orientation of a curve

Question

In the eponymous wikipedia article we read the following definition of curve orientation:

In the case of a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections), the curve is said to be positively oriented or counterclockwise oriented, if one always has the curve interior to the left (and consequently, the curve exterior to the right), when traveling on it.

That got me thinking. When I draw a closed curve on a sheet of paper (a two dimensional plane), I can imagine myself walking along the curve, on top of the paper, so to speak. With this visualization I can determine which is the positive and which is the negative orientation of the curve as per the definition given above.

But I could also imagine myself, as a tiny ant, walking upside-down on the other side of the paper (the side that touches the table) and then the positive and negative orientations would reverse. So the definition of the orientation depends on selecting an "up" side of the plane from the point of view of the observer. But the observer exists outside the 2D space defined by the plane and I have never, so far, imagined that the Euclidean 2D plane has an "up" side. Can someone clear up the confusion?

Answer 1

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 $S$.