## 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 \(s \neq 0\).

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