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

Uniform scale transformation of a ray

A ray \((O,\vec{d})\) in \(\mathbb{R}^3\) subjected to a uniform scale transformation represented by a scalar \(s \in \mathbb{R}\) yields a ray \((O',\vec{d}')\) where \(O' := s O\) and \(\vec{d}' := \vec{d}\).

Scale transformation of a ray

An ray \((O,\vec{d})\) in \(\mathbb{R}^3\) subjected to a scale transformation represented by a vector \(\vec{s} \in \mathbb{R}^3\) yields a ray \((O', \vec{d})\) where \(O' := \vec{s} \cdot O \) and \(\vec{d}' := \vec{d} \).

Translate transformation of a ray

A ray \((O,\vec{d})\) in \(\mathbb{R}^3\) subjected to translate transformation represented by a vector \(\vec{t} \in \mathbb{R}^3\) yields a ray \((O',\vec{d}')\) where \(O' := O + \vec{t}\) and \(\vec{d}' := \vec{d}\).