View source for Complex of lines

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241501.png" /> of straight lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241502.png" />-dimensional (projective, affine, Euclidean) space depending on three parameters. A straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241503.png" /> is called a ray of the complex. Through each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241504.png" /> of the ambient space there passes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241505.png" />-parameter family of rays of the complex, called the cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241506.png" /> and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241507.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241508.png" /> of the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c0241509.png" /> is the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415010.png" /> tangent to the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415011.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415012.png" />. This correspondence is called the normal correlation. Each plane of the space contains a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415013.png" />-parameter family of rays of the complex enveloping plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415014.png" />. By a centre of inflection of a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415015.png" /> one means a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415016.png" /> at which the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415017.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415018.png" /> corresponding to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415020.png" /> on a generator of it at which the tangent plane of the surface is the same as the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415021.png" /> corresponding to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415023.png" /> one has an invariant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415024.png" /> (the centre of the ray) at which the vector of the normal to the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415025.png" /> that corresponds to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415026.png" /> in the normal correlation is orthogonal to the normal to the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415027.png" /> corresponding to the ideal point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415028.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024150/c02415029.png" />-parameter family) of planes, conics, quadrics, and other figures (see [[Manifold of figures (lines, surfaces, spheres)|Manifold of figures (lines, surfaces, spheres,...)]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Finikov,   "Theorie der Kongruenzen" , Akademie Verlag  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Kovantsov,   "Theory of complexes" , Kiev  (1963)  (In Russian)</TD></TR></table>