Octant (solid geometry)

Short description: One of eight divisions of a Euclidean 3D coordinate system

Three axial planes (x=0, y=0, z=0) divide space into eight octants. The eight (±,±,±) coordinates of the cube vertices are used to denote them. The horizontal plane shows the four quadrants between x\- and y-axis. (Vertex numbers are little-endian balanced ternary.)

An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. It is similar to the two-dimensional quadrant and the one-dimensional ray.[1]

The generalization of an octant is called orthant.

## Naming and numbering

Two representations of a right-hand coordinate system. The first one corresponds to the cube image.

A convention for naming an octant is to give its list of signs, e.g. (+,−,−) or (−,+,−). Octant (+,+,+) is sometimes referred to as the first octant, although similar ordinal name descriptors are not defined for the other seven octants. The advantages of using the (±,±,±) notation are its unambiguousness, and extensibility for higher dimensions.

The following table shows the sign tuples together with likely ways to enumerate them. A binary enumeration with − as 1 can be easily generalized across dimensions. A binary enumeration with + as 1 defines the same order as balanced ternary. The Roman enumeration of the quadrants is in Gray code order, so the corresponding Gray code is also shown for the octants.

Octants  | Gray
code  | x | y | z | Binary  | Balanced
ternary

|
|
|
|
|

− as 1  | \+ as 1
< | > | < | > | < | >
0  | + | + | \+  | 0 | 0 | 7 | 7  | 13 | 13
1  | − | + | \+  | 1 | 4 | 6 | 3  | 11 | −5
3  | + | − | \+  | 2 | 2 | 5 | 5  | 7 | 7
2  | − | − | \+  | 3 | 6 | 4 | 1  | 5 | −11
7  | + | + | −  | 4 | 1 | 3 | 6  | −5 | 11
6  | − | + | −  | 5 | 5 | 2 | 2  | −7 | −7
4  | + | − | −  | 6 | 3 | 1 | 4  | −11 | 5
5  | − | − | −  | 7 | 7 | 0 | 0  | −13 | −13
Quadrants for comparison  | Roman  | x | y | Binary  | Balanced
ternary

|
|
|
|

− as 1  | \+ as 1
< | > | < | > | < | >
I  | + | \+  | 0 | 0 | 3 | 3  | 4 | 4
II  | − | \+  | 1 | 2 | 2 | 1  | 2 | −2
IV  | + | −  | 2 | 1 | 1 | 2  | −2 | 2
III  | − | −  | 3 | 3 | 0 | 0  | −4 | −4

Little- and big-endian are marked by "<" and ">".

Verbal descriptions are ambiguous, because they depend on the representation of the coordinate system. In the two depicted representations of a right-hand coordinate system, the first octant could be called right-back-top or right-top-front respectively.

## See also

* Orthant
* Octant (plane geometry)
* Octree

## References

1. ↑ Weisstein, Eric W.. "Octant". http://mathworld.wolfram.com/Octant.html.

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