Some variations of dual Euler-Rodrigues formula with an application to point-line geometry

This paper examines the Euler-Rodrigues formula in dual 3-space D3 by analyzing its variations such as vectorial form, exponential map, point-line theory and quaternions which have some intrinsic relations. Contrary to the Euclidean case, dual rotation in dual 3-space corresponds to a screw motion in Euclidean 3-space. This paper begins by explaining dual motion in terms of the given dual axis and angle. It will then go on to express dual Euler-Rodrigues formula with algebraic methods. Furthermore, an application of dual Euler-Rodrigues formula to point-line geometry is accomplished and point-line displacement operator is obtained by dual Euler-Rodrigues formula. Finally, dual Euler-Rodrigues formula is presented with the help of dual Euler-Rodrigues parameters that is expressed as a dual quaternion.