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

Uniform scale transformation of a sphere

A sphere \(\left\{ \mathbf{p} \in \mathbb{R}^3 \;|\; |\mathbf{p} - \mathbf{c}| \leq r \right\}\) in \(\mathbb{R}^3\) subjected to a uniform scale transformation represented by a scalar \(s \in \mathbb{R}\) is a sphere \(\left\{ \mathbf{p} \in \mathbb{R}^3 \;|\; |\mathbf{p} - \mathbf{c}'| \leq r' \right\}\) with \(\mathbf{c}' = s\mathbf{c}\) and \(r' = |s|r\).

Reasoning: For the case of \(s \neq 0\), the idea is now to pick a point on the hull of the sphere \(\mathbf{q}\) in the direction \(\group{s,s,s}\) from the center e.g. \(\mathbf{q} = r\frac{\vec{s}}{|\vec{s}|} + \mathbf{c}\). \(\mathbf{q}\) and \(\mathbf{c}\) are subjected to the scale transformation which gives the center of the new sphere \(\mathbf{c}' = s\mathbf{c}\) and a point \(\mathbf{q}' = s\mathbf{q}\) on the hull of the new sphere such that the radius of the new sphere is \(r' = \norm{\mathbf{q}' - \mathbf{c}'}\). For the case of \(s = 0\), the idea is the same. However, we pick \(\mathbf{q} = \mathbf{c}\) as the point on the hull of the sphere.

In the above definition, \(r'\) is defined as \(r' = |s|r\) instead of \(r' = \norm{\mathbf{q}' - \mathbf{c}'}\). We show that both definitions are equivalent.

For the case of \(s = 0\) it is easy to see that \(|s|r\) and \(\norm{\mathbf{q}' - \mathbf{c}'}\) both yield the same result \(0\).

Let's assume \(s \neq 0\).

\[\begin{aligned} &&\norm{\mathbf{q}' - \mathbf{c}'} \end{aligned}\]
By the definition of \(\mathbf{q}'\) and \(\mathbf{c}'\).
\[\begin{aligned} &=\norm{s\mathbf{q} - s\mathbf{c}}\\ &=\norm{s\group{\mathbf{q} - \mathbf{c}}} \end{aligned}\]
Property of norms \(\norm{s\vec{v}} = |s|\norm{\vec{v}}\).
\[\begin{aligned} &=|s|\norm{\mathbf{q} - \mathbf{c}} \end{aligned}\]
Definition of \(\mathbf{q}\).
\[\begin{aligned} &=|s|\norm{r\frac{\vec{s}}{|\vec{s}|} + \mathbf{c} - \mathbf{c}}\\ &=|s|\norm{r\frac{\vec{s}}{|\vec{s}|}}\\ \end{aligned}\]
Property of norms \(\norm{s\vec{v}} = |s|\norm{\vec{v}}\).
\[\begin{aligned} &=|s||r|\norm{\frac{\vec{s}}{|\vec{s}|}}\\ \end{aligned}\]
As \(\frac{\vec{s}}{|\vec{s}|}\) is unit and the Euclidean norm of a unit vector is \(1\).
\[\begin{aligned} &=|s||r|1\\ &=|s|r \end{aligned}\]

Translate transformation of a sphere

A sphere \(\left\{ \mathbf{p} \in \mathbb{R}^3 \;|\; |\mathbf{p} - \mathbf{c}| \leq r \right\}\) in \(\mathbb{R}^3\) subjected to a translation transformation represented by a vector \(\vec{t} \in \mathbb{R}\) is a sphere \(\left\{ \mathbf{p} \in \mathbb{R}^3 \;|\; |\mathbf{p} - \mathbf{c}'| \leq r \right\}\) with \(\mathbf{c}' = \mathbf{c} + \vec{t}\).