On geodesics of the rotation group SO(3)

Abstract

The simplest model with which to examine the dynamics of the human eye consists of a rigid body which is free to rotate about a fixed point. Two classical laws governing monocular vision, which are known as Listing's law and Donders' law, can be enforced in this model using a single holonomic constraint. While there has been considerable attention paid to the kinematics of the eye, the dynamics of the eye predicted by rigid body models has not received the same level of attention. In the present paper, the unforced dynamics of the resulting rigid body model are examined with particular emphasis placed on the geodesics of the configuration manifold. A comprehensive portrait of these motions is presented, and the insight gained is related to the dynamics of the gaze direction and saccadic motions of the eye. Among our results, we find that modeling the eye as an asymmetric rigid body produces a nonintegrable system of governing equations and that the geodesics on the configuration manifold provide a wealth of potential motions for the gaze direction.

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part II: Non‐classical energy‐minimizing microrotations in 3D and their computational validation***

“…For a general introduction to analysis on smooth manifolds which includes smooth coverings, see, e.g., [47]. For a dynamical systems approach to quaternions, see, e.g., [72] which nicely demonstrates the usefulness of quaternions for mechanics applications with constraints. A more algebraic approach to quaternions with historical remarks is given in [25], and, finally, for those who enjoy the classics, see [37] and [38].…”

“…For some interesting recent applications to strain measures in mechanics, see, e.g., [64]. Another interesting recent usecase for geodesics on the group of unit quaternions is the simulation of eye movements, see [72].…”

Geodesic Motions in SO(2,1)

“…where the (X I , Y I ) → T race {X I .Y I } map is called the Killing form of SO(3) [9]. Since h has nondegenerate, symmetric, bilinear form, h will be a semi-Riemann metric on SO(2, 1).…”

“…Moreover, they described the kinetic energy of a rotating particle in terms of the unit quaternion. They showed that kinetic energy of the rotating particle is constant along the geodesics of the special orthogonal group [9].…”

“…Moreover, they described the kinetic energy of a rotating particle in terms of the unit quaternion. They showed that kinetic energy of the rotating particle is constant along the geodesics of the special orthogonal group [9]. Jaferi and Yaylı studied the generalized quaternions and they have indicated how unit generalized quaternions can be used to describe rotation in 3-dimensional space E 3 αβ [6].…”