orientation of a curve

What you have described can be translated into precise mathematical language, something like this:

Given a 2-d surface $S$ in a 3-d space $M$, given a sense of orientation of that 3-d space $M$, and given an "up" direction transverse to $S$ in $M$, there is an induces sense of orientation of that 2-d space $S$.

For example, often in our ordinary 3-d space $M$ we impose Cartesian $x,y,z$ coordinates giving $M=\mathbb R^3$, we study the $x,y$ plane $S$ where $z=0$, and we use the "right hand rule" for defining an orientation of $M$: extend the thumb, forefinger, and middle finger in three different directions. Now rotate the right hand so that the middle finger points "upward" in the $z$-direction: the thumb and forefinger now define an orientation of the plane $S$.

However, you might be able to discern from this description that there is also a "right hand rule" procedure designed to work entirely intrinsically in the 2-d space $S$: simple extend your thumb and forefinger and lay them down on $S$.

All this business about ants, right hands, thumbs, etc. is a bit too biological for a mathematician, and so all of this has to be formalized in a purely mathematical language. This can be done using some tools of differential topology. What happens, in fact, is that there is a purely intrinsic notion of "$n$-dimensional orientation" for any $n$-dimensional space. Also, given an $n-1$ dimensional subspace $S$ of an $n$-dimensional space $M$, there is a mathematical formula which inputs a choice of $n$ dimensional orientation of $M$ together with a choice of "upward" direction transverse to $S$ in $M$, and outputs an induced $n-1$ dimensional orientation of $S$.