What does it mean to fix an orientation on the plane?

Question

I have an exercise in geometry that works with plane curves. For starters, it fixes an orientation on the vector plane $\stackrel{\to}{P}$ and states that $J: \stackrel{\to}{P} \to \stackrel{\to}{P}$ is the 90 degrees rotation in the sense of orientation.

I wonder what to "fix an orientation means", is it to fix a basis and give an order between the basis?.

The exercise then ask me to prove that tangent $t(s) = \alpha'(s)$ and normal $J(t(s))$ form a positive orthonormal basis of $\stackrel{\to}{P}$.

To see that the basis is orthonormal is easy. But how can I show the basis is oriented positively with respect to the original one?

Answer 1

Orientation only makes sense here with respect to ordered bases. Ordered bases divide into two equivalence classes based on the sign of the determinants of the change-of-basis matrices between them. Each class consists of all of the ordered bases that are related by a change-of-basis matrix with positive determinant. (You can’t always use the determinant of the matrix of basis vectors since you might not be talking about the entire parent space—that matrix won’t always be square.) We say that all of the bases in a class have the same orientation, but which of the two orientations is “positive,” “right-handed,” or whatever term you want to use is an arbitrary choice that might be made based on other factors extrinsic to this division.

When you fix an orientation, you choose one of these equivalence classes to be the “positive” one. In practice, this will entail choosing a member of the class as the “reference” basis. You can then compare the orientation of any other ordered basis to that one by examining the determinant of the change-of-basis matrix, or, if these bases span the entire parent space, comparing the signs of the determinants of the matrices formed from the basis vectors.

Answer 2

Yes, to fix an orientation means to choose an ordered basis $E=[e_1,e_2]$. Once this is chosen, another oriented basis $F=[f_1,f_2]$ is called positive if the change of basis matrix $A$ such that $E=FA$ has positive determinant.

For your problem about a plane curve, you can proceed as follows. Let $[e_1,e_2]$ be an orthonormal basis; choosing this order: first $e_1$, then $e_2$, fixes an orientation on the plane. The rotation $J$ acts as follows: $Je_1 = e_2$ and $Je_2 = -e_1$.

The unit tangent vector $t(s)$ can be expressed as $t(s) = a_1(s)e_1 + a_2(s) e_2$. Use the linearity of $J$ to express $Jt(s)$ in terms of the basis. You will be able to write down the matrix $A$, and you will see its determinant is $a_1(s)^2 + a_2(s)^2 = \|t(s)\|^2 = 1 > 0$. And so, $[t(s),Jt(s)]$ has the same orientation as $[e_1,e_2]$, i.e. it is positively oriented.