in order to obtain Divide both sides of this equation by the identity to obtain This is Rodrigues' formula for the screw axis of a composite displacement defined in terms of the screw axes of the two displacements. He derived this formula in 1840. The three screw axes A, B, and C form a spatial triangle and the dual angles at these vertices between the common normals that form the sides of this triangle are directly related to the dual angles of the three spatial displacements. Matrix form of dual quaternion multiplication The matrix representation of the quaternion product is convenient for programming quaternion computations using matrix algebra, which is true for dual quaternion operations as well. The quaternion product AC is a linear transformation by the operator A of the components of the quaternion C, therefore there is a matrix representation of A operating on the vector formed from the components of C. Assemble the components of the quaternion into the array . Notice that the components of the vector part of the quaternion are listed first and the scalar is listed last. This is an arbitrary choice, but once this convention is selected we must abide by it. The quaternion product AC can now be represented as the matrix product The product AC can also be viewed as an operation by C on the components of A, in which case we have The dual quaternion product ÂĈ = (A, B)(C, D) = (AC, AD+BC) can be formulated as a matrix operation as follows. Assemble the components of Ĉ into the eight dimensional array Ĉ = (C1, C2, C3, c0, D1, D2, D3, d0), then ÂĈ is given by the 8x8 matrix product As we saw for quaternions, the product ÂĈ can be viewed as the operation of Ĉ on the coordinate vector Â, which means ÂĈ can also be formulated as, More on spatial displacements The dual quaternion of a displacement D=([A], d) can be constructed from the quaternion S=cos(φ/2) + sin(φ/2)S that defines the rotation [A] 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=([A], d) is given by Let the Plücker 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