23. A Geometrical Approach to Model Circular Rotations

Here we introduce an elegant way to geometrically model the rotation of a rigid body in vector form. Typically, to perform a rotation in Euclidian space ℜ³ one uses rotation matrices based on a given sequence of Euler angles. Another approach is to use quaternions. A matrix exponent is often used to describe rotations in mathematical expressions and derivations, i.e., the exponential map from so(3) to SO(3). However, the nine elements of the rotation matrix are still exclusively used for calculating rotations in Euclidian space. The axis/angle representation in terms of quaternions and Rodrigues’ rotation formula are alternative approaches. However, hidden geometrical properties, or the complexity of using quaternion algebra are the stumbling blocks that lead to the situation that rotation matrices are still almost exclusively used nowadays. Here we introduce the spherical orthodrome rotation that describes a rotation purely geometrically in a highly transparent way as an orthodrome, or a great arc on a sphere. The application of such transparent geometrical rotations in vector form has many advantages compared to any other rotation. Here we introduce spherical rotation and show basic geometrical properties, i.e., the use of vector algebra to very efficiently perform rotation of a vector in Euclidian space or to describe any orientation. Thus, this approach could be used to model Earth orientation and rotation as well as the attitude of a satellite. We also show that this geometrical rotation approach could be used in orbit modeling, since orbit perturbations can be represented by circular rotations with an axis of rotation very close to the main axis of the satellite orbit.