# CoordTrans - coordinate system handedness

Dear Antonio,

The 3D coordinate system (X, Y, Z) is a right-handed coordinate system and the corresponding rotation transformation parameters (Rx, Ry, and Rz) are according to the right-hand-rule, i.e., positive in the counterclockwise direction. Is this correct?

CoordTrans transforms points from one system to another one, if both systems are left-handed or right-handed systems. The rotation sequence is parameterized by an axis of rotation and a rotation angle expressed by a quaternion. The basic transformation equations are published in:

Lösler, M., Eschelbach, C.: Zur Bestimmung der Parameter einer räumlichen Affintransformation. avn - Zeitschrift für alle Bereiche der Geodäsie und Geoinformation, 121(7), S. 273-277, 2014.

The PDF is freely available.

A quaternion can be expressed as a rotation matrix. Having such a matrix, it is possible to determine Euler-angles. However, 12 rotation sequences exist, for instance, X-Y-Z, X-Y-X, etc. For that reason, the angles are useless without knowing the underlying sequence. CoordTrans obtains the angles from the quaternion using the following equations:

$r_x = \arctan2(r_{23}, r_{33}),$
$r_y = \arcsin(-r_{13}),$
$r_z = \arctan2(r_{12}, r_{11}).$

Here, $r_{ij}$ is an element of the rotation sequence defined by the quaternion, i.e.,

$r_{11} = 2 q0^2-1+2q1^2,$
$r_{12} = 2(q1q2-q0q3),$
$r_{13} = 2(q1q3+q0q2),$
$r_{23} = 2(q2q3-q0q1),$
$r_{33} = 2q0^2-1+2q3^2.$

Kind regards
Micha

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