Octant (solid geometry)

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

The coordinate system on the left is used in the image above.

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.

x y z Binary Balanced
− as 1 + as 1
0 +++ 0077 1313
1 ++ 1463 11−5
3 ++ 2255 77
2 + 3641 5−11
7 ++ 4136 −511
6 + 5522 −7−7
4 + 6314 −115
5 7700 −13−13
Quadrants for comparison
Roman x y Binary Balanced
− as 1 + as 1
I ++ 0033 44
II + 1221 2−2
IV + 2112 −22
III 3300 −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

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