## Lectures in Geometry – Semester 1 Analytic Geometry – Postnikov

In this post, we will see the book Lectures in Geometry – Semester 1 Analytic Geometry by M. Postnikov. This book is the first one of a five part Lectures in Geometry series. So far we have volumes 1, 2 and 5, volumes 3 and 4 are missing.

This textbook comprises lectures read by the author to the first-year students of mathematics at Moscow State University. The book is divided into two parts containing the texts of lectures read in the first and second semesters, respectively. Part One contains 29 lectures and read in the first semester.
The subject matter is presented on the basis of vector axiomatics
of geometry with special emphasis on logical sequence in introduction of the basic geometrical concepts. Systematic exposition and application of bivectors and trivectors enables the author to successfully combine the above course of lectures with the lectures of the following semesters. The book is intended for university undergraduates majoring in mathematics.

The book was translated from Russian by by Vladimir Shokurov and was published in 1982 by Mir.

You can get the book here.

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# Contents

Preface to the Russian edition 11
Preface to the English edition 13

## Lecture 1 15

The subject-matter of analytic geometry. Vectors. Vector addition. Multiplication of a vector by a number. Vector spaces. Examples. Vector spaces over an arbitrary field

## Lecture 2 23

The simplest consequences of the vector space axioms. Independence of the sum of any number of vectors on brackets arrangement. The concept of a family

## Lecture 3 28

Linear dependence and linear independence. Linearly independent sets. The simplest properties of linear dependence. Linear-dependence theorem

## Lecture 4 35

Collinear vectors. Coplanar vectors. The geometrical meaning of collinearity and coplanarity. Complete families of vectors, bases, dimensionality. Dimensionality axiom. Basis criterion. Coordinates of a vector. Coordinates of the sum of vectors and those of the product of a vector by a number

## Lecture 5 43

Isomorphisms of vector spaces. Coordinate isomorphisms. The isomorphism of vector spaces of the same dimension. The method of coordinates. Affine spaces. The isomorphism of affine spaces of the same dimension. Affine coordinates, Straight lines in affine space. Segments –

## Lecture 6 54

Parametric equations of a straight line. The equation of a straight line in a plane. The canonical equation of a straight line in a plane. The general equation of a straight line in a plane. Parallel lines. Relative position of two straight lines in a plane. Uniqueness theorem. Position of a straight line relative to coordinate axes. The half-planes into which a straight line divides a plane

## Lecture 7 63

An intuitive notion of a bivector. A formal definition of the bivector. The coincidence of the two definitions. A zero bivector. Conditions for the equality of bivectors. Parallelism of the vector and the bivector. The role of the three-dimensionality condition. Addition of bivectors

## Lecture 8 71

The correctness of the definition of a bivector sum. The product of a bivector by a number. Algebraic properties of external product. The vector space of bivectors. Bivectors in a plane and the theory of areas. Bivectors in space

## Lecture 9 82

Planes in space. Parametric equations of a plane. The general equation of a plane. A plane passing through three noncollinear points

## Lecture 10 87

The half-spaces into which a plane divides space. Relative positions of two planes in space. Straight lines in space. A plane containing a given straight line and passing through a given point. Relative positions of a straight line and a plane in space. Relative positions of two straight lines in space. Change from one basis for a vector space to another

## Lecture 11 99

Formulas for the transformation of vector coordinates. Formulas for the transformation of the affine coordinates of points. Orientation. Induced orientation of a straight line. Orientation of a straight line given by an equation. Orientation of a plane in space

## Lecture 12 112

Deformation of bases. Sameness of the sign bases. Equivalent bases and matrices. The coincidence of deformability with the sameness of sign. Equivalence of linearly independent systems of vectors. Trivectors. The product of a trivector by a number. The external product of three vectors

## Lecture 13 123

Trivectors in three-dimensional vector space. Addition of trivectors. The formula for the volume of a parallelepiped. Scalar product. Axioms of scalar multiplication. Euclidean spaces. The length of a vector and the angle between vectors. The Cauchy-Buniakowski inequality. The triangle inequality. Theorem on the diagonals of a parallelogram. Orthogonal vectors and the Pythagorean theorem

## Lecture 14 133

Metric form and metric coefficients. The condition of positive definiteness. Formulas for the transformation of metric coefficients when changing a basis. Orthonormal families of vectors and Fourier coefficients. Orthonormal bases and rectangular coordinates. Decomposition of positive definite matrices. The Gram-Schmidt orthogonalization process. Isomorphism of Euclidean spaces. Orthogonal matrices. Second-order orthogonal matrices. Formulas for the transformation of rectangular coordinates

