Rotors for Moving Frames in Projective Geometric Algebra

Abstract

Projective Geometric Algebra (PGA) is a relatively new mathematical model to describe rigid-body motions. As such, there has yet to be in-depth research on using PGA in conjunction with differential geometry. In particular, PGA's ability to describe rigid-body motions appears to be well-suited to describe the instantaneous motion of moving frames associated with parametric curves. In this thesis, we derive rotors that perform the instantaneous motion of a parametric curve for the translation frame, Bishop frame and Frenet-Serret frame in 3 dimensions. We demonstrate the use of these rotors by iteratively applying them to the corners of a square to construct a square tube fitted around a helix. We also generalize the construction of the rotors associated with moving frames to an arbitrary number of dimensions. We thus further develop the use of PGA in the field of differential geometry.