We consider two 3D isomorphic coordinates system S and P with different origin. We have n points (I guess 4 is minimum) on which we know :
- their 3D cartesian coordinates x y z on system S
- the two spherical coordinates inclination and azimuth on system R but not the radius.
Using those data, we would like to know the transformation to get any point on a system to the other. It can be either the matrix transformation or quaternion or just equations.
The problem which seems to have no easy solution is to determine the location of the origin of system P in the system S.
EDIT : This problem seems equivalent to find intersection points of 3 spindle tori, but I do not see any solution to compute those intersection points.
Origin of the problem :
Using moving head fixture, I want to be able to select a specific place on a scene and converting the XY location to the Pan/tilt inputs of my fixture. In order that, I can calibrate the fixture by orienting them on the 4 corner of the scene and thus determining the pan/tilt of those 4 corner. We can suppose that we know the dimension of the scene, but not the exact location of the fixture nor its orientation.
This problem could also be used to locate yourself in the interstellar space using the known location of a few stars and by measuring the angle between them. So I guess somebody have already think of it.
Given the position of two points in space and the apparent angle between them specifies a unique sphere of points from which this apparent angle appears. Given the position of four points specifies 6 such spheres. If the positions of the points are general and the angular data are consistent, there will be a unique point of intersection of the spheres.