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The Universe of Quadrics

Boris Odehnal, Hellmutch Stachel, and Georg Glaeser
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Tushar Das
, on
The Universe of Quadrics (UQ) is the beautifully written sequel to the authors’ 2016 The Universe of Conics (UC), which was also a pleasure to review. In the authors’ words: “The ulterior motive was to write two compendia containing important geometric knowledge that seems in danger of getting lost.”  Throughout UQ there is constant reference to UC and I would recommend readers interested in diving in these waters to have both texts close. 
The narrative moves from understanding planar loci
\( Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 \) 
in the first volume to studying more general spatial loci
\( Ax^{2} + By^{2} + Cz^{2}  +Dxy + Exz +F yz + Gx + Hy +Iz +J=0 \)
including ellipsoids, hyperboloids, paraboloids as well as singular surfaces (quadratic cones and cylinders) and degenerate cases (planes or pairs of planes), while always keeping an eye to connections and contrasts with the material exposed in UC. The study begins in the second chapter with an analytic geometry approach via the standard equations (as may be familiar to some students between precalculus and multivariable calculus) and soon moves to a linear algebra approach in the third chapter. Reading assignments from this material in an undergraduate linear algebra course would provide students with a chance at glimpsing several attractive images of the geometric knowledge that seeks to be preserved by the authors.
For the fourth chapter (and later) there is an expectation that the reader be familiar with some notions of planar projective geometry before setting out the rudiments first in three dimensions.  The necessary background is in UC, and the authors helpfully reference the relevant chapters.  The chapter closes with a brief section on projective models of non-Euclidean geometries, which would have been nice to explore further (to help avoid the “danger of getting lost”). The fifth chapter presents a classification of pencils of quadrics in projective 3-space while shying away from the classification in affine 3-space.  
At almost 80 pages, Chapter 6 focuses on cubic and quartic space curves (of the first kind) that arise from the intersection of two quadrics. The next chapter on confocal quadrics presents Ivory’s Theorem as well as an introductory discussion of its relationship to the rigidity/flexibility of (bar) frameworks in discrete geometry, and ends with a description of Staude’s “string constructions” of quadrics. The eighth chapter is a potpourri exploring anamorphic images, movable conics on quadrics, as well as rational parameterizations of quadrics. Chapter 9 is another long chapter - almost 100 pages - making connections to the classical differential geometry of curves and surfaces. I especially liked the last section on (mainly algebraic and rational) geodesics on quadrics, and felt that expanding this section would make for several connections to prior material as well as to other areas of general mathematical interest.
The tenth chapter on line geometry and sphere geometry exposes three important models (Plücker’s, Lie’s, and Study’s quadrics) while touching on the wide scope of applications in kinematics and allied areas. The last chapter is a smorgasbord of various generalizations of quadrics, including a fun section on superquadrics (I fondly recall learning about “squircles” from my students!) and another on multifocal surfaces in three-dimensional Euclidean space. The latter, which are the loci of points whose sum/product of distances to a fixed set of foci is constant, seem especially amenable to various explorations in topology/combinatorics/algebra with students via graphing software like Geogebra 3D Calculator.
Both UC and UQ were published in the Springer Spektrum series. The binding, paper, and printing of text and hi-res images are all to be commended. All lavish praise for the wonderfully produced figures in UC, may equally be applied to those in UQ.  My remarks about the figures that appeared in UC may be reapplied to UQ:
Sadly, the reader is left completely in the dark regarding the production of the beautiful and instructive figures that adorn almost every page. A separate resource on how to create such figures may be extremely well-received, and poised to revitalize classes on geometry, especially those aimed at training teachers at the elementary, middle and high school levels. Such a text/course may radically enhance the creation and use of manipulative materials and technology in the setting of geometry education. We hope that the authors will provide some documentation and further aids (perhaps on their websites) to empower readers to create and manipulate such figures on their own.
I hope the authors create a resource to help teachers and students in this regard. Such technical training in drawing (with the aid of software) may also be “in danger of getting lost”. Quibbles apart, I do recommend the UC and UQ pair highly. Talk to your library liaison and see if your school can get them for you and your students to dip into!

Tushar Das is a Professor of Mathematics at the University of Wisconsin–La Crosse. His website page is at