Oriented box

An oriented box is a 3D object that is defined by a center C,three orthonormal axes U_i that form a right handed coordinate system and three positive non zero extents e_i for i = 0, 1, 2. All points X within the box or on the surface of the box are defined by the equation:

X = C + RY

where the matrix R = [U₀, U₁, U₂] and Y = (y₀, y₁, y₂) with |y_i| ≤ e_i.

A special case of the oriented box is the axis-aligned box. In such a box the axes U_i are all parallel to one of the coordinate system axes.

Ex. The properties of an axis-aligned cube with the side 1 centered at the origin:

C = (0, 0, 0)

U₀ = (1, 0, 0)
U₁ = (0, 1, 0)
U₂ = (0, 0, 1)

e_i = 0.5, i = 0, 1, 2

Rerence:
3D Game Engine Design: A Practical Approach to Real-Time Computer Graphics by David H. Eberly