Congruence of lines

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A set $C$ of lines in a three-dimensional space (projective, affine or Euclidean) depending on two parameters. A line $l\in C$ is said to be a ray of the congruence. The order of a congruence is the number of lines in it passing through an arbitrary point of the space; its class is the number of lines in an arbitrary plane.

Rays in a congruence can be decomposed in two ways into a one-parameter family of torses (developable surfaces, cf. Developable surface) such that through every ray $l\in C$ there pass two torses that are real and different (the case of a hyperbolic ray), or imaginary (an elliptic ray), or real and coincident (a parabolic ray). The points of contact of a ray $l\in C$ with the edges of regression of these torses are called the foci of $l$. The surfaces formed by the foci of the rays of a congruence are called its focal surfaces. The tangent planes to the focal surfaces passing through a ray $l$ of the congruence are called the focal planes of $l$. The torses of a congruence intersect every focal surface in a net of lines called a focal net of the congruence. The focal net of lines on every focal surface is conjugate. In a hyperbolic domain, a congruence is the set of common tangents to two focal surfaces; in the elliptic case, a congruence is formed by real common tangents to two conjugate imaginary surfaces; in the parabolic case, a congruence is formed by the tangents to a family of asymptotic lines of a unique focal surface. The centre of a ray of a congruence is the middle of the segment determined by the foci of the ray. The surface described by the centres of the rays is called the mean surface of the congruence. The bases of the common perpendiculars to two adjacent rays $l(u,v)$ and $l'(u+du,v+dv)$ occupy a segment on the ray $l$ whose end points are called the boundary points of the ray. The planes perpendicular to the direction of the common perpendiculars at the boundary points are called the principal planes; the ruled surfaces whose striction lines intersect the rays in their boundary points are called the principal surfaces. The set of boundary points of a ray is called the boundary surface.

Examples of congruences: a $W$-congruence, in which the asymptotic lines on the focal surfaces correspond to one another; a linear congruence, that is, the set of straight lines in the space intersecting two given lines called the directrices; a normal congruence, that is, the set of normals to some surface; an isotropic congruence, that is, a congruence with indefinite principal surfaces.

Together with congruences of lines, congruences (two-parameter families) of planes, conics, quadrics and other figures (see Manifold of figures (lines, surfaces, spheres,...)) have been considered. Congruences of arbitrary lines (curves) in a space are called curvilinear congruences.


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



[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 2 , Chelsea, reprint (1972)
How to Cite This Entry:
Congruence of lines. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.S. Malakhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article