In geometry, an octant of a sphere is a spherical triangle with three right angles and three right sides. It is sometimes called a trirectangular (spherical) triangle. It is one face of a spherical octahedron.
For a sphere embedded in three-dimensional Euclidean space, the vectors from the sphere's center to each vertex of an octant are the basis vectors of a Cartesian coordinate system relative to which the sphere is a unit sphere. The spherical octant itself is the intersection of the sphere with one octant of space.
Uniquely among spherical triangles, the octant is its own polar triangle.
The octant can be parametrized using a rational quartic Bézier triangle.
The solid angle subtended by a spherical octant is π/2 steradian or one-eight of a spat, the solid angle of a full sphere.
See also
Notes
References
Legendre, Adrien-Marie (1858). Davies, Charles (ed.). Elements of Geometry and Trigonometry. New York: A. S. Barnes & Co. p. 197. /wiki/Adrien-Marie_Legendre ↩
Stillwell, John (1992). Geometry of Surfaces. Universitext. New York: Springer-Verlag. p. 68. doi:10.1007/978-1-4612-0929-4. ISBN 0-387-97743-0. MR 1171453. 0-387-97743-0 ↩
Coxeter, H. S. M. (1982). "Rational spherical triangles". The Mathematical Gazette. 66 (436): 145–147. doi:10.2307/3617755. JSTOR 3617755. /wiki/Doi_(identifier) ↩
Farin, G.; Piper, B.; Worsey, Andrew J. (1987). "The octant of a sphere as a non-degenerate triangular Bézier patch". Computer Aided Geometric Design. 4 (4): 329–332. doi:10.1016/0167-8396(87)90007-0. /wiki/Doi_(identifier) ↩
"octant". PlanetMath.org. 2013-03-22. Retrieved 2024-10-21. https://planetmath.org/octant ↩