# Continuous R^2xR^2xR^2/E^+(2) -> R^3 injection

• pmsrw3
In summary, the problem discussed in this conversation involves finding a way to represent the translation and rotation-independent aspects of an animal's motion in two dimensions. The goal is to have a continuous injection from the group of rigid-body motions in two dimensions (E+(2)) to ℝ^3. It has been discovered that there are continuous bijections that achieve this, despite initial doubts. This problem relates to the Borsuk-Ulam theorem in topology.
pmsrw3
This is a question that comes from my research. I know next to nothing about 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)).

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 believe, is impossible. Am I right? I have a feeling this is basically Borsuk–Ulam, but like I said, I'm pretty ignorant of topology.

Thanks for any help.

Got some answers to this on mathoverflow.net. It turns out there ARE continuous bijections ℝ^2×ℝ^2×ℝ^2/E+(2) -> ℝ^3.

## 1. What is "Continuous R^2xR^2xR^2/E^+(2) -> R^3 injection"?

"Continuous R^2xR^2xR^2/E^+(2) -> R^3 injection" refers to a specific mathematical function that maps a three-dimensional space (R^3) to a two-dimensional space (R^2xR^2xR^2/E^+(2)). This function is continuous, meaning that small changes in the input will result in small changes in the output.

## 2. How is "Continuous R^2xR^2xR^2/E^+(2) -> R^3 injection" used in scientific research?

This function is commonly used in fields such as physics, engineering, and computer science to model and analyze complex systems in three-dimensional space. It can help researchers understand the behavior of these systems and make predictions about their future states.

## 3. What is the difference between an injection and a continuous injection?

An injection is a type of function that maps each element in the domain to a unique element in the range. A continuous injection is a specific type of injection where small changes in the input result in small changes in the output. This property is important for studying the behavior of systems over time.

## 4. Can "Continuous R^2xR^2xR^2/E^+(2) -> R^3 injection" be reversed?

No, this function is not reversible. This means that it is not possible to map the output back to the original input. This is because the function maps a three-dimensional space to a two-dimensional space, resulting in a loss of information.

## 5. What are some real-life applications of "Continuous R^2xR^2xR^2/E^+(2) -> R^3 injection"?

This function has numerous applications in fields such as computer graphics, robotics, and signal processing. It can be used to model and analyze the movement of objects in three-dimensional space, create realistic 3D images, and track the motion of objects in videos.

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