Dual Quaternion - Matrix Form of Dual Quaternion Multiplication

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 C=c0+C into the array C=(C1, C2, C3, c0). 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

AC = C = \begin{bmatrix} a_0 & -A_3 & A_2 & A_1 \\ A_3 & a_0 & -A_1 & A_2 \\ -A_2 & A_1 & a_0 & A_3 \\ -A_1 & -A_2 & -A_3 & a_0 \end{bmatrix} \begin{Bmatrix} C_1 \\ C_2 \\ C_3 \\ c_0 \end{Bmatrix}.

The product AC can also be viewed as an operation by C on the components of A, in which case we have

AC = A = \begin{bmatrix} c_0 & C_3 & -C_2 & C_1 \\ -C_3 & c_0 & C_1 & C_2 \\ C_2 & -C_1 & c_0 & C_3 \\ -C_1 & -C_2 & -C_3 & c_0 \end{bmatrix} \begin{Bmatrix} A_1 \\ A_2 \\ A_3 \\ a_0 \end{Bmatrix}.

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

\hat{A}\hat{C} = \hat{C} = \begin{bmatrix} A^+ & 0 \\ B^+ & A^+ \end{bmatrix}\begin{Bmatrix} C \\ D\end{Bmatrix}.

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,

\hat{A}\hat{C} = \hat{A} = \begin{bmatrix} C^- & 0 \\ D^- & C^- \end{bmatrix}\begin{Bmatrix} A \\ B\end{Bmatrix}.

Read more about this topic:  Dual Quaternion

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