An oriented box is a 3D object that is defined by a center C,three orthonormal axes Ui that form a right handed coordinate system and three positive non zero extents ei 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 |yi| ≤ ei.

A special case of the oriented box is the axis-aligned box. In such a box the axes Ui 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)

ei = 0.5, i = 0, 1, 2

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