The purpose of this book is to examine both the properties and applications of quaternions, and, in particular, to explore visual representations that help to develop our intuition about quaternions and their exploitation. Of all the natural advantages of quaternions, several stand out clearly above all others:
Normalized quaternions are simply Euclidean four-vectors of length one, and thus are points on a unit hypersphere (known to mathematicians as the three-sphere) embedded in four dimensions. Just as the ordinary unit sphere has two degrees of freedom, e.g., latitude and longitude, the unit hypersphere has three degrees of freedom.
There is a relationship between quaternions and three-dimensional rotations that permits the three rotational degrees of freedom to be represented exactly by the three degrees of freedom of a normalized quaternion.
Because quaternions relate three-dimensional coordinate frames to points on a unit hypersphere, it turns out that quaternions provide a meaningful and reliable global framework that we can use to measure the distance or similarity between two different three-dimensional coordinate frames.
Finally, again because they are points on a hypersphere, quaternions can be used to define optimal methods for smooth interpolation among sampled sets of three-dimensional coordinate frames.
These features in fact characterize essential properties—an appealing geometric interpretation, the existence of meaningful similarity measures, and interpolatability of sampled data—that are advantageous in general for mathematical representations that are used to model physical actions and states. Part I of the book will focus on understanding these basic properties and how they can be useful to us.
For the reader who is looking for more novel ways to exploit the properties of quaternions, we turn in Part II of the book to a number applications and insights that go well beyond conventional quaternion methods. Among the more sophisticated applications of quaternions presented in the text, I emphasize particularly the exploitation of quaternion manifolds, which have appeared in scattered technical articles, but have no equivalent systematic treatment elsewhere. There are three basic types of quaternion manifolds whose properties can be briefly summarized as follows:
Quaternion Curves. Quaternion curves are familiar in the computer graphics literature as the means to implement smooth interpolations of three-dimensional coordinate frames; however, there are many other applications that lead to natural curves in quaternion space representing continuous sequences of orientation frames. In a typical application from elementary differential geometry, moving frames are attached to a curve, constrained so that a particular frame axis follows the curve’s tangent. This leads to classical constructs such as the Frenet–Serret frame; however, there are other equally important, but less familiar, frames that complement the Frenet–Serret equations, and can be used when the latter fail. By studying the family of all possible frames using the quaternion representation, we are led to an elegant system of quaternion differential equations describing the evolution of curve framings in general; these equations, known since the 19th century, correspond in essence to the square root of the conventional equations. The dynamic quaternion frame equations are now commonly used to replace the more traditional Euler-angle gyroscope dynamics equations in modern approaches to physically-based modeling for computer graphics. The entire family of concepts is amenable to depiction as curves embedded in quaternion space, with a variety of possible visualizations, applications, and intuitive interpretations.
Quaternion Surfaces. Computer graphicists and others with some mathematical background may be familiar with the Gauss map, which is essentially a mesh on a unit sphere, each point of which is the normal direction of the corresponding vertex on a meshed surface; the Gauss map contains deep hints of the fundamental nature of Riemannian geometry. In graphical models, surfaces are almost never adequately described simply by their vertex positions and normals; typically, in order to apply a texture, for example, a complete frame is required. I have proposed the quaternion Gauss map [70] as a natural and fundamental approach to the surface frame problem. The properties of surface frame fields have no consistent representation in terms of Euler angles, but can be studied as a surface with meaningful metric properties using the quaternion Gauss map. Many fundamental properties of differential geometry, e.g., the absence of a global set of coordinate frames on an ordinary unit sphere, are inescapably exposed by the quaternion Gauss map and the associated quaternion surfaces. For those with mathematical inclinations, we present a novel quaternion extension of the Weingarten equations for the classical differential geometry of surfaces.
Quaternion Volumes. Since the unit quaternions that are the central focus of this book are themselves three-dimensional, and the entire quaternion space is a hypersphere, the only remaining spaces that can usefully be constructed from quaternions consist of bounded volumetric objects. Quaternion volumes have found a new and essential application in the construction of orientation domains; in particular, a human or robotic joint with three full degrees of freedom can have any state represented as a quaternion point. The collection of all such states is a volume, normally continuous, and the permissible space of states is delineated by the boundary of that volume, thereby giving an elegant means of clamping errorful orientation measurements or commands to the permitted domain. Quaternion volumes incorporate a natural method for determining consistent distances and optimal paths from an invalid state to the allowed orientation domain.
Finally, in Part III of the book, we briefly look at the issues involved in attempting to generalize the properties of quaternions to dimensions other than three.
Intended Audience: I assume initially that the reader has a passing familiarity with complex variables, linear algebra, and analytic geometry, and has tried to transform a vertex with a matrix a few times. Certain topics, such as complex variables, will be reviewed thoroughly since they will serve as a framework for producing analogies that will be used to study quaternions and their properties. As the text moves on to more complex concepts, and particularly in the later chapters, the mathematics becomes more challenging, though I have made substantial efforts to add intuitive remarks and self-explanatory notes so that most readers should still be able to enjoy and assimilate essential features of the material.
My intent is that the earlier chapters should be informative to almost anyone with the noted minimal background, and that these readers need not trouble themselves with the more technical chapters. However, those who can potentially benefit from some of the more advanced mathematical concepts that arise in the study of quaternions should find ample material to keep them occupied in the later chapters.
