Non-Convex Coordinate Transform Problem Rotating Frame

In summary, the goal is to find the transform between the two frames, given a set of 3D points at a given time in one frame, the angle of rotation between the frames, and new 3D points at a subsequent time. The problem can be solved using plane equations and Lorentz transformations, but the unknown plane normal and distance in one frame pose a challenge. It may be possible to approximate a solution using this method.
  • #1
phil0stine
5
0
I am sure this is not the best description of the problem, so let me know how I can clarify.

Say there are 2 coordinate systems, with one orbiting around the other. Call one fixed ƒ and the other rotating ρ. The goal is to find the transform between the two frames.

What's known is
1) A set of 3D points at a given time τ in frame ρ
2) The angle θ which frame ρ is subsequently rotated about the z-axis of frame ƒ between time τ and τ+1
3) New 3D points at time τ+1 in frame ρ, and so forth.
4) All points from all views lie in a single plane in frame ƒ (this does not mean the points at time τ and time τ+1 are the same, just that they are coplanar)

I want to set up the problem to use the plane equation in frame ƒ to solve an overdetermined system, but I run into the problem that I do not know the plane normal η and distance d in frame ƒ.

Is there any way this can be solved (even approximately)? Thanks
 
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  • #2
I would start small and go from one dimension, then two dimensions to finally three dimension. A sketch would help a lot of understand the problem better.
 
  • #3
phil0stine said:
I am sure this is not the best description of the problem, so let me know how I can clarify.

Say there are 2 coordinate systems, with one orbiting around the other. Call one fixed ƒ and the other rotating ρ. The goal is to find the transform between the two frames.

What's known is
1) A set of 3D points at a given time τ in frame ρ
2) The angle θ which frame ρ is subsequently rotated about the z-axis of frame ƒ between time τ and τ+1
3) New 3D points at time τ+1 in frame ρ, and so forth.
4) All points from all views lie in a single plane in frame ƒ (this does not mean the points at time τ and time τ+1 are the same, just that they are coplanar)

I want to set up the problem to use the plane equation in frame ƒ to solve an overdetermined system, but I run into the problem that I do not know the plane normal η and distance d in frame ƒ.

Is there any way this can be solved (even approximately)? Thanks
Lorentz transformations concepts can be used to solve the puzzle.
 
Last edited by a moderator:

1. What is the Non-Convex Coordinate Transform Problem in a Rotating Frame?

The Non-Convex Coordinate Transform Problem in a Rotating Frame is a mathematical challenge that arises when trying to convert coordinates between two rotating reference frames. It involves finding the optimal transformation that minimizes the error between the two frames, taking into account the rotation of both frames.

2. Why is the Non-Convex Coordinate Transform Problem difficult to solve?

The Non-Convex Coordinate Transform Problem is difficult to solve because it is a non-convex optimization problem. This means that there are multiple local minima, making it challenging to find the global minimum. Additionally, the problem is highly nonlinear and involves complex mathematical equations.

3. What are the applications of the Non-Convex Coordinate Transform Problem in a Rotating Frame?

The Non-Convex Coordinate Transform Problem has various applications in fields such as robotics, computer graphics, and navigation. It is used to convert coordinates between different rotating reference frames in order to accurately track the position and orientation of objects or vehicles in motion.

4. How do scientists approach solving the Non-Convex Coordinate Transform Problem?

Scientists use various mathematical and computational methods to solve the Non-Convex Coordinate Transform Problem. This may involve gradient descent algorithms, numerical optimization techniques, or analytical solutions based on specific assumptions about the rotating frames and their relationship.

5. What are some current research efforts in solving the Non-Convex Coordinate Transform Problem?

Current research efforts in solving the Non-Convex Coordinate Transform Problem focus on developing more efficient and accurate algorithms, as well as exploring new applications in fields such as virtual reality and augmented reality. Some scientists are also investigating ways to incorporate uncertainty and noise into the transformation process for better real-world performance.

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