| ||Preface to the Russian edition Preface to the English edition|
| ||Lecture 1||
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||
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||
Linear dependence and linear independence. Linearly independent sets. The simplest properties of linear dependence. Linear-dependence theorem|
| ||Lecture 4||
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||
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||
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||
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||
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||
Planes in space. Parametric equations of a plane. The general equation of a plane. A plane passing through three noncollinear points|
| ||Lecture 10||
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||
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||
Deformation of bases. Sameness of the sign bases. Equivalent bases and matrices. The coincidence of deformabil-ity 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||
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||
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 he transformation of rectangular coordinates|
| ||Lecture 15||
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||
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 hn space|
| ||Lecture 17||
The parabola. The ellipse. The focal and directorial properties of the ellipse. The hyperbola. The focal and directorial properties of the hyperbola|
| ||Lecture 18||
The equations of ellipses, parabolas and hyperbolas leferred 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||
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||
Introductory remarks. The centre of a second-degree curve. Centres of symmetry. Central and nonceniral curves of second degree. Straight lines of non-asymptotic direction. Tangents. Straight lines of asymptotic direction|
| ||Lecture 21||
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 rioncentral curve|
| ||Lecture 22||
Second-degree curves in the complex affine 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||
Ellipsoids. Imaginary ellipsoids. Second-degree imaginary cones. Hyperboloids of two sheets. Hyperboloids of one sheet. Rectilinear generators of a hyperboloid of one sheet.
Second-degree cones. Elliptical paraboloids. Hyporbolic paraboloids. Elliptical cylinders. Other second-degree surfaces. The statement of the classification theorem|
| ||Lecture 24||
Coordinates of a straight line. Pencils of straight lines. Ordinary and ideal pencils. Extended planes. Models of projective-affine geometry|
| ||Lecture 25||
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 afftne coordinates in the bundle of straight lines. Formulas for the transformation of homogeneous affine coordinates. Projective coordinates. Second-degree curves in the projec-tive plane|
| ||Lecture 26||
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||
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||
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||
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. Classification of linear fractional transformations. Stereographic projection formulas. Rotations of a sphere as linear fractional transformations of a plane. Isometries of a cube
This book is a faithful record of Lectures which thd author delivered in the first semester of a two-year course in geometry at the Mathematics-Mechanics Faculty of Moscow State University to students studying mathematics. The contents of these Lectures were determined by the curriculum, by the established traditions of the Faculty's Department of Higher Geometry and Topology, by the needs of the second semester's course and by the author's personal aims. The sequence of presentation was governed by the necessity of agreement with the concurrently delivered courses in algebra and analysis, by the requirements of the assistants conducting seminars and by other similar considerations of no fundamental consequence but of paramount practical importance. For example, the decision to consider some question or other at one of the last Lectures was dictated by the impracticability of consolidating the material of those Lectures with the aid of exercises. The contents of the very last Lecture were determined by the fact that owing to the postponement of some Lectures because of the intervening holidays it would often fall on the examination period and is sometimes not delivered at all for lack of time, etc. etc.
Only two features of the book seem to deserve special mention. The first is that from the outset the exposition is based on axioms and geometric visualization is made use of solely for propaedeutic purposes. For obvious reasons, from the many possible systems of axioms the "vector-point" one developed by H. Weyl has been chosen. This accounts for the unusually early introduction of the general concept of
ector space. Experience has shown that as a rule students learn this material without difficulty.
The other, more controversial, feature of the book is a systematic development and use of bivectors and trivectors.
This makes it possible to separate distinctly the affine part of the theory from its metric part and provides a background for a general theory of multivectors in the second semester.
Each "Lecture" in the book is really a two hours' discourse, as a rule. This explains why a previous topic often gives way to a new one in the middle of a Lecture. One exception is the last, 28th, Lecture which is a combination of two different versions of the concluding Lecture. Because of the specific character of oral and written forms of presentation, the "isochronous" Lectures have turned out to be of different lengths in the book. Their number is accounted for by the fact that although the curriculum assigns 36 Lectures to a course in analytic geometry, in practice it has to be ended as early as the 28th Lecture or earlier.
I also wish to express my gratitude to T-P. Aldatova for her prompt and excellent typing of the original manuscript.
After the Russian edition of this book appeared some of my fellow Lecturers asserted that many of the Lectures in the book are far too long to be physically delivered during the allowed two teaching periods. By the right of friendship I had to remind them that a Lecturer must prepare for his Lectures—even if he has been lecturing for over a dozen years— and make in advance an elaborate, practically minute-by-minute plan of every Lecture. It is necessary to consider eforehand the rhythm of the Lecture to be delivered—what portions of it are to be read slowly, almost at dictation speed, and what may be said quicker—and its pattern of intonation—where to raise the voice and where to lower it. One also needs a joke somewhere about the middle of the Lecture to rouse the tired students and it should be prepared yet at home, and in every detail, up to a play of facial muscles. It goes without saying that one must plan in advance what to write on the blackboard and in what order and where, and when to delete anything, and coordinate all this with everything else. It is surprising how all this extends the limits of Lecture time and how much it is then possible to say in an outwardly unhurried and thorough manner, with numerous repetitions and explanations.
Some reviewers have reproached me for a systematic use of bivectors and trivectors saying that one may well do without them. Some well-known physicist, Max Planck, I think, once said that new ideas (he meant scientific ideas but this can be fully applied to methodical ideas as well), could win only when their opponents have retired from the stage as a result of a natural change of generations. An ex-
cellent example illustrating this thesis is the introduction of vectors into the courses of analytic geometry half a century ago. Now only a few people remember the fierce discussions concerning this matter and the present generation does not know how many a lance was broken and how much ink split in attempts to prove that vectors were a harmful thing because replacing three equations in coordinates by one vector equation they saved paper but proportionally hampered comprehension. The last of the authoritative opponents of vectors in the USSR died soon after the war but some ten years more passed before diffidently excusatory reservations disappeared altogether from vectorial presentations of geometry (as well as from mechanics and physics where, however, this happened a little earlier). Now bivectors and tri-vectors are awaiting their turn.
I have taken the opportunity to introduce some minor improvements in the text. The most serious one is perhaps a simpler construction of the complexification of an affine space in Lecture 19. It is true that it contains a certain element of arbitrariness (which was what restrained me at first) but experience has shown that this arbitrariness is perfectly harmless. Besides, the last Lecture has been divided into two since two versions of the concluding Lecture were combined in it for purely technical, internal editorial reasons. So the book contains 29 Lectures now.
As far as I can judge with my poor knowledge of English the translation is well done and conveys all the nuances of my thought.
M.M. Postnikov May 1, 1980