Any two lines of the same set may be taken as bases of two projective rows, or of two projective pencils which generate the surface. They are cut by the lines of the other set in two projective rows.
The plane at infinity like every other plane cuts the surface either in a conic proper or in a line-pair. In the first case the surface is called an Hyperboloid of one sheet, in the second an Hyperbolic Paraboloid.
The latter may be generated by a line cutting three lines of which one lies at infinity, that is, cutting two lines and remaining parallel to a given plane.
Quadric Surfaces
§ 91. The conics, the cones of the second order, and the ruled quadric surfaces complete the figures which can be generated by projective rows or flat and axial pencils, that is, by those aggregates of elements which are of one dimension (§§ 5, 6). We shall now consider the simpler figures which are generated by aggregates of two dimensions. The space at our disposal will not, however, allow us to do more than indicate a few of the results.
§ 92. We establish a correspondence between the lines and planes in pencils in space, or reciprocally between the points and lines in two or more planes, but consider principally pencils.
In two pencils we may either make planes correspond to planes and lines to lines, or else planes to lines and lines to planes. If hereby the condition be satisfied that to a flat, or axial, pencil corresponds in the first case a projective flat, or axial, pencil, and in the second a projective axial, or flat, pencil, the pencils are said to be projective in the first case and reciprocal in the second.
For instance, two pencils which join two points S1 and S2 to the different points and lines in a given plane π are projective (and in perspective position), if those lines and planes be taken as corresponding which meet the plane π in the same point or in the same line. In this case every plane through both centres S1 and S2 of the two pencils will correspond to itself. If these pencils are brought into any other position they will be projective (but not perspective).
The correspondence between two projective pencils is uniquely determined, if to four rays (or planes) in the one the corresponding rays (or planes) in the other are given, provided that no three rays of either set lie in a plane.
Let a, b, c, d be four rays in the one, a′, b′, c′, d′ the corresponding rays in the other pencil. We shall show that we can find for every ray e in the first a single corresponding ray e′ in the second. To the axial pencil a (b, c, d ...) formed by the planes which join a to b, c, d ..., respectively corresponds the axial pencil a′ (b′, c′, d′ ... ), and this correspondence is determined. Hence, the plane a′e′ which corresponds to the plane ae is determined. Similarly the plane b′e′ may be found and both together determine the ray e′.