Theory

Quaternion Analysis

The essays in this section deal with the basic ideas that underlie the applied studies. They range from a fairly simple introduction to a number of papers that deal with basic, but sophisticated, concepts related to rotations in three-dimensional space.

	Anatomical Movement and Quaternions [ Word \| PDF ] In this essay, the basic concepts of quaternion analysis are developed and related to the description of anatomical movement. This probably a good place to start learning about the nature of the approach that is used in all the other essays on this site.
	Introduction to Quaternions [ Word \| PDF ] This essay is intended to introduce the concepts of quaternions and framed vectors and the subsidiary concept of frames of reference. The presentation has minimal equations and those presented are used principally to illustrate how one might apply the concepts to describe an anatomical structure and deduce quantitative consequences of movement in such structures.
	Movements of Orientable Objects [ Word \| PDF ] This essay builds up the basis for quaternion as an extension of vectors and reviews their basic attributes. It follows the line of reasoning in Joly's A Manual of Quaternions. It is probably the most through presentation of the logic of quaternions.
	Introduction to Quaternion Analysis [ Word \| PDF ] The approach in this essay is more mathematically sophisticated than that in the first essay, but the material presented is much the same.
	Commentary on Joly's A Manual of Quaternions [ Word \| PDF ] This is a commentary to follow the text of Joly's A Manual of Quaternions, which is probably the most accessible text on quaternions from the original studies of quaternions, contemporaneous with Hamilton. An effort has been made to render the concepts into a more modern interpretation and symbolism. A PDF copy of Joly's A Manual of Quaternions may be accessed by clicking on the highlighted words.
	Structural Ratios [ Word \| PDF ] This is an exploration of the nature of mathematical ratios, progressing from ratios of integers to ratios of vectors and ratios of oriented planes. It forms the background for several essays that involve applications of quaternion analysis to anatomical structures.
	Compound Movements [ Word \| PDF ] This essay considers the manner in which rotations and translations combine
	Catastrophes: Abutment and Tethering [ Word \| PDF ] When two tethered surfaces intersect, the point of contact becomes the new center of rotation and the shift of forces is extremely abrupt. This essay considers this situation and suggests that it may be the basis of back injuries in many instances.
	On Ligaments [ Word \| PDF ] This is a consideration of the forms that ligaments take and the implications for the types of tethering and abutment one may expect from each type of ligament.
	Quaternion Shear [ Word \| PDF ] This is a consideration of the nature of shear and how one might begin to describe shear in the context of quaternion analysis. This is an active research area and the material considered here is apt to be superceded by better analysis in the near future.
	Quaternion Aphorisms [ Word \| PDF ] These are brief statements that encapsulate the special attributes of quaternions and the mathematical objects that are associated with them. See the essay on structural ratios for a development of these ideas.

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