There are two reasons for me to ask this question:
- I want to know if my understanding on this issue is correct.
- To clarify a doubt I have.
I want to change the co-ordinate system of a set of points (Old cartesian coordinates system to New cartesian co-ordinate system). This transformation will involve Translation as well as Rotation. This is what I plan to do:
(Kindly refer to the image attached here )
With respect to this image I have a set of points which are in the XYZ coordinate system (Red). I want to change it with respect to the axes UVW (Purple). In order to do so, I have understood that there are two steps involved: Translation and Rotation.
When I translate, I only change the origin. (say, I want the UVW origin at (5,6,7). Then, for all points in my data, the x co-ordinates will be subtracted by 5, y by 6 and z by 7. By doing so. I get a set of Translated data.)
Now I have to apply a rotation transform (on the Translated data). The Rotation matrix is shown in the image. The values Ux, Uy and Uz are the co-ordinates of a point on the U axis which has unit distance from origin. Similarly, the values Vx, Vy and Vz are the coordinates of a point on the V axis which has a unit distance from origin. (I want to know if I am right here.)
Given this information, how can I calculate the value of Wx, Wy and Wx (a point lying on the W axis with unit distance from the origin)? I will know the co-ordinates of the origin {Obviously, (0,0,0), and I have the co-ordinates of points lying at unit distance from origin on the U and V axes. Also, U, V an W are orthonormal.} How can I calculate the 3rd row of rotation matrix R?
Is my approach correct? My ultimate goal is to transform the co-ordinate system. I can calculate points on the U and V axes, but not on the W axis. Keeping this in mind, please provide me with inputs.
(As far as I know, with the Point U (Ux, Uy, Uz), Point V (Vx, Vy and Vz) and the origin, I can define a plane. With this, I can find a point which is at unit distance from (0, 0, 0) in a direction along the plane's normal. How should I do this?)
(Also, if it serves any purpose, I would like to let you know that I am using MATLAB.)