New MAA Book: Geometric Transformations IV

September 15, 2009

Geometric Transformations IV: Circular Transformations
I.M. Yaglom (translated by A. Shenitzer)
293 pp., paperback, 2009. Series: Anneli Lax New Mathematical Library #44
ISBN: 978-0-88385-648-2

This book, which concludes I.M. Yaglom's four-part Geometric Transformations series, can be studied independently of Parts I, II, and III. It develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed.

The plane geometry of secondary school—figures composed of lines and circles—takes on a new life when it is viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. Applications include ruler-and-compass constructions and the Poincaré model of hyperbolic geometry. The straightforward, direct presentation assumes background only in elementary geometry and trigonometry.

Highlights:

1. Numerous exercises lead the reader to a mastery of the methods and concepts
2. The second half of the book contains detailed solutions of all the problems
3. Accessible to undergraduates and assumes only familiarity with elementary geometry and trigonometry

Excerpt:

Contents:
1. Reflection in a circle (inversion). 2. Application of inversions to the solution of constructions. 3. Pencils of circles. The radical axis of two circles. 4. Inversion (concluding section). 5. Axial circular transformations. A. Dilatation. B. Axial inversion. Supplement: Non-Euclidean Geometry of Lobachevskii-Bolyai, or Hyperbolic Geometry. Solutions.

About the Author:
Isaac Moisevitch Yaglom (1921–1988) graduated from Sverdlovsk University in 1942 and received his Candidate's Degree in 1945. A born educator, he was professor of geometry at Moscow State Pedagogical Institute from 1957–1968. All the volumes of his Geometric Transformations are part of the MAA's Anneli Lax NML series.

About the Translator:
Abe Shenitzer, born in Warsaw in 1921, is professor emeritus of mathematics at York University, in Toronto. He attended Brooklyn College and received his Ph.D. in 1954 from New York University.

Order Geometric Transformations IV from the MAA Bookstore or call 1.800.331.1622.
List: $ 44.95
MAA Member Price: $ 34.95