Curvilinear coordinates question

[mnb96]

Hi,
if we consider a transformation of coordinates Cartesian$\rightarrow$Polar, it is straightforward to derive $r = (x^2 + y^2)^{1/2}$ and $\theta = atan2(y/x)$, because we actually know what our new coordinate system should be like.

Now let's pretend we have never seen polar coordinates, and we have no idea how to convert from cartesian to polar.
However, all we know is that we have the following family of linear transformations in $\mathcal{R}^2$,
$$(\begin{array}{ccc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array})$$
which are actually rotation matrices acting on 2D vectors, and we would like to find a new frame of reference in which the parameter $\theta$ becomes one coordinate.

Is it possible from this knowledge only, to arrive at the formulas above for $r,\theta$?

 

[wofsy]

the flow on the plane that the action of the rotation group determines (for positive angles) and its unit length orthogonal complement (outward pointing) integrate to polar coordinates.

 

[mnb96]

Hi!
thanks for the answer. I actually had the same idea you wrote but only as a visual intuition. Could you elaborate more at least the formalization of the process (not necessarily all the steps).
It is not very clear to me the first step: considering the flow determined by the rotation group.

 

[wofsy]

Starting at any point in the plane positive rotations map generates a circle centered at the origin in the plane. this circles ia a parameterized curve and its derivatives are vectors of length 1 tangent to the circle. Once you have these vectors you can find their outward pointing orthogonal complements. This gives you the flow you want.