iOS How to change a point to another orientation?

[idonotliketostu]

Example:

grid1 is a regular ( x y z ) grid
grid2 is like grid 1 but rotated +45 degrees along the z axis (a yaw of 45)
grid 1 and 2 touch at origin

a dot grid 2 has points (1,1,500);
x = 1
y = 1
z = 500;


the same point viewed from a "global perspective" on grid 1 would have points
x = sqrt2
y = 0;
z = 500;


My solution was to rotate grid2 one axis at a time until it looked like grid1 while updating the changing points. (rotate around z axis changes points x and y; rotate around y axis changes points z and x)

My question is..
Is there a known formula to change the point from grid 2 to grid 1?

 

[chown33]

3D affine transform.

It's like a 2D affine transform, but for 3 dimensions instead of 2, so it's a 4x4 matrix instead of a 3x3 for 2D:
http://en.wikipedia.org/wiki/Transformation_matrix

If all you're doing is a rotation, then all you need is a 3D rotation matrix:
http://en.wikipedia.org/wiki/Rotation_matrix