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Perspective and Projective Geometry

Annalisa Crannell, Marc Frantz, and Fumiko Futamura
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Fenton
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This is an unusual book, in several ways. The content is unusual, being a blend of art techniques and significant mathematics. The writing is unusual; the book is designed as a collection of interactive modules to lead students through the content. The physical layout is unusual in that the pages are perforated so that they can be torn out. I found the book interesting and challenging, and I enjoyed my experience working through it.
The content is as advertised: the techniques of linear perspective and the projective geometry that grew from and informs perspective drawing. One-, two-, and three-point perspective are presented. The notion of perspective collineation is developed, leading to the projective plane and Desargues’ Theorem. I liked the geometric definition of harmonic set based upon the theorems of Ceva and Menelaus. I especially liked the authors’ definition of cross-ratio. It was reminiscent of Ceva’s Theorem and made much more sense than other presentations I have seen. Later topics include Eves’ Theorem, Casey’s Theorem, and homogeneous coordinates. These are applied to certain challenges in perspective drawing. (I confess that I struggled with using Casey’s Theorem to properly represent angles in a drawing.) The final chapter ventures into the topology of the projective plane, though there is little about art in this chapter.
Each module consists of activities interspersed with exposition, then followed by conceptual exercises, art homework, and proof/counterexample questions. This design requires interactive study and the activities encourage experimentation and exploration. GeoGebra is assigned as a tool for exploration. I began using GeoGebra much earlier than the authors indicated, even for some of the drawing exercises. The art projects were challenging, at least for me, and some of the geometry questions were as well. I am still not sure about using side and top views to determine the observation point; this plagued me in many places. I found only one significant typographical error: on page 195, the authors refer to angle OAB when I believe they mean angle AOB.
The pages in the paperback edition are perforated to be easily torn out so that ruler-&-pencil work can be carried out on the diagrams. I was tempted to do so many times, though I was afraid of losing loose pages. For this reason, the e-book version would be difficult to use. I wish the publisher would create a loose-leaf version with punched holes.
This book would support a lively, interactive course. But what course would that be? There is enough trigonometry required to intimidate many (most?) art majors. A general education course is a possibility, though the mathematics content is substantial and I wonder if the typical college student would feel overwhelmed by it. A mathematics major elective course from this book, perhaps with some supplementary content, would be great fun for students and the instructor. The book could provide a good independent study course for the right student. If someone is curious about how linear perspective works and how mathematics relates to drawing, this book is an excellent way to explore those questions. 


Bill Fenton is professor of mathematics at Bellarmine University in Louisville, KY. He is co-author, with Sr. Barbara Reynolds, of College Geometry Using The Geometer’s Sketchpad.
The table of contents is not available.