This is an introductory textbook on geometry (affine, Euclidean and projective) suitable for any undergraduate or first-year graduate course in mathematics and physics. In particular, several parts of the first ten chapters can be used in a course of linear algebra, affine and Euclidean geometry by students of some branches of engineering and computer science. Chapter 11 may be useful as an elementary introduction to algebraic geometry for advanced undergraduate and graduate students of mathematics. Chapters 12 and 13 may be a part of a course on non-Euclidean geometry for mathematics students. Chapter 13 may be of some interest for students of theoretical physics (Galilean and Einstein's general relativity). It provides full proofs and includes many examples and exercises. The covered topics include vector spaces and quadratic forms, affine and projective spaces over an arbitrary field; Euclidean spaces; some synthetic affine, Euclidean and projective geometry; affine and projective hyperquadrics with coefficients in an arbitrary field of characteristic different from 2; Bézout's theorem for curves of P^2 (K), where K is a fixed algebraically closed field of arbitrary characteristic; and Cayley-Klein geometries.

A comparative presentation of affine, euclidean and projective geometries An elementary but complete proof of the strong Bézout's theorem for curves of P2 (K) A detailed exposition of Cayley-Klein plane geometries

Autorentext
Lucian Silvestru B**descu (deceased): Graduation from University of Bucharest, Department of Mathematics, (1967), Ph. D in Mathematics, University of Bucharest (1971) with the thesis Rational contractions of algebraic varieties. Permanent position: Full professor at University of Bucharest, Department of Mathematics, until 2002, full profesor at Università degli Studi di Genova, Dipartimento di Matematica, until 2014. Field of interest: Algebric Geometry (singularities, hyperplan section and classification of projective varieties, deformation theory of singularities, arithmetic rang of projective submanifolds, algebraic surfaces, projective and formal geometry).
**Ettore Carletti obtained a degree in Mathematics from the University of Genoa in 1976; he earned his Diploma di Perfezionamento in 1983. He has been a researcher in Algebra and Geometry at the University of Genoa from 1985 until 2018 when he retired. His main research focus has been zeta functions in number theory and geometry.

Inhalt
1 Linear Algebra.- 2 Bilinear and quadratic forms.- 3 Affine Spaces.- 4 Euclidean Spaces.- 5 Affine hyperquadrics.- 6 Projective Spaces.- 7 Desargues' Axiom.- 8 General Linear Projective Automorphisms.- 9 Affine Geometry and Projective Geometry.- 10 Projective hyperquadrics.- 11 Bezout's Theorem for Curves of P^2(K).- 12 Absolute plane geometry.- 13 Cayley-Klein Geometries