Quaternion and normalization

Question

I know that quaternions need to be normalized if I want to rotate a vector.

But are there any reasons to not automatically normalize a quaternion? And if there are,

Answer 1

Funnily enough, building rotation matrices is one operation where normalizing quaternions is NOT needed, saving you one sqrt:

M = [w*w+x*x-y*y-z*z, 2*(-w*z+x*y),    2*(w*y+x*z);
     2*(w*z+x*y),     w*w-x*x+y*y-z*z, 2*(-w*x+y*z);
     2*(-w*y+x*z),    2*(w*x+y*z),     w*w-x*x-y*y+z*z] / (w*w+x*x+y*y+z*z)

(in a MATLAB-ish notation) for the quaternion w+x*i+y*j+z*k.

Moreover, if you are working with homogeneous coordinates and 4x4 transformation matrices, you can also save some division operations: just make a 3x3 rotation part as if the quaternion was normalized, and then put its squared length into the (4,4)-element:

M = [w*w+x*x-y*y-z*z, 2*(-w*z+x*y),    2*(w*y+x*z),     0;
     2*(w*z+x*y),     w*w-x*x+y*y-z*z, 2*(-w*x+y*z),    0;
     2*(-w*y+x*z),    2*(w*x+y*z),     w*w-x*x-y*y+z*z, 0;
     0,               0,               0,               w*w+x*x+y*y+z*z].

Multiply by a translation matrix, etc., as usual for a complete transformation. This way you can do, e.g.,

[xh yh zh wh]' = ... * OtherM * M * [xold yold zold 1]';
[xnew ynew znew] = [xh yh zh] / wh.

Normalizing quaternions at least occasionally is still recommended, of course (it may also be required for other operations).