Definition 1 (Parametrised curve).
A parametrised curve in $\mathbb R^n$ is a smooth map $c: M \to \mathbb R^n$, where $M \subset \mathbb R$ is an interval.
Definition 2 (Reparametrisation).
A parametrised curve $d: [e, f] \to \mathbb R^n$ is a reparametrisation of $c: [a,b] \to \mathbb R^n$ if there exits a bijective smooth map $\varphi: [e, f] \to [a,b]$ with $d = c \circ \varphi$ and if $\varphi'(t) \ne 0$ holds for all $t \in [e, f]$.
Definition 3 (Preserving orientation).
A reparametrisation $\varphi: [a,b] \to [c, d]$ is orientation preserving if $\varphi'(t) > 0$ holds for all $t \in [a,b]$ and orientation-reversing otherwise.
Example (Half circle)
Consider the curves $c: (-1,1) \to \mathbb R$, $t \mapsto (t, \sqrt{1 – t^2})$ and $d: (0, \pi) \to \mathbb R$, $t \mapsto (\cos(t), \sin(t))$.
Then $d$ is a reparametrisation of $c$ with $\phi: (0, \pi) \to (-1,1)$, $t \mapsto \cos(t)$, which is strictly decreasing.
As $\phi'(t) = \sin(t) > 0$ for all $t \in (0, \pi)$, this reparametrisation is orientation preserving.
My Question
The curve $d$ parametrises the half circle positively (i.e. from right to left) and $c$ the other way around. Why is the reparametrisation then called orientation preserving?
Could it be that the definitions are not quite correct? As in that in definition 2 the curve is called reparametrisation and in the other the bijective map is referred to as reparametrisation?
Best Answer
The derivative of $\cos(t)$ is $-\sin(t)$, so this is $<0$ and everything is consistent. You can also see that $\phi$ maps $0$ to $1$ and $\pi$ to $-1$, which shows that is is orientation reversing.
Edit (to adress the comment): There is a certain amount of abuse of language in the combination of Definitions, which is rather common in the area. My preferred solution is to use the following line of phrasing: One introduces the notion of a differemorphism $\phi$ between two intervals. Then one observes that there are two classes of such diffeomorphisms, which one calls orientation preserving and orientation reversing, respectively. Then one says that $d$ is a reparametrization of $c$ if $d=c\circ\phi$ for a diffeomorphism $\phi$. This immedately shows that there are two classes of reparametrizations according to the two classes of diffeomorphisms.
This makes the wording a bit more complicated but avoids the problem you point out. In my opinion, it certainly makes sense to introduce the more complicated wording, if one intends to go furter towards surfaces or more general types of manifolds. If one intends to stick to curves, I am not sure.