Octant of a sphere
{{short description|Spherical triangle with three right angles}}
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.{{cite book |last= Legendre |first= Adrien-Marie |date= 1858 |title= Elements of Geometry and Trigonometry |author-link= Adrien-Marie Legendre |editor-last= Davies |editor-first= Charles |editor-link= Charles Davies (professor) |location= New York |publisher= A. S. Barnes & Co. |url= https://books.google.com/books?id=ywhKAAAAMAAJ |page= 197}} It is one face of a spherical octahedron.{{cite book
| last = Stillwell | first = John
| doi = 10.1007/978-1-4612-0929-4
| isbn = 0-387-97743-0
| mr = 1171453
| page = 68
| publisher = Springer-Verlag | location = New York
| series = Universitext
| title = Geometry of Surfaces
| year = 1992}}
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.{{cite journal |last=Coxeter |first=H. S. M. |title=Rational spherical triangles |journal=The Mathematical Gazette |volume=66 |number=436 |year=1982 |pages=145–147 |doi=10.2307/3617755 |jstor=3617755 }}
The octant can be parametrized using a rational quartic Bézier triangle.{{cite journal |last1=Farin |first1=G. |first2=B. |last2=Piper |first3=Andrew J. |last3=Worsey |title=The octant of a sphere as a non-degenerate triangular Bézier patch |journal=Computer Aided Geometric Design |volume=4 |number=4 |year=1987 |pages=329–332 |doi=10.1016/0167-8396(87)90007-0 }}
The solid angle subtended by a spherical octant is {{pi}}/2 steradian or one-eight of a spat, the solid angle of a full sphere.{{cite web | title=octant | website=PlanetMath.org | date=2013-03-22 | url=https://planetmath.org/octant | access-date=2024-10-21}}