Maths - Rotation about any point - forum discussion

By: Nobody/Anonymous - nobody
Error in 3D rotation around a point matrices  
2005-05-26 08:03

On this webpage, I believe I have found an error: 
 
https://www.euclideanspace.com/maths/geometry/affine/aroundPoint/index.htm 
 
It says: 
 
So matrix representing rotation about a given point is: 
 
[R] = [T]-1 * [R0] * [T] 
 
However with the matrices given, you have to use column vectors and calculate the rotated point by; 
 
[New] = [R] * [Old] 
 
You can see by looking at this and the equation for [R] that it will give you the wrong results. In order to make it work, you need to swap the translate and inverse translate matrices, so you would re-write your equation like this: 
 
[R] = [T] * [R0] * [T]-1 
 
And then re-work you matrix products to correct them. Alternatively, you could use row vectors and leave your equation as it is, but transpose your elementary (T, R0, and T-1) matrices and re-work the products. 
 
Regards, 
Richard 
SirRichard@fascinationsoftware.com 

By: Martin Baker - martinbaker
RE: Error in 3D rotation around a point matri  
2005-06-02 11:12

Richard, 
 
Thank you very much for this, I agree with what you say and I'll update the page. 
 
I think perhaps what I did wrong is after saying that rotate about arbitrary point is equivalent to the following 3 steps: 
 
1) translate the arbitrary point to the origin (which should be translate by -Px,-Py,-Pz) 
2) rotate about the origin  
3) then translate back (which should be translate by +Px,+Py,+Pz) 
 
Then if the arbitrary point is (Px,Py,Pz) the first translation is -Px,-Py,-Pz, then the rotation, then add +Px, +Py, +Pz. 
 
So if we are using the global frame-of-reference (as explained here: https://www.euclideanspace.com/maths/geometry/rotations/for/ 
) then, 
 
[resulting transform] = [third transform] * [second transform] * [first rotation] 
 
[resulting transform] = [+Px,+Py,+Pz] * [rotation] * [-Px,-Py,-Pz] 
 
Which, as you say, inverts the translation component of the resulting transform. 
 
Thanks, 
 
Martin