Quaternions

Eric Bainville - Mar 2007

Unit quaternions and rotations

A unit quaternion is a quaternion q of norm |q|=1. The set of unit quaternions is a non-commutative multiplicative group. Given a unit quaternion q, let's define the function r_q operating on 3-vectors as follows:

r_q(u) = q × u × q*

r_q has the following properties:

r_q(u) is a 3-vector: r_q is a transformation of R³.
r_q(a.u + b.v) = a.r_q(u)+b.r_q(v): r_q is a linear transformation of R³.
<r_q(u),r_q(v)> = <u,v> (preserves the dot product): r_q is an orthogonal transformation of R³.
r_q = r_-q: q and -q correspond to the same transformation.
r₁ is the identity.
r_q^-1 = r_q*: q and q* represent inverse transformations.
r_p(r_q(u)) = r_p×q(u): p × q corresponds to r_p o r_q.
If x is the vector part of q, then r_q(x) = x: x is invariant by r_q.

Developping the product, we can obtain the 3×3 matrix M_q of r_q, for q=(s,u):

s²+u_x²-u_y²-u_z²	2(u_x.u_y-s.u_z)	2(u_x.u_z+s.u_y)
2(u_x.u_y+s.u_z)	s²-u_x²+u_y²-u_z²	2(u_y.u_z-s.u_x)
2(u_x.u_z-s.u_y)	2(u_y.u_z+s.u_x)	s²-u_x²-u_y²+u_z²

After reduction, it appears that det(M_q)=|q|⁶=1: r_q is a rotation.

Up to the identification of q and -q, r_q is an isomorphism between the group of rotations in R³, and the group of unit quaternions.

Computing the product of two rotations requires 16.mul+12.add in quaternion form, and 27.mul+18.add in matrix form. Computing the image of a vector by a rotation requires 28.mul+20.add in quaternion form (using two consecutive quaternion products), and 9.mul+6.add in matrix form. Computing M_q requires 10.mul+21.add.

Using quaternions is a faster and more stable way of composing a large number of rotations. To apply the rotation itself to any number of vectors (even to one vector, unless it is known to have two zero coordinates), it is more convenient to precompute the matrix form, then compute matrix-vector products.