I am interested in the set of oriented rectangles, that are centered on the origin, and can be described by their width, height and angle.
I am looking for a "good" parameterization that is non-ambiguous (a rectangle should correspond to only one set of parameters) and continuous (close rectangle...
This is an interesting question. (I never understand why some questions that serious thought has gone into don't get upvotes.) But I think you need to be a bit more precise. Apparently you wouldn't mind the angle wrapping around from $2\pi$ to $0$, even though this is also a discontinuity in the mapping from the rectangle space to the parameter space. Presumably you don't mind this because you consider an angle parameter as an element of $U(1)$ (or a point on the unit circle), where this discontinuity doesn't exist. ...
... But then you could treat the entire parameter space that way and make identifications that remove the discontinutities. So perhaps what you want is a parametrization in terms of numbers and/or elements of $U(1)$, without more complicated identifications in the parameter space?
@joriki You're absolutely right. I assumed that I could parameterize 2D rotations in a continuous way, typically using a point on the unit circle as you said. However, I am unable to find a good parameterization for oriented rectangles that way. Now that I think of it, maybe a good embedding in a larger space would work if there is a simple enough constrain to describe the manifold (like for $U(1)$).
@user209974: But then you need to somehow distinguish what you want to do from the trivial solution of parametrizing the rectangle space by the rectangle space, which is also a manifold that's relatively simple to describe.
@user209974: You wrote "maybe a good embedding in a larger space would work if there is a simple enough constraint to describe the manifold". I'm saying that if you allow manifolds beyond $\mathbb R$ and $U(1)$, you need to delimit precisely which ones you allow, since the space of rectangles itself that your parametrizing is also a manifold and could trivially be used to parametrize itself. So there must be a line somewhere between $\mathbb R$, $U(1)$ and "larger spaces with simple enough constraints to describe the manifold" on the one hand and the space of rectangles itself on the other.
@joriki Ah, I think I know what you mean, but the space of rectangles is not really a good parameter space for me if I cannot describe it with numbers. By embedding into a larger space, I was implicitely referring to $R^n$ (similarly to $U(1)$ being represented in $R^2$ as a circle).
@user209974: Sure you can. You can parametrize the rectangles by the coordinates of their corners, thus embedding the space of rectangles in $\mathbb R^8$.
Yes, I think you should specify in the question something like that you want a homeomorphism between the space of rectangles and a manifold embedded in some R^n.
It's a great question. math.SE can be a bit frustrating in that way, that questions that obviously have more content than so many of the homework questions and the like just don't get upvoted.
Unfortunately I don't know enough about homotopy and homology, which I think might be the tools needed to prove that this is impossible, if indeed it is.
Discussion between joriki and user209974
Discussion between joriki and user209974