Comments on how to mathematically express a relationship in which a vector can be any 3D unit vector
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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: $$\mathbf{e} = \begin{cases} \mathbf{s}/\lVert \mathbf{s} \rVert, & \lVert \mathbf{s} \rVert \neq 0 \\ \mathbf{x} ~~~ | ~~~ \mathbf{x} \in \mathbb{R}^3, ~\lVert \mathbf{x} \rVert = 1, & \lVert \mathbf{s} \rVert = 0 \end{cases}$$
What is the best way to express the relationship/mapping from $\mathbf{s}$ to $\mathbf{e}$ mathematically (using the equations above or something else)?
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If I understand the situation correctly, the problematic issue is the zero rotation, and the issue is that if you do not rotate things at all, than that corresponds to a zero rotation around any axis.
If this is true, then I could understand any of your three ideas. The third one might be marginally harder to read because of the spacing, but that can be dealt with; it is typography, not mathematics.
In idea two you might want to add the word «or».
But generally, I would use whichever is clearest and uses as similar notations as you use for similar ideas in the rest of the text.
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