Concept – Dual Quaternions

HALCON ScriptCpp++.NETPythonC

This chapter contains operators for handling dual quaternions.

Introduction to Dual Quaternions

A dual quaternion \(\hat{q} = q_{r} + \varepsilon q_{d}\) consists of the two quaternions \(q_{r}\) and \(q_{d}\), where \(q_{r}\) is the real part, \(q_{d}\) is the dual part, and \(\varepsilon\) is the dual unit number (\(\varepsilon^2=0\)). Each quaternion \(q=w+ix+jy+kz\) consists of the scalar part \(w\) and the vector part \({\mathbf v}=(x,y,z)\), where \((1,i,j,k)\) are the basis elements of the quaternion vector space.

For information how dual quaternions can be used for the description of rigid 3D transformations and their relation to Plücker coordinates, see “Solution Guide III-C - 3D Vision”.

Representing Dual Quaternions in HALCON

In HALCON, a dual quaternion is represented by a tuple with eight values \([w_{r}, x_{r}, y_{r}, z_{r}, w_{d}, x_{d}, y_{d}, z_{d}]\), where \(w_{r}\) and \({\mathbf v}_{r}= (x_{r},y_{r},z_{r})\) are the scalar and the vector part of the real part and \(w_{d}\) and \({\mathbf v}_{d}= (x_{d},y_{d},z_{d})\) are the scalar and the vector part of the dual part.