The theory of turns

Properties (a)-(e) are just those necessary to ensure that T is a group, called by Biedenham and Louck (1981) Hamilton s group of turns.  

Let QP be a diameter of the unit sphere then, since great circles defined by Q and P are not unique, all turns TK defined by pairs of opposite points are equivalent. Since rTn can be chosen on any great circle, it commutes with any turn T. The operation of adding I, to a turn T is described as conjugation, 

Turns of length n/2 have some unique properties and are denoted by the special symbol E. 

Exercise 12.4-1 (a) Show that 1), T . (b) Show that E° = —E. (c) Prove that any turn T 

The above analysis shows that the set of turns To Tn E, E-, i = 1,2,3, provides a geometric realization of the quaternion group and thus establishes the connection between the quaternion units and turns through rc/2, and hence rotations through % (binary rotations). This suggests that the whole set of turns might provide a geometric realization of the set of unit quaternions. Section 12.5 will not only prove this to be the case, but will also provide us with the correct parameterization of a rotation. 

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