The book keeps several different levels of readers in mind, with signposts indicating when a more advanced background may be needed to benefit fully from a particular section. I therefore hope that readers with any number of diverse backgrounds and interests, plus enough intellectual curiosity to follow a given train of thought through to its inevitable conclusions, can absorb the arguments in the sections that are relevant to their interests, find them as fascinating as I do, and perhaps generate some of their own original insights by asking “why” one time too many!
The basic material presented here covers first the needs of those wishing to deepen their intuitive understanding of the relationship between quaternions and rotations, quaternion-based animation, and moving coordinate frames. The next level of material deals with concepts of 3D curves and surfaces appearing extensively in computer graphics and scientific visualization applications. Finally, there is a selection of topics addressing the needs of those familiar with advanced problems and research, including an attempt to address some of the deeper theoretical underpinnings of quaternions, such as the question of what aspects of quaternions are and are not generalizable to other numbers systems and other dimensions. While we do not focus extensively on practical implementations or attempt to provide a “cookbook” for the graphics game developer, we do supply selected software examples to illustrate various points, and summarize the core body of code in an appendix.
Illustrations: The illustrations form an essential part of the text, and my goal has been to provide whatever visual cues are at my disposal to meet the objective of allowing the reader to truly “visualize quaternions.” There are three main styles of graphics that complement the text: the first consists of simple line drawings, many formatted originally in the xfig package, and largely redrawn by the publisher’s artists to correspond to my original drawings and sketches. The major portion of the mathematically precise illustrations were obtained directly from their equations using Mathematica™; high-resolution 3D shaded models were produced by the MeshView package [84,85], my own quaternion-friendly 3D-plus-4D visualization tool that accepts a wide variety of elementary modeling data. The actual model files used by MeshView were themselves typically computed and created using Mathematica.
Outline: The history of quaternions has been recounted many times, and in many ways. Our introduction will pause only momentarily to dwell on the historical aspects and development of the field, referring the reader to a number of sources more concerned with this material than we are. The book is divided into several major thematic units. First we look into various ways in which rotations, and hence quaternions, enter into everyday experience, and then begin to study 2D and 3D rotations. We attempt to draw fruitful analogies between ordinary complex variables and quaternions, exploiting the surprising richness of two-dimensional rotations and following an inevitable path to the relationship of quaternions to three-dimensional orientation spaces. Next, we examine the quaternion description of moving frames of coordinate-axis triples on curves and surfaces, revealing many essential insights that are almost completely neglected in the standard literature. By introducing the quaternion Gauss map, we are able to do for frames what the Gauss map historically accomplished for surface normals, and then we explore the idea of characterizing properties of curves and surfaces by analyzing the quaternion fields of their moving frames. A family of vexing problems in modeling and geometry is solved by defining a method that constructs optimal moving frames. Finding optimal continuous relations among frames leads us to the historically-important study of quaternion applications to orientation splines and energy-constrained choices of smooth quaternion orientation paths. The application of quaternion volumes to delimit the orientation spaces of biological or mechanical joints completes the family of dimensions that can be examined as quaternion manifolds. A selection of particular techniques that helps exploit all these quaternion manifold applications includes overviews of spherical modeling primitives and spherical geometry. The final chapters of our treatment outline the larger mathematical framework of division algebras and Clifford Algebras, in which quaternions play an essential part, and thus shed light on the ways in which quaternion methods may (and may not) be extended to higher dimensions.
Musings. The presentation style of this book is a somewhat personal and idiosynchratic one. It has evolved over many years of writing about mathematics, and takes a form that makes it easy for me personally to comprehend and retain essential facts. It may not satisfy everyone. In particular, the mathematics is quite pragmatic, completely neglecting theorems and proofs; but for this I offer no apologies. My choices of notation and mathematical exposition are clearly rooted in my background as a classically trained theoretical physicist, not a mathematician or an engineer. However, the desire to make mathematical things visible has deep psychological roots, possibly artistic in nature, and certainly owes a debt to a year of painfully inadequate accomplishment but inspirational hands-on study of 3D sculpture squeezed into my education as an undergraduate science major. For me, visual representations have always formed a more essential part of any intuitive understanding of science than mathematical formulas. One must of course be wary that graphical representations cannot often substitute for a facility with the mathematics itself. Nevertheless, visual representations can certainly make the mathematics simpler to recall and reconstruct, aid in the recognition of essential features, and enhance the likelihood that correct intuitive insights can be generated and applied.
Notations indicating advanced or supplementary material. This book contains a wide variety of information. We have attempted to separate the basic material into the first set of Chapters in Part I, and reserve the later Chapters in Part II and Part III for more advanced topics and applications. The main text in Part I is intended to follow a fairly sequential train of thought, so that an “average” reader with strong computer graphics background should be able to follow the main sections. However, some material of an advanced nature is included in various places in Part I for completeness, and is marked with a dagger (†) to indicate that this material requires additional background knowledge and can be skipped if desired. In Parts II and III, the † symbol is generally omitted, since nearly every section requires some specialized knowledge or background. Extremely important points, relating concepts and ideas that convey major themes of the book, are singled out by placing the text inside a 
. Finally, examples of elementary computer programs are given at critical points throughout the text, with a summary in an appendix.
URL for demonstration software. The software fragments listed in the tables can be found at http://www.visualizingquaternions.com, along with several demonstration systems that support the interactive visualization of quaternions and supplement many of the concepts presented in the text.