Rotation matrix and translation vector¶

A pose can also be defined by a rotation matrix $R$ and a translation vector $T$ .

$R = \left(\begin{array}{ccc} r_{00} & r_{01} & r_{02} \\ r_{10} & r_{11} & r_{12} \\ r_{20} & r_{21} & r_{22} \end{array}\right), \qquad T = \left(\begin{array}{c} X \\ Y \\ Z \end{array}\right).$

The pose transformation can be applied to a point $P$ by

$P' = R P + T.$

Conversion from rotation matrix to quaternion¶

The conversion from a rotation matrix (with $det(R)=1$ ) to a quaternion $q=(\begin{array}{cccc}x & y & z & w\end{array})$ can be done as follows.

$x &= \text{sign}(r_{21}-r_{12}) \frac{1}{2}\sqrt{\text{max}(0, 1 + r_{00} - r_{11} - r_{22})} \\ y &= \text{sign}(r_{02}-r_{20}) \frac{1}{2}\sqrt{\text{max}(0, 1 - r_{00} + r_{11} - r_{22})} \\ z &= \text{sign}(r_{10}-r_{01}) \frac{1}{2}\sqrt{\text{max}(0, 1 - r_{00} - r_{11} + r_{22})} \\ w &= \frac{1}{2}\sqrt{\text{max}(0, 1 + r_{00} + r_{11} + r_{22})}$

The $\text{sign}$ operator returns -1 if the argument is negative. Otherwise, 1 is returned. It is used to recover the sign for the square root. The $\text{max}$ function ensures that the argument of the square root function is not negative, which can happen in practice due to round-off errors.

Conversion from quaternion to rotation matrix¶

The conversion from a quaternion $q=(\begin{array}{cccc}x & y & z & w\end{array})$ with $||q||=1$ to a rotation matrix can be done as follows.

$R = 2 \left(\begin{array}{ccc} \frac{1}{2} - y^2 - z^2 & x y - z w & x z + y w \\ x y + z w & \frac{1}{2} - x^2 - z^2 & y z - x w \\ x z - y w & y z + x w & \frac{1}{2} - x^2 - y^2 \end{array}\right)$