Complex of lines

A set $ K $ of straight lines in $ 3 $- dimensional (projective, affine, Euclidean) space depending on three parameters. A straight line $ l \in K $ is called a ray of the complex. Through each point $ M $ of the ambient space there passes a $ 1 $- parameter family of rays of the complex, called the cone of $ M $ and denoted by $ K _ {M} $. A complex of lines defines a correspondence between the points of a ray of the complex and the planes passing through this ray: Corresponding to each point $ M $ of the ray $ l $ is the plane $ \Pi $ tangent to the cone $ K _ {M} $ at the point $ M $. This correspondence is called the normal correlation. Each plane of the space contains a $ 1 $- parameter family of rays of the complex enveloping plane curve $ s $. By a centre of inflection of a ray $ l \in K $ one means a point $ M \in l $ at which the curve $ s $ of the plane $ \Pi $ corresponding to the point $ M $ in the normal correlation has a cusp. On each ray of a complex there are, in general, four centres of inflection. A point of tangency of a ruled surface of a complex is a point $ M $ on a generator of it at which the tangent plane of the surface is the same as the plane $ \Pi $ corresponding to the point $ M $ in the normal correlation. On every ruled surface of a complex there are, in general, precisely two points of tangency. The lines described by these points are called lines of tangency of the ruled surface. The principal surfaces of a complex are the ruled surfaces for which the lines of tangency are asymptotic lines of them. A projective classification of complexes can be realized in terms of the multiplicity of the centres of inflection of their rays.

In Euclidean space, on each ray $ l $ one has an invariant point $ C $( the centre of the ray) at which the vector of the normal to the plane $ \Pi $ that corresponds to the point $ C $ in the normal correlation is orthogonal to the normal to the plane $ \Pi $ corresponding to the ideal point of $ l $. Examples of a complex are: a special complex, i.e. the set of all tangents to a given surface; a linear complex, defined by a linear homogeneous equation with respect to the Grassmann coordinates of the rays of the complex; and a special linear complex, i.e. the set of straight lines of three-dimensional space that intersect a given straight line.

Besides a complex of lines it is possible to consider a complex (a $ 3 $- parameter family) of planes, conics, quadrics, and other figures (see Manifold of figures (lines, surfaces, spheres,...)).

#### References[edit]

[1] |  S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)

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[2] |  N.I. Kovantsov, "Theory of complexes" , Kiev (1963) (In Russian)