== Composition of rotations == 

 : (Copied from my talk page).    

 Hi Oleg, 

 About compositions of rotations: you wrote 

 If however one performs rotation around a point (axis) followed by rotation around another point (axis), the overall movement may be a translation rather than a rotation. 

 I think our editing reverts have been because we agree but have opposite points of view:  you want to point out 
 that two rotations may be a translation (or rotation), but I want to point out that two rotations in general do not represent a translation or rotation. 

 Thus, perhaps 

 If however one performs rotation around a point (axis) followed by rotation around another point (axis), the overall movement may be a translation rather than a rotation (but in general is neither). 

  , formerly MrMoto, but this account seems to have been eaten up. ( 

 : Can you prove that the composition of two rotations around different centers in the plane may be something else than a rotation or a translation? Because I can prove that it is either a rotation or a translation.     
 :: Never mind, I think I am wrong. (As always, best ideas come in the shower ).    

 ::: Heh. ok )  You are correct in the context of planar rotations, but in R3, for example, the axes of rotation should intersect.   

 :::: If I may interject, rotations are direct isometries that leave a point fixed. In 2D we have only two choices for direct isometries (aside from the identity): rotations and translations. The latter leave no point fixed. In 3D we get a new option, screws. These can always be cast in the form of a rotation around an axis parallel to a translation. In 4D a rotation generally does not have an axis, and leaves only a single point fixed. Translations and screws still exist, of course. In a vector space we always rotated around a fixed origin, with no possibility of translation or screws. Most folks forget that this is not the only option, since it's the most common way to describe rotations. With a fixed origin, rotations form a group; without one, they are not even closed under composition. 
 :::: Perhaps it would help readers to see explicit examples in 2D and 3D. I'd suggest pictures and words and matrices, to make the point memorable. 
 :::::  

 :::::  
 :::: The fixed points for the leftmost 3D rotation have the form (x−3z,y,3x+z)/2, while the composition of the two rotations is a screw unless z = −2x.   

 Thank you KSmrq. Would be nice to have this somewhere. Maybe not the full formulas in the general purpose rotation article, but at least the ideas.    

 : (Copied from my talk page).