This is a question that comes from my (biological) research. I'm very weak in topology, so I'm not able to assure myself of the answer. The problem is this: I'm watching an animal move in two dimensions. At three successive points in time I have three positions, (x1,y1), (x2,y2), (x3,y3). But there are three uninteresting degrees of freedom in these numbers: two that say where it all happened and one that gives the angle you're looking at it from. In other words, I am only interested in translation and rotation-invariant aspects of the motion. Thus, the three positions are best understood not as being a point in ℝ^2×ℝ^2×ℝ^2, but in the orbit space ℝ^2×ℝ^2×ℝ^2/E+(2), E+(2) being the group of rigid-body motions in two dimensions, acting uniformly on all three positions, i.e. e in E+(2) acts on ((x1,y1), (x2,y2), (x3,y3)) to produce (e(x1,y1), e(x2,y2), e(x3,y3)). You can use the translation degree of freedom to reduce this to ℝ^2×ℝ^2/SO(2).


I want to get three numbers that contain all the rotation and translation-independent information in (x1,y1), (x2,y2), (x3,y3). This is easy. I would also like the mapping to be continuous. That is, I would like to have a continuous injection from ℝ^2×ℝ^2×ℝ^2/E+(2) -> ℝ^3. This, I suspect, is impossible. Am I right? 


Thanks for any help.



continuous R^2xR^2xR^2/E^+(2) -> R^3 injection?
By translation, fix the first point to be at the origin. Consider the other two points as complex numbers, and take their quotient. As long as the other two points are not both at the origin, this continuously gives an element of $\mathbb{CP}^1\cong S^2$. Up to rotation, the other two points are determined by this quotient together with a scale parameter, such as the sum the sides of the triangle. A point on the sphere together with a nonzero scale parameter gives you a point in $\mathbb{R}^3\setminus\{0\}$. Thus we have a continuous injection (in fact, homeomorphism) from your space (except for the point where all three points coincide) to $\mathbb{R}^3\setminus\{0\}$.


What about the case when all three points coincide? Well, a sequence in your space will converge to the case when all three points coincide iff the scale parameter converges to 0. Thus we can continuously extend the map to send that point to $0\in\mathbb{R}^3$. We thus get that your space is actually homeomorphic to $\mathbb{R}^3$.


