The difference between the sum of the angles , , and of a spherical triangle
and
radians ( ),
The notation
is sometimes used for spherical excess instead of , which can cause confusion since it is also frequently used
to denote the surface area of a spherical
triangle (Zwillinger 1995, p. 469). The notation is also used (Gellert et al. 1989, p. 263).
The value of the excess is the solid angle (in steradians ) subtended by the spherical
triangle , as proved by Thomas Hariot in 1603 (Hopf 1940).
The equation for the spherical excess in terms of the side lengths , , and is known as l'Huilier's theorem ,
where
is the semiperimeter .
See also Angular Defect ,
Descartes Total Angular Defect ,
Girard's
Spherical Excess Formula ,
L'Huilier's Theorem ,
Spherical Triangle ,
Tetrahedron
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References Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR
Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold,
1989. Harris, J. W. and Stocker, H. Handbook
of Mathematics and Computational Science. New York: Springer-Verlag, p. 109,
1998. Hopf, H. "Selected Chapters of Geometry." ETH Zürich
lecture, pp. 1-2, 1940. http://www.math.cornell.edu/~hatcher/Other/hopf-samelson.pdf . Todhunter,
I. and Leathem, J. G. "Spherical Trigonometry: For the Use of Colleges
and Schools." London: Macmillan, p. 101, 1901. Zwillinger,
D. (Ed.). CRC
Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 469,
1995. Referenced on Wolfram|Alpha Spherical Excess
Cite this as:
Weisstein, Eric W. "Spherical Excess."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalExcess.html
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