So i am looking for a proof of this claim. I am not even sure how to start because I do not know what the rigorous definition of "touch" is. But by drawing a picture i can see that it is when two curves have a single intersection point and then kind of go into different directions. But I do not know how to formalize this. Is the definition "an intersection point where the two curves have equal tangents"? I think i am struggling to even articulate what i am asking.
Thanks
I'm not sure if this answers your question, but if you are looking at graphs of real valued-functions, you could describe the graphs of $f$ and $g$ as "touching" at a point $x_0$ if there is a neighborhood $I$ of $x_0$ with the property that $f(x) \ge g(x)$ for all $x \in I$ and $f(x_0) = g(x_0)$. Thus the graphs intersect but can't "cross" as the value of $g$ never exceeds that of $f$.
Interestingly if $f$ and $g$ are differentiable and their graphs touch at $x_0$ in this sense then it must follow that $f'(x_0) = g'(x_0)$ simply because $f - g$ attains its minimum value at $x_0$.
I never though about an analogy to space curves but now I'm curious.