Orthant

An orthant around a point denotes a subset of the - dimensional space in geometry , which lies on exactly one side of the axis-parallel hyperplanes running through . To be more precise, an orthant is the intersection of half-spaces of the that are parallel to an axis and run through . Formal: ${\displaystyle p}$ ${\displaystyle d}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle p}$ ${\displaystyle d}$ ${\displaystyle p}$ ${\displaystyle \mathbb {R} ^{d}}$

${\displaystyle \Omega }$is an orthant around if and only if there are numbers ( ) with . ${\displaystyle (p_{1},p_{2},\ldots ,p_{d})\in \mathbb {R} ^{d}}$ ${\displaystyle d}$ ${\displaystyle e_{i}\in \{-1,+1\}}$ ${\displaystyle i=1,\ldots ,d}$ ${\displaystyle \Omega =\left\{(x_{1},x_{2},\ldots ,x_{d})\in \mathbb {R} ^{d}\mid \forall i=1,\ldots ,d\colon \quad e_{i}p_{i}\leq e_{i}x_{i}\right\}}$

It follows from this that there is an exactly orthant around a point . ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle 2^{d}}$

Examples

• In the four quadrants of the Cartesian coordinate system correspond to the orthants around the origin . ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle (0,0)}$

• The eight orthants around the origin are called octants . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle (0,0,0)}$