I am trying to understand a notion given in this paper, summarized as the following:
Let $c \colon I \to \mathbb{R}^3$ be a smooth, regular, arc-length parametrized curve with image $C$, where $I$ is a compact interval of length $a$.
For a smooth function $\xi\colon I \to \mathbb{R}^3$ with $\xi(t) \times c'(t) \neq 0$ for all $t \in I$, we call $$f \colon I \times (- \varepsilon, \varepsilon) \to \mathbb{R}^3, (t,v) \mapsto c(t) + v \xi(t) $$ a ruled strip, where $\varepsilon > 0$ is fixed and chosen, such that $f$ is an embedding.
Two ruled strips $f$ and $g$ along a curve $C$ are said to have the same image (as map germs along $C$), if there exists $\delta \in (0, \varepsilon]$, such that $$f(I \times (- \delta,\delta) ) \subseteq g(I \times (- \varepsilon,\varepsilon) ) $$ holds.
I am a bit confused by this, because I only know of the definition of a germ based at a point, so I don't understand what a "map germ along $C$" is supposed to mean. In the paper, it also often done to call ruled strips "equal as map germs" without mentioning a base point like in the definition of a germ (which I think should mean again along $C$ in this case). The only thing I could come up with while reading the paper are the following two interpretations of mine:
Map germs along $C$ means they are equal as germs at every point "along" $C$, i.e. we have $f,g$ are equal as germs at every point $(t,0)$ for $t \in I$.
Having the same image as map germs and being equal as map germs means the same thing, as a way of abbreviating wording.
Can anyone confirm this or give a proper definition? Is there in general maybe a notion of saying that two functions are germs at a given set?