$$% Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}}$$

Definition of an axis aligned box

An axis aligned box in $$\mathbb{R}^3$$ is defined by a point vector $$\mathbf{min}$$ and a point vector $$\mathbf{max}$$ such that $$\forall_{i=1}^3 \mathbf{min}_i \leq \mathbf{max}_i$$. $$\mathbf{min}$$ is called the minimal point of the axis aligned box, $$\mathbf{max}$$ is called the maximal point of the axis aligned box. The axis aligned box is the set of points $$\left\{ \mathbf{p} \in \mathbb{R}^3 \;|\; \forall_{i=1}^3 \mathbf{min}_i \leq \mathbf{p}_i \leq \mathbf{max}_i \right\}$$.