Quaternions

Eric Bainville - Mar 2007

Quaternion from rotation matrix

We have seen in the previous chapter how to obtain the 3×3 rotation matrix corresponding to unit quaternion q. We see here how to obtain q=(s,u_x,u_y,u_z), one of the two opposite unit quaternions corresponding to a given rotation matrix M:

M_xx	M_xy	M_xz
M_yx	M_yy	M_yz
M_zx	M_zy	M_zz

We first compute the four values:

4.s² = 1+M_xx+M_yy+M_zz,
4.u_x² = 1+M_xx-M_yy-M_zz,
4.u_y² = 1-M_xx+M_yy-M_zz,
4.u_z² = 1-M_xx-M_yy+M_zz.

Comparing these values (all between 0 and 4), we can identify the largest component of q in magnitude. Then we pick the three products involving this component in the following list to build a vector proportional to q. q is then obtained by normalizing this vector.

4.s.u_x = M_zy - M_yz,
4.s.u_y = M_xz - M_zx,
4.s.u_z = M_yx - M_xy,
4.u_x.u_y = M_xy + M_yx,
4.u_y.u_z = M_yz + M_zy,
4.u_z.u_x = M_zx + M_xz.

For example, suppose the largest component is u_x, we should normalize the vector (4.s.u_x,4.u_x²,4.u_x.u_y,4.u_x.u_z). This guarantees the best stability in all cases, and the unit norm of the result.