in the direction X through the point P be given by, Then the dual quaternion of the displacement of this body transforms Plücker coordinates in the moving frame to Plücker 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. Dual quaternions and 4×4 homogeneous transforms It might be helpful, especially in rigid body motion, to represent unit dual quaternions as homogeneous matrices. As given above a dual quaternion can be written as: where r and d are both quaternions. The r quaternion is known as the real or rotational part and the quaternion is known as the dual or displacement part. The rotation part can be given by where is the angle of rotation about the direction given by unit vector . The displacement part can be written as . The dual-quaternion equivalent of a 3D-vector is and its transformation by is given by . These dual quaternions (or actually their transformations on 3D-vectors) can be represented by the homogeneous transformation matrix where the 3×3 orthogonal matrix is given by For the 3D-vector the transformation by T is given by Connection to Clifford algebras Besides being the tensor product of two Clifford algebras, the quaternions and the dual numbers, the dual quaternions have two other formulations in terms of Clifford algebras. First, dual quaternions are isomorphic to the Clifford algebra generated by 3 anticommuting elements , , with and . If we define and , then the relations defining the dual quaternions are implied by these and vice versa. Second, the dual quaternions are isomorphic to the even part of the Clifford algebra generated by 4 anticommuting elements with For details, see Clifford algebras: dual quaternions. Eponyms Since both Eduard Study and William Kingdon Clifford used and wrote about dual quaternions, at times authors refer to dual quaternions as "Study biquaternions" or "Clifford biquaternions". The latter eponym has also been used to refer to split-biquaternions. Read the article by Joe Rooney linked below for view of a supporter of W.K. Clifford's claim. Since the claims of Clifford and Study are in contention, it is convenient to use the current designation dual quaternion to avoid conflict. See also Screw theory Rational motion Quaternions and spatial rotation Conversion between quaternions and Euler angles Olinde Rodrigues Dual-complex number References Notes Sources A.T. Yang (1963) Application