Journal of Space Exploration, Propulsion and Aerospace Systems
The R3 Spin Group Rotation Vectors in Three-Dimensional Space
Abstract
James Buckeyne
Three-dimensional rotations have three degrees of freedom, yet the standard representations used in computation often hide that simplicity behind sequential parameterizations, redundant constraints, or less direct geometric coordinates. This monograph develops a Rotation Vector framework in which a rotation is represented by a vector in R3 whose direction gives the axis and whose magnitude gives the angle.
The framework is constructive throughout. A Rotation Vector may be applied directly to points through Rodrigues’ rotation formula, converted to a matrix when needed, or mapped to the unit quaternions through the exponential map. More importantly, rotations may be composed directly in Rotation Vector form, without treating quaternion or matrix conversion as the primary computational path. This yields a compact description of identity, inverse, relative rotation, sequential composition, and additive combination of simultaneous torques, while keeping the geometry visible.
The presentation also clarifies the relation of this framework to the classical theory of rotations. Rotation Vectors are used here as the natural pure-rotation logarithmic coordinates of Spin(3) and therefore sit naturally in the Lie-algebraic description of rigid rotation, while still remaining accessible as ordinary vectors. The result is not a rejection of matrices or quaternions, but a re-centering of rotation theory around a representation that matches the axis-angle content of the motion itself.