Line Geometry

Mapping on Hyperspace.

If the pik (or the xi) are thought of as homogeneous point co-ordinates in space of five dimensions, the quadratic identity P=o becomes the equation of a quadric variety, hence line geometry may be interpreted as the geometry on a quadric variety in five-way space. A linear complex is then the section of the quadric by a linear four-way space, and hence it defines a three-dimensional quadratic variety in space of four dimensions. By stereographic projection this can be mapped on the points of three-way space, hence there is a one-to-one cor respondence between the points of ordinary space and the lines of any linear complex.

The axes of the complexes of a pencil describe a ruled cubic surface, the cylindroid, which is of value in the application of line geometry to dynamics, as is shown in the treatise of Ball. It may be generated as follows : Given a line d, and on it two points P, P'. Through P draw any line p perpendicular to d, and through P' draw a line p' perpendicular to d and to p. Now as a variable point G moves along d from P to P', associate with it a line g always passing through G, perpendicular to d, and such that the distance from P proportional to the cosine of double the angle that g makes with the plane d, p. At P, g coincides with p, at P', g coincides with p' and when G is in the middle of the segment PP', g makes an angle of 45° with the plane d, p. Then think of a second line g' doing the same thing, but winding in the other direction. The line d is a double line on the resulting sur face. The planes containing the pairs of intersecting lines g, g' are all parallel; they intersect in an ideal line d' which also lies on the surface.

By taking d for the z axis, x—y=o, z=h for p, x+y=o, z=—h for p', the equation has the form = 2hxy.

Every ruled cubic with two distinct directrix lines can be projected into this form.

The General Complex.

An algebraic equation homogeneous and of order it in pik, together with the identity P= o defines a complex of order n. By holding one point (y) fixed, the points (x) which satisfy the equation all lie on a cone of order n, having (y) as vertex. Similarly, the lines of the complex which lie in a fixed plane all touch a curve of class n. For an arbitrary point, the complex cone has no multiple or cuspidal generators, and in an arbitrary plane the complex curve has no double or inflexional tangents. A point (y) which is the vertex of a cone having a double generator is called a singular point. The locus

of the singular points is a surface. Similarly, the plane (u) which contains a complex curve with a double or inflexional tangent is called a singular plane. The envelope of the singular planes is identical with the locus of the singular points. It is called the surface of singularities, and it may be simply a curve.

A congruence may be the complete or partial intersection of two complexes. If it is algebraic the number of its lines passing through an arbitrary point is called its order, the number in an arbitrary plane is called its class. Let p be any line of a con gruence ; the distance from p to consecutive lines of the con gruence is a certain differential expression of the first order, which vanishes to the third order for two points P, P' on p. These are called foci, and their locus, as p describes the given congruence, is the focal surface, to which p is a bitangent. P and P' may describe the same surface, or different surfaces, or either or both may describe a curve. In the last case the congruence consists of the lines which intersect both curves.

Congruences of lines are of particular importance in differential geometry (q.v.), both metric and projective, e.g., that formed as lines of curvature, asymptotic lines, etc., and that formed by the tangents to one parameter systems of curves on a surface, by the normals to a surface. They are also of importance in optics.

The complete or partial intersection of three complexes is a ruled surface. The number of its lines which meet a given line is its order, which is also its class. For every order greater than two, a con-conical ruled surface must have one or more double curves. Every plane section is either proper or, if composite, must consist of one proper curve and of straight lines, counted simply or multiply. All the plane sections, apart from straight line components, are birationally equivalent, hence if the surface is algebraic all have the same genus, which may be called the genus of the ruled surface. These ruled surfaces of maximum genus for a given order are those contained in a linear congruence. If all the generators of a ruled surface are tangents to a curve C, the surface is called a developable surface. Then C appears on the surface as a cuspidal curve. The surface also contains double curves, if the order of the curve C is greater than 3.