230, MR 1967890