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The main asset of this delightful book is its methodical and unencumbered presentation of the most basic mathematics, vector and matrix operations from the first page. Specifically, it illustrates basic algebraic field theory and generalizes complex numbers into quaternions in an uncomplicated way. The fluid presentation encourages the reader to continue through the necessarily lengthy introduction of the classical rotation operators (as detailed use of quaternions doesn't start until about 100 pages, in Chapter 5).
I appreciated the fact its introductory nature is honestly clarified by the subtitle: it is a self-declared primer. It is also one of the few textbooks I have seen making extensive use of a marginal gloss (explanatory notes in the margin), which seems much more efficient than footnotes or appendices. Many facts are repeated - noticeable but not too annoying, and handled well in the gloss. This level of presentation will certainly benefit most readers new to the subject. Anyone writing a technically oriented textbook should consider reviewing this title for its format alone.
The book defines a quaternion as a 3-D vector plus a scalar. Defining the quaternion with these more conventional mathematical notions makes the very concept more approachable. But it is not clear whether this (and other) notation is truly unique to this book or otherwise widely acknowledged in literature. For example, most of the notation adopted for classic rotation operators seemed unnecessarily different (and therefore slightly confusing) compared to those few other engineering and science textbooks I've been able to reference on the subject. And a few terms, such as "kyperplane", appear unique to this book alone.
Considering that this is an introductory textbook, the recommended "further reading" list was by far the most disappointing aspect of this title. Out of sixteen (16) meager references provided, 1/4 are Prof. Kuipers' own patent declarations; the rest are mostly hard-to-get Air Force reports, out of print books, and a few specialty journal articles. The lack of specific references is especially bothersome when facts or theorems are cited without support or proof, such as "Euler's Theorem" (p. 83).
Engineers and engineering students should also be aware that some of the "applications to orbits and aerospace" (from the subtitle) appear to be more for academic or illustrative purposes than for immediate, practical application. For example, the publisher's on-line table of contents identifies "Chapter 11 - Quaternion Calculus for Kinematics and Dynamics." However, this chapter doesn't really cover the conventional transformations of relative velocity or accelerations with respect to rotating frames of reference, which is essential to the study of dynamics and kinematics of air and space vehicles. In the preface, the author acknowledges that "It was difficult knowing where to stop, since the subject deserves much more attention and greater depth." As a result, the book may have slightly more appeal to those interested in 3-D programming and visualization.
God bless the author, who at age 80 apparently supplied the textbook copy in camera ready form. Unfortunately, my 3rd printing still contains many obvious typographical errors, which is the publisher's responsibility (who holds the copyright). A lack of editorial review normally implies that less obvious errors are lurking in those all-important equations, but thankfully Prof. Kuipers is kind enough to provide an errata sheet if the reader requests it via email. However, the reader should be aware that his printed book is still be published uncorrected, and no official errata appears at the publisher's website at this time.
In summary, I would recommend this primer for the engineering student or programmer with a novice to intermediate level of familiarity with rotational sequences. The book's style of presentation is commendable, and the extensive gloss makes the subject matter more understandable to the beginner. Discussions of some engineering applications, as well as specific topics such as orbital mechanics, gravitational theory, etc., are presented with far less detail, clarity, and rigor. While disappointing, this is forgivable as the author seemingly intends to illustrate, rather than develop rigorously complete relationships, for these applications. However, the lack of modern, easily obtained references and some seemingly unique notation may give this title less longevity as a research or reference text.