Dual Quaternion - More On Spatial Displacements

More On Spatial Displacements

The dual quaternion of a displacement D=(, d) can be constructed from the quaternion S=cos(φ/2) + sin(φ/2)S that defines the rotation and the vector quaternion constructed from the translation vector d, given by D = d₁i + d₂j + d₃k. Using this notation, the dual quaternion for the displacement D=(, d) is given by

Let the Plucker coordinates of a line in the direction x through a point p in a moving body and its coordinates in the fixed frame which is in the direction X through the point P be given by,

Then the dual quaternion of the displacement of this body transforms Plucker coordinates in the moving frame to Plucker coordinates in the fixed frame by the formula

Using the matrix form of the dual quaternion product this becomes,

This calculation is easily managed using matrix operations.