Line Geometry

a branch of geometry in which lines are considered as the elements of space. A line in space is determined by four constants—the coefficients a, b, p, and q in the equations x = az + p and y = bz + q. Consequently, the quantities a, b, p, and q may be considered as the coordinates of the line. If these coordinates are functions of one, two, or three parameters, then the corresponding sets of lines form, respectively, ruled surfaces and congruences and complexes of lines. These geometric entities are the objects studied in line geometry. A hyper-boloid of one sheet is an example of a ruled surface, the set of common tangents to two surfaces is an example of a congruence, and the set of tangents to a surface is as an example of a complex of lines.

In order to study ruled surfaces, congruences, and complexes of lines from a unified point of view, we use in line geometry so-called linear homogeneous coordinates of a line. Specifically, given two points M₁ (x₁, y₁, z₁) and M₂(x₂, y₂, z₂), we call six numbers proportional (or equal) to the numbers

ξ₁ = x₁ - x₂, ξ₂ = y₁ - y₂, ξ₃ = z₁ - z₂, ξ₄ = y₁z₂ - y₂z₁, ξ₅ = x₂z₁ - x₁z₂
ξ₆ = x₁y₂ - x₂y₁

linear homogeneous coordinates of the line that passes through these points.

The numbers ξ₁, ξ₂, and ξ₃ are the components of the vector and ξ₄, ξ₅, and ξ₆ are the components of the moment of this vector with respect to the origin. It can be easily verified that the numbers ξ_i satisfy the relation

(1) ξ₁ξ₄ + ξ₂ξ₅ + ξ₃ξ₆ = 0

Thus, to every line there correspond six numbers ξ_i that are determined up to a multiplicative constant and satisfy relation (1), and, conversely, numbers ξ_i (not all equal to zero) that are related by condition (1) uniquely define a line (as its coordinates in the sense indicated above).

One homogeneous linear equation

defines a linear complex—a set of lines that fill space in such a way that through every point in space there passes a plane pencil of lines. Thus, to every point (“pole”) in space is associated a plane (“polar plane”) that contains all the lines of the complex passing through that point. This correspondence is called a null system; it is analogous to the correspondence between the poles and polar planes of a quadratic surface. If the polar planes of all points in space pass through one line (the axis), then the complex consists of all lines that intersect the axis; such a complex is called a special linear complex. In this case, the coefficients of equation (2) satisfy the condition a₁a₄ + a₂a₅ + a₃a₆ = 0.

A system of two homogeneous linear equations of the form (2) defines a linear congruence—the set of lines that intersect two given lines (which may be imaginary). Three homogeneous linear equations define a ruled surface, which in this case is either a hyperboloid of one sheet or a hyperbolic paraboloid.

Linear homogeneous coordinates of a line were introduced by J. Plücker in 1846. He also studied in detail the theory of linear complexes. Line geometry subsequently was developed in the works of F. Klein and the Russian mathematician A. P. Kotel’nikov. The differential geometry of congruences initiated by E. Kummer in 1860 was developed extensively in the works of the Italian mathematicians L. Bianchi and G. Sannia and the French mathematician A. Ribaucour. The theory of ruled surfaces and congruences was developed by the Soviet mathematician D. N. Zeiliger on the basis of the “spiral” calculus devised in 1895 by Kotel’nikov. The projective theory of congruences was constructed in 1927 by the Soviet mathematician S. P. Fini-kov.