Hyperrectangular

Hyperrectangular is an adjective describing objects or regions that have the shape of a hyperrectangle. In mathematics, a hyperrectangle is the Cartesian product of intervals in n-dimensional space. The standard axis-aligned hyperrectangle in R^n is the set [a1,b1] × [a2,b2] × ... × [an,bn], with ai < bi for all i. This generalizes the rectangle in the plane and the rectangular prism in three-dimensional space; if all side lengths are equal, the shape is a hypercube. The term is often used interchangeably with “box” in computational contexts such as axis-aligned bounding boxes (AABBs).

Geometric properties include that there are 2^n vertices, obtained by choosing either ai or bi for each coordinate. The center is the n-vector of midpoints ((ai+bi)/2). The side length along axis i is li = bi - ai. The n-dimensional volume is V = product of the li. The (n−1)-dimensional surface measure is S = 2 × sum over i of the product of all lengths except li. The shape is convex and axis-aligned with respect to the coordinate axes; non-axis-aligned boxes are typically called parallelepipeds.

Related terms include the hypercube, when all li are equal, and the orthotope, used in some literature as another name for a hyperrectangle. In practice, hyperrectangular shapes are used to bound regions in multivariate integration and optimization with box constraints, and as bounding boxes in computer graphics and machine learning. They also underpin Cartesian-product constructions in probability theory and the discretization of multi-dimensional spaces.