## Lecture 15 148

Trivectors in oriented Euclidean space. Triple product of three vectors. The area of a bivector in Euclidean space. A vector complementary to a bivector in oriented Euclidean space. Vector multiplication. Isomorphism of spaces of vectors and bivectors. Expressing a vector product in terms of coordinates. The normal equation of a straight line in the Euclidean plane and the distance between a point and a straight line. Angles between two straight lines in the Euclidean plane

## Lecture 16 160

The plane in Euclidean space. The distance from a point to a plane. The angle between two planes, between a straight line and a plane, between two straight lines. The distance from a point to a straight line in space. The distance between two straight lines in space. The equations of the common perpendicular of two skew lines in space ‘

## Lecture 17 166

The parabola. The ellipse. The focal and directorial properties of the ellipse. The hyperbola. The focal and directorial properties of the hyperbola

## Lecture 18 176

The equations of ellipses, parabolas and hyperbolas referred to a vertex. Polar coordinates. The equations of ellipses, parabolas and hyperbolas in polar coordinates. Affine ellipses, parabolas, hyperbolas. Algebraic curves. Second-degree curves and associated difficulties. Complex affine geometry and its insufficiency

## Lecture 19 187

Real-complex vector spaces. Their dimensionality. Isomorphism of real-complex vector spaces. Complexification. Real-complex affine spaces. The complexification of affine spaces. Real-complex Euclidean spaces. Real and imaginary curves of second degree.

## Lecture 20 193

Introductory remarks. The centre of a second-degree curve. Centres of symmetry. Central and noncentral curves of second degree. Straight lines of non-asymptotic direction. Tangents. Straight lines of asymptotic direction

## Lecture 21 203

Singular and nonsingular directions. Diameters. Diameters and centres. Conjugate directions and conjugate diameters. Simplification of the equation of the second-degree central curve. Necessary refinements. Simplification of the equation of the second-degree noncentral curve

## Lecture 22 215

Second-degree curves in the complex aífine plane. Second-degree curves in the real-complex affine plane. The uniqueness of the equation of a second-degree curve. Second-degree curves in the Euclidean plane. Circles

## Lecture 23 228

Ellipsoids. Imaginary ellipsoids. Second-de imaginary cones. Hyperboloids of two sheets. Hyperboloids of one sheet. Rectilinear generators of a hyperboloid of one sheet. Second-degree cones. Elliptical paraboloids. Hyperbolic paraboloids. Elliptical cylinders. Other second-degree surfaces. The statement of the classification theorem

## Lecture 24 254

Coordinates of a straight line. Pencils of straight lines. Ordinary and ideal pencils. Extended planes. Models of projective-affine geometry

## Lecture 25 264

Homogeneous affine coordinates. Equations of straight lines in homogeneous coordinates. Second-degree curves in the projective-affine plane. Circles in the projective-Euclidean real-complex plane. Projective planes. Homogeneous affine coordinates in the bundle of straight lines. Formulas for the transformation of homogeneous affine coordinates. Projective coordinates. Second-degree curves in the projective plane

## Lecture 26 274

Coordinate isomorphisms of vector spaces. Coordinate isomorphisms of affine spaces. Projective-affine spaces. Projective spaces. Pencils of planes. Bundles of planes. Extending space with ideal elements. Orthogonal, affine and projective transformations

## Lecture 27 288

Expressing an affine transformation in terms of coordinates. Examples of affine transformations. Factorization of affine transformations. Orthogonal transformations. Motions of a plane. Symmetries and glide symmetries. A motion of a plane as a composition of two symmetries. Rotations of a space

## Lecture 28 302

The Desargues theorem. The Pappus-Pascal theorem. The Fano theorem. The duality principle. Models of the projective plane. Models of the projective straight line and of the projective space. The complex projective straight line

## Lecture 29 315

Linear fractional transformations. Linear transformations. Inversion. Inversions and linear fractional transformations. Two properties of linear fractional transformations. Fixed points of linear fractional transformations. Parabolic, elliptical, hyperbolic and loxodromic linear fractional transformations. The three-point theorem. The multiplier of linear fractional nonparabolic transformation. Classiffcation of linear fractional transformations. Stereographic projection formulas. Rotations of a sphere as linear fractional transformations of a plane. Isometries of a cube

Subject index 341