In this post, we will see the book Geometric Transformations Volume 2 Projective Transformations by P.S. Modenov; A.S. Parkhomenko.

About the book

This book is intended for use in geometry seminars at universities and teacher-training colleges. It may also serve as supplementary reading for high school teachers seeking to extend their knowledge. Additionally, many sections can be used as source material for school mathematics clubs under a teacher’s guidance.

The subject matter covers transformations of the plane that preserve fundamental geometric figures: straight lines and circles. Specifically, it discusses orthogonal, affine, projective, and similarity transformations, as well as inversions. The treatment is elementary, though coordinate methods are used where a synthetic approach is more cumbersome. A little vector algebra is also employed, but the text is self-contained.

This is the second volume of a two-volume translation of the Russian book *Geometric Transformations* by Modenov and Parkhomenko. It contains translations of Chapters V (with Appendix) and VI of the original, presented here as Chapters I and II, respectively.

The greater portion of this volume focuses on projective transformations, which are collinearity-preserving transformations of the projective plane. Many mappings of the ordinary plane that preserve collinearity are best understood as defined on an extended object called the projective plane. Notably, the affine transformations discussed in Chapter IV of the first volume (*Euclidean and Affine Transformations*) are conveniently regarded as projective transformations that fix the ideal line.

Chapter I begins with the motivation for constructing the projective plane, followed by various alternative constructions. It then proves most of the basic facts and outlines some applications. Chapter II addresses an independent topic at the same level of sophistication. However, the appendices to Chapter I introduce more advanced concepts without detailed motivation or treatment. Readers who are unprepared should not hesitate to read through them, absorbing what they can and taking the rest on faith. These sections provide glimpses into the methods and concerns of modern geometry and algebra.

Overall, this book is probably suitable for undergraduate readers. However, interest and ability are more important for benefiting from it than the quantity of previous knowledge.

You can get the book here.
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