{{short description|Spherical triangle with three right angles}} [[File:Octant of a sphere.png|thumb|An octant of the sphere in orthographic projection]]
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'''.<ref>{{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}}</ref> It is one face of a spherical octahedron.<ref>{{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}}</ref>
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.<ref>{{cite journal |last=Coxeter |first=H. S. M. |author-link=H. S. M. Coxeter |title=Rational spherical triangles |journal=The Mathematical Gazette |volume=66 |number=436 |year=1982 |pages=145–147 |doi=10.2307/3617755 |jstor=3617755 }}</ref>
The octant can be parametrized using a rational quartic Bézier triangle.<ref>{{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 }}</ref>
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.<ref name="u421">{{cite web | title=octant | website=PlanetMath.org | date=2013-03-22 | url=https://planetmath.org/octant | access-date=2024-10-21}}</ref>
== See also == * Hemisphere (geometry) * Trirectangular tetrahedron
== Notes == {{reflist}}
{{geometry-stub}} Category:Spherical geometry