RULED QUADRIC SURFACES
Â§ 89. We have considered hitherto projective rows which lie in the same plane, in which case lines joining corresponding points envelop a conic. We shall now consider projective rows whose bases do not meet. In this case, corresponding points will be joined by lines which do not lie in a plane, but on some surface, which like every surface generated by lines is called a ruled surface. This surface clearly contains the bases of the two rows.
If the points in either row be joined to the base of the other, we obtain two axial pencils which are also projective, those planes being corresponding which pass through corresponding points in the given rows. If A', A be two corresponding points, a, a' the planes in the axial pencils passing through them, then AA' will be the line of intersection of the corresponding planes a, a' and also the line joining corresponding points in the rows.
If we cut the whole figure by a plane this will cut the axial pencils in two projective flat pencils, and the curve of the second order generated by these will be the curve in which the plane cuts the surface. Hence
The locus of lines joining corresponding points in two projective rows which do not lie in the same plane is a surface which contains the bases of the rows, and which can also be generated by the lines of intersection of corresponding planes in two projective axial pencils. This surface is cut by every plane in a curve of the second order, hence either in a conic or in a linepair. No line which does not lie altogether on the surface can have more than two points in common with the surface, which is therefore said to be of the second order or is called a ruled quadric surface.
That no line which does not lie on the surface can cut the surface in more than two points is seen at once if a plane be drawn through the line, for this will cut the surface in a conic. It follows also that a line which contains more than two points of the surface lies altogether on the surface.
Â§ 90. Through any point in space one line can always be drawn cutting two given lines which do not themselves meet.
If therefore three lines in space be given of which no two meet, then through every point in either one line may be drawn cutting the other two.
If a line moves so that it always cuts three given lines of which no two meet, then it generates a ruled quadric surface.
Let a, b, c be the given lines, and p, q, r . . . lines cutting them in the
points A, A', A" . .; B, B', B" . . ; C, C', C" . . . respectively ; then
the planes through a containing p, q, r, and the planes through b containing the same lines, may be taken as corresponding planes in two axial pencils which are projective, because both pencils cut the line c in the same row, C, C', C" . . .; the surface can therefore be generated by projective axial pencils.
Of the lines p, q, r . . . no two can meet, for otherwise the lines a, b, c which cut them would also lie in their plane. There is a single infinite number of them, for one passes through each point of a. These lines are said to form a set of lines on the surface.
If now three of the lines p, q, r be taken, then every line d cutting them will have three points in common with the surface, and will therefore lie altogether on it. This gives rise to a second set of lines on the surface. From what has been said the theorem follows:
A ruled quadric surface contains two sets of straight lines. Every line of one set cuts every line of the other, but no two lines of the same set meet.
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 linepair. 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.
End of Article: RULED QUADRIC 

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