The points in which the lines in S1 cut the planes corresponding to them in S2 are therefore the same as the points in which the lines in S2 cut the planes corresponding to them in S1.
The locus of these points is a surface which is cut by a plane in a conic or in a line-pair and by a line in not more than two points unless it lies altogether on the surface. The surface itself is therefore called a quadric surface, or a surface of the second order.
To prove this we consider any line p in space.
The flat pencil in S1 which lies in the plane drawn through p and the corresponding axial pencil in S2 determine on p two projective rows, and those points in these which coincide with their corresponding points lie on the surface. But there exist only two, or one, or no such points, unless every point coincides with its corresponding point. In the latter case the line lies altogether on the surface.
This proves also that a plane cuts the surface in a curve of the second order, as no line can have more than two points in common with it. To show that this is a curve of the same kind as those considered before, we have to show that it can be generated by projective flat pencils. We prove first that this is true for any plane through the centre of one of the pencils, and afterwards that every point on the surface may be taken as the centre of such pencil. Let then α1 be a plane through S1. To the flat pencil in S1 which it contains corresponds in S2 a projective axial pencil with axis a2 and this cuts α1 in a second flat pencil. These two flat pencils in α1 are projective, and, in general, neither concentric nor perspective. They generate therefore a conic. But if the line a2 passes through S1 the pencils will have S1 as common centre, and may therefore have two, or one, or no lines united with their corresponding lines. The section of the surface by the plane α1 will be accordingly a line-pair or a single line, or else the plane α1 will have only the point S1 in common with the surface.
Every line l1 through S1 cuts the surface in two points, viz. first in S1 and then at the point where it cuts its corresponding plane. If now the corresponding plane passes through S1, as in the case just considered, then the two points where l1 cuts the surface coincide at S1, and the line is called a tangent to the surface with S1 as point of contact. Hence if l1 be a tangent, it lies in that plane τ1 which corresponds to the line S2S1 as a line in the pencil S2. The section of this plane has just been considered. It follows that—
All tangents to quadric surface at the centre of one of the reciprocal pencils lie in a plane which is called the tangent plane to the surface at that point as point of contact.
To the line joining the centres of the two pencils as a line in one corresponds in the other the tangent plane at its centre.
The tangent plane to a quadric surface either cuts the surface in two lines, or it has only a single line, or else only a single point in common with the surface.
In the first case the point of contact is said to be hyperbolic, in the second parabolic, in the third elliptic.