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** = [

**U**_{0},

**U**_{1},

**U**_{2}] and

**Y** = (

*y*_{0},

*y*_{1},

*y*_{2}) 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)
**U**_{0} = (1, 0, 0)

**U**_{1} = (0, 1, 0)

**U**_{2} = (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