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

## 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 $$\mathbf{c}' = \mathbf{c} + \vec{t}$$.