# Projective geometry

The branch of geometry in which one studies properties of figures that do not change under projective transformations (cf. Projective transformation), e.g. under projection. Such properties are said to be projective; the property of points to be on one line (collinearity), the order of algebraic curves, etc., are such properties.

Under projection of points of one plane $\Pi_0$ onto another $\Pi_1$, not every point of $\Pi_1$ need have a pre-image in $\Pi_0$ and not every point of $\Pi_0$ need have an image in $\Pi_1$. This leads to the necessity of completing affine space by so-called infinitely-distant elements (improper points, lines or planes) and to the formation of a new geometric object — three-dimensional projective space. Here, each line is completed by one improper point, each plane by one improper line and the whole space by one improper plane. Parallel lines are completed by one and the same improper point, non-parallel lines by distinct improper points, parallel planes by one improper line, and non-parallel planes by distinct lines. The improper points by which a plane is completed belong to the improper line completing this plane. All improper points and lines belong to the improper plane. By completing affine space to projective space, projection becomes one-to-one. A similar procedure applies to an $n$-dimensional space.

There are various ways of axiomatically constructing projective space. A modification of the axiom system proposed in 1899 by D. Hilbert for the foundation of elementary geometry (cf. Hilbert system of axioms) is often used (cf. [10]). Projective space is regarded as a set of elements of three kinds: points, lines and planes, between which an incidence relation, basic for projective geometry, satisfying appropriate axioms, is established. These axioms differ from the corresponding group of axioms of elementary geometry in that every two lines in one plane are required to have a common point, and on each line there should be, at least, three distinct points. In order to obtain, in concrete situations, a "richer" projective geometry, this group of axioms is completed by order axioms and the axiom of continuity (for real projective space), the Pappus axiom (for projective geometry over commutative skew-fields), the Fano postulate (for projective geometry over skew-fields of characteristic $\neq 2$), etc.

The duality principle takes a remarkable position in projective geometry. One says that a point and a line (a line and a plane, a point and a plane) are incident if the point lies on the line (the line passes through the plane, etc.). Now, if a certain proposition $\mathcal{T}$ concerning points, lines and planes in projective space, formulated only in terms of incidence between them, is valid, then the dual proposition $\mathcal{T}'$, obtained from $\mathcal{T}$ by replacing the word "point" with "plane" and vice versa and retaining the word "line", is also valid.

The Desargues assumption plays an important role in projective geometry. Its truth is necessary and sufficient for the introduction, by projective means, of a system of projective coordinates, composed from the elements of a skew-field $K$ which is naturally attached to the points of the projective line (cf. Projective algebra).

The foundations of projective geometry were laid in the 17th century by G. Desargues (in relation to the theory of perspective developed by him) and B. Pascal (in relation to the study of certain properties of conic sections). The work of G. Monge (second half of the 18th century and beginning of the 19th century) was of great value for the subsequent development of projective geometry. J. Poncelet (beginning of the 19th century) expounded projective geometry as an independent discipline. The merit of Poncelet lies in the isolation of projective properties of figures in a separate class and in establishing a correspondence between metric and projective properties of these figures. The work of J. Brianchon is of the same period. The subsequent development of projective geometry was through the works of J. Steiner and M. Chasles. The work of Ch. von Staudt, in which the essence of an axiomatic construction of projective geometry is present, also played an important role in the development of projective geometry. All these scientists tried to prove theorems in projective geometry by synthetic means, after having obtained an exposition of projective properties of figures. The analytic direction in projective geometry can be noticed in the work of A. Möbius. The work of N.I. Lobachevskii on the creation of non-Euclidean geometry had an influence on the development of projective geometry; it made it possible for A. Cayley and F. Klein to consider various geometric systems from the point of view of projective geometry. The development of analytic methods of ordinary projective geometry and the construction, on that basis, of complex projective geometry (J. Plücker, E. Study, E. Cartan) posed the problem of the dependence of some projective properties on the skew-field over which the geometry was constructed. A.N. Kolmogorov and L.S. Pontryagin created topological projective geometry.

#### References

 [1] N.A. Glagolev, "Projective geometry" , Moscow-Leningrad (1963) (In Russian) [2] D. Hilbert, S.E. Cohn-Vossen, "Anschauliche Geometrie" , Springer (1932) [3] H.S.M. Coxeter, "The real projective plane" , Cambridge Univ. Press (1961) [4] D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) pp. Appendix VIII [5] R. Hartshorne, "Foundations of projective geometry" , Benjamin (1967) [6] N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian) [7] P.S. Aleksandrov, "Lectures on analytical geometry" , Moscow (1968) (In Russian) [8] N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian) [9] E. Artin, "Geometric algebra" , Interscience (1957) [10] O. Veblen, J.W. Young, "Projective geometry" , 1–2 , Blaisdell (1938–1946) [11] W. Blaschke, "Projektive Geometrie" , Wolfenbütteler Verlagsanstalt (1947)