Clarification: What does it mean when "$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points"

Question

"$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points"

Does this mean:

(A) $\phi:[a,b]\to U$ and $\psi:[a,b]\to U$

(B) $\phi:[a,b]\to U$ and $\psi:[c,d]\to U$ s.t. $\phi(a)=\psi(c)$ and $\phi(b)=\psi(d)$

(C) Something else?

Answer 1

If you consider both curves up to (order-preserving) reparametrization (i.e., the parametrization is a diffeomorphism with strictly positive derivative), then (A) and (B) are equivalent, since any two closed intervals in the Real line are homeomorphic (by an order-preserving homeomorphism). Now, take a homeomorphism $h: [c,d] \rightarrow [a,b]$. Then $\phi':=\phi o h$ is a reparametrization of $\phi$, with $\phi': [a,b] \rightarrow \mathbb R^2$.

So you can always reduce case (B) to case (A), and then, up to reparametrization you can just consider the case (A).

Answer 2

It must be (B), since (A) does not require that $\phi(a)=\psi(a)$ and $\phi(b)=\psi(b)$. The curves are subsets of $U$, and for them to have the same endpoints, the function values on the endpoints of the domain must be equal.