- #1
mnb96
- 715
- 5
Hello,
I posted a similar question long time ago, but after working on it I am still unable to arrive at a solution.
Let's have a group of linear transformations (rotations in the xy-plane):
[tex]R_\theta=\{ (\begin{array}{ccc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}) \\ : \\ \theta \in [0,2\pi] \}[/tex]
The question is: How can I construct an orthogonal curvilinear coordinates system, in which the parameter [itex]\theta[/itex] works as one coordinate?
What I am supposed to get as a result are essentially the equations defining the cartesian-to-polar transformation.
----------------
My attempt:
Observe that given any vector x, the orbit [tex]R_{\theta}(\mathbf{x})[/tex] is a parametric curve which is obviously a circle.
The (gradient) vectors
[tex]e_\theta=\frac{\partial R_{\theta}(\mathbf{x})}{\partial \theta}[/tex] are tangent to the curve, so if we consider their orthogonal complement [tex]e_\theta^*[/tex] (which is easy to find), we have already found a family of local orthogonal bases.
How can I continue from this point?
I am supposed to get: [itex]r = (x^2 + y^2)^{1/2}[/itex] and [itex]\theta = atan2(y/x) [/itex], but I don't know how to arrive at that.
I posted a similar question long time ago, but after working on it I am still unable to arrive at a solution.
Let's have a group of linear transformations (rotations in the xy-plane):
[tex]R_\theta=\{ (\begin{array}{ccc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}) \\ : \\ \theta \in [0,2\pi] \}[/tex]
The question is: How can I construct an orthogonal curvilinear coordinates system, in which the parameter [itex]\theta[/itex] works as one coordinate?
What I am supposed to get as a result are essentially the equations defining the cartesian-to-polar transformation.
----------------
My attempt:
Observe that given any vector x, the orbit [tex]R_{\theta}(\mathbf{x})[/tex] is a parametric curve which is obviously a circle.
The (gradient) vectors
[tex]e_\theta=\frac{\partial R_{\theta}(\mathbf{x})}{\partial \theta}[/tex] are tangent to the curve, so if we consider their orthogonal complement [tex]e_\theta^*[/tex] (which is easy to find), we have already found a family of local orthogonal bases.
How can I continue from this point?
I am supposed to get: [itex]r = (x^2 + y^2)^{1/2}[/itex] and [itex]\theta = atan2(y/x) [/itex], but I don't know how to arrive at that.