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how to mathematically express a relationship in which a vector can be any 3D unit vector

I'm currently doing some work with 3D rotations and exponential coordinates. Exponential coordinates $\mathbf{s} \in \mathbb{R}^3$ are a rotation parameterization defined as
$$\mathbf{s} = \theta \mathbf{e}$$
where $\mathbf{e} \in \mathbb{R}^3$ is a unit-length axis of rotation, and $\theta \in [0,\pi]$ is an angle of rotation. The set of all axes is equivalent to the 2-sphere, $\mathcal{S}^2$:
$$\mathcal{S}^2 = \{ \mathbf{x} \in \mathbb{R}^3 ~~~ | ~~~ \lVert \mathbf{x} \rVert = 1 \} .$$

Given a set of exponential coordinates $\mathbf{s}$, the angle can be recovered as follows:
$$\theta = \lVert \mathbf{s} \rVert .$$
If $\lVert \mathbf{s} \rVert \neq 0$, then the axis can be recovered as follows:
$$\mathbf{e} = \mathbf{s}/\lVert \mathbf{s} \rVert .$$
However, if $\lVert \mathbf{s} \rVert = 0$, then any unit-length axis can be used as the axis $\mathbf{e}$.

I'm wondering how to express this relationship as an equation. Below are some of my ideas, but none are entirely satisfactory to me:


**idea 1:**
$$\mathbf{e} = \begin{cases} \mathbf{s}/\lVert \mathbf{s} \rVert, & \lVert \mathbf{s} \rVert \neq 0 \\ \text{any 3D unit vector}, & \lVert \mathbf{s} \rVert = 0 \end{cases}$$

**idea 2:**
\begin{align}
\mathbf{e} &= \mathbf{s}/\lVert \mathbf{s} \rVert, & \lVert \mathbf{s} \rVert &\neq 0\\\\
\mathbf{e} &\in \mathcal{S}^2, & \lVert \mathbf{s} \rVert &= 0
\end{align}

**idea 3:**
$$\mathbf{e} = \begin{cases} \mathbf{s}/\lVert \mathbf{s} \rVert, & \lVert \mathbf{s} \rVert \neq 0 \\ \mathbf{x} ~~~ | ~~~ \lVert \mathbf{x} \rVert = 1, & \lVert \mathbf{s} \rVert = 0 \end{cases}$$

**What is the best way to express the relationship/mapping from $\mathbf{s}$ to $\mathbf{e}$ mathematically (using the equations above or something else)?**