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

Uniform scale transformation of an axis aligned box

An axis aligned box \((Min,Max)\) in \(\mathbb{R}^3\) subjected to a uniform scale transformation represented by a scalar \(s \in \mathbb{R}\) yields an axis aligned box \((Min',Max')\) where \(Min'_i = \min\{ s Min_i, s Max_i \}\) and \(Max'_i = \max\{ s Min_i, s Max_i \}\).

Scale transformation of an axis aligned box

An axis aligned box \((Min,Max)\) in \(\mathbb{R}^3\) subjected to a scale transformation represented by a vector \(\vec{s} \in \mathbb{R}^3\) yields an axis aligned box \((Min',Max')\) where \(Min'_i = \min\{ s_i Min_i, s_i Max_i \}\) and \(Max'_i = \max\{ s_i Min_i, s_i Max_i \}\).

Translate transformation of an axis aligned box

An axis aligned box \((Min,Max)\) in \(\mathbb{R}^3\) subjected to a translate transformation represented by a vector \(\vec{t} \in \mathbb{R}^3\) yields an axis aligned box \((Min',Max')\) where \(Min' = Min + t\) and \(Max' = Max + t\).