Foundations of Projective Geometry by Robin Hartshorne (PDF)

Description

This text is designed for a one-semester undergraduate course in projective geometry. In incorporates a synthetic approach starting with axioms from which the general theory is deduced, together with an analytic approach using the real projective plane as a model. The first is refined as the second is generalized until the two coincide via the introduction of coordinates in an abstract projective plane. Special attention is paid to the role of Desargues’ and Pappus’ axioms in the theory. At the end of the book is a list of problems that can be used as exercises while reading. The emphasis on the various groups of transformations that arise in projective geometry introduces the reader to group theory in a practical context. While the book does not assume any previous knowledge of abstract algebra, some familiarity with group theory would be useful. First published in 1967 and long out of print, this book is now reissued with a new preface, an appendix on the simple group of order 168, which appears as the group of automorphisms of a projective plane of seven points, and a list of errata.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Great book from Hartshorne for introduction to projective geometry.

⭐As the preface says, this book approaches the subject from two different directions: analytic and synthetic. The synthetic approach seems to contain both (1) the parts of the subject best suited to students who know only high-school math, and (2) the parts ill-suited to those who are not “pure” mathematicians. An example of the former is a high-school-geometry-style proof that any three points on any line may be transformed to any three points on any other line via a composition of two perspectivities. An example of the latter is the projective plane that is freely generated by a configuration. The inclusion of things like this latter example is the reason why the word “foundations” is in the title; “foundations” does not mean that which should be learned before everything else; perhaps “foundations” should be defined as “that which will annoy those who want to apply projective geometry to computer graphics and which will delight `pure’ mathematicians who want to know how the subject interacts with abstract algebra.” The latter group will like this book and will find it very readable.This book is short and omits some topics found in many books. Conics and polarities are not dealt with.

⭐As the preface says, this book approaches the subject from two different directions: analytic and synthetic. The synthetic approach seems to contain both (1) the parts of the subject best suited to students who know only high-school math, and (2) the parts ill-suited to those who are not “pure” mathematicians. An example of the former is a high-school-geometry-style proof that any three points on any line may be transformed to any three points on any other line via a composition of two perspectivities. An example of the latter is the projective plane that is freely generated by a configuration. The inclusion of things like this latter example is the reason why the word “foundations” is in the title; “foundations” does not mean that which should be learned before everything else; perhaps “foundations” should be defined as “that which will annoy those who want to apply projective geometry to computer graphics and which will delight `pure’ mathematicians who want to know how the subject interacts with abstract algebra.” The latter group will like this book and will find it very readable.This book is short and omits some topics found in many books. Conics and polarities are not dealt with.

⭐代数的な公理からスタートしているため、すっきりしていて、爽快な感じがする。５０年も前に、バーコフ・マクレーンを初めて読んだ時の感動に近いものを感じた。ベブレン・ヤングなどより垢抜けしていると思う。

⭐

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