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

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.

posted almost 2 years ago

CC BY-SA 4.0

2y ago

tommi‭

435 reputation 8 17 55 6

What makes the most sense to do somewhat depends on the context of this expression.

What it seems you are really trying to do is defined $\mathbf{e}$ as a function of $\mathbf{s}$. What's making it awkward is that $\mathbf{e}$ isn't a function of $\mathbf{s}$ but merely a total (multi-valued) relation on $\mathbf{s}$. Thus the Right solution is to address this head-on. You could do this by having $\mathbf{e}$ be a set-valued function or by having something like an $\mathsf{axis}\mathsf{\_}\mathsf{of}$ relation. Specifically, $$\mathbf{e}(\mathbf{s})=\{\begin{array}{ll}\{\mathbf{s}/\Vert \mathbf{s}\Vert \},& \Vert \mathbf{s}\Vert \ne 0\\ {\mathcal{S}}^2,& \Vert \mathbf{s}\Vert =0\end{array}$$ or $$\mathsf{axis}\mathsf{\_}\mathsf{of}(\mathbf{e},\mathbf{s})⟺(\Vert \mathbf{s}\Vert =0\wedge e\in {\mathcal{S}}^2)\vee (\Vert \mathbf{s}\Vert \ne 0\wedge e=\mathbf{s}/\Vert \mathbf{s}\Vert )$$ The latter formula could be simplified in various ways, e.g. $$\mathbf{e}\in {\mathcal{S}}^2\wedge (\Vert \mathbf{s}\Vert \ne 0⟹\mathbf{e}=\mathbf{s}/\Vert \mathbf{s}\Vert )$$ Incidentally, the definition of $\mathsf{axis}\mathsf{\_}\mathsf{of}$ is a direct answer to your title question.

Now I will talk about various less Right options. Why do this? First, it is perfectly valid and quite doable to do things the way I described above. But it will also feel weird and often be clumsy^[1]. And all this for one particular edge case. Essentially, the (non-functional) relational nature of $\mathbf{e}$ pollutes everything it touches.

One only slightly less Right approach is to make an arbitrary choice. Unlike the situation with division of numbers (where we have a non-functional relation as well but in that case a partial, single-valued one), you have options. One way of doing this is to only partially axiomatize/specify the $\mathbf{e}$ function. This is easy enough to do informally, but I'll also explain how it would be done formally. Essentially, you add $\mathbf{e}$ as a new function symbol^[2] and add axioms $\mathrm{\forall }\mathbf{s}\in \mathcal{E}.\mathbf{e}(\mathbf{s})\in {\mathcal{S}}^2$ and $\mathrm{\forall }\mathbf{s}\in \mathcal{E}.\Vert \mathbf{s}\Vert \ne 0⟹\mathbf{e}(\mathbf{s})=\mathbf{s}/\Vert \mathbf{s}\Vert$ where $\mathcal{E}$ will be the set of exponential coordinates^[3]. The value of $\mathbf{e}(\mathbf{0})$ is not (logically) decidable from these axioms but it is a particular but unknown value constrained to lie in ${\mathcal{S}}^2$. That is, $\mathbf{e}(\mathbf{0})=\mathbf{e}(\mathbf{0})$ is true^[4]. Note how $1/0=1/0$ is not true. This seeming equation between expressions is actually shorthand for a predicate unrelated to equality. Of course, you could just specify the value, say $(1,0,0)$ (making clear that this choice is arbitrary). The only real downside to this is that it is technically possible to write theorems that depend on this choice.

Finally, you can do what is done for division of numbers: implicitly add constraints to restrict the relation to the domain on which it is single-valued and thus a function. That said, the situation isn't nearly as bad as for division. Per the previous paragraph, as long as $\mathbf{e}(\mathbf{0})$ is treated as having a consistent if unknown value — which is the natural thing to do — there won't be any problems.

posted almost 2 years ago

CC BY-SA 4.0

2y ago by trichoplax‭

Derek Elkins‭

1604 reputation 2 48 165 25