Geometric Transformations: Volume 4, Circular Transformations Anneli Lax New Mathematical Library Series

The familiar plane geometry of secondary school - figures composed of lines and circles - takes on a new life when 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. This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III. The present Part IV 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. Applications include ruler-and-compass constructions and the Poincaré model of hyperbolic geometry. The straightforward, direct presentation assumes only some background in elementary geometry and trigonometry.

1. Reflection in a circle (inversion); Notes to Section 1; 2. Application of inversions to the solution of construction; Problems: constructions with compass alone; Problems involving the construction of circles; Notes to Section 2; 3. Pencils of circles. The radical axis of two circles; Notes to Section 3; 4. Inversion (concluding section); Notes to Section 4; 5. Axial circular transformations; A. Dilatation; B. Axial inversion; Notes to Section 5; Supplement I; Non-euclidean geometry of Lobachevskii-Bolyai, or hyperbolic geometry; Notes to Supplement I; Solutions; Section 1; Section 2; Section 3; Circular transformations; Section 4; Section 5; Supplement II; Notes to Supplement II.

I. M. Yaglom authored many books which have since become academic standards of reference. These include Complex Numbers in Geometry, Geometric Transformations, A Simple Non-Euclidean Geometry and its Physical Basis, and Probability and Information. He was Professor of Mathematics at Yaroslavl State University from 1974–83 and a technical consultant at the Academy of Pedagogical Sciences from 1984–88.