Non-Convex Coordinate Transform Problem Rotating Frame

[phil0stine]

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

 

[fresh_42]

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.

 

[satishkumar]

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.