Page:EB1911 - Volume 11.djvu/725

PROJECTIVE] corresponding which meet the plane 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′.

Similarly the correspondence between two reciprocal pencils is determined if for four rays in the one the corresponding planes in the other are given.

§ 93. We may now combine—  1. Two reciprocal pencils.

 Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.

 2. Two projective pencils.

 Each plane cuts its corresponding plane in a line, but a ray as a rule does not cut its corresponding ray. The locus of points where a ray cuts its corresponding ray is a twisted cubic. The lines where a plane cuts its corresponding plane are secants.

 3. Three projective pencils.

 The locus of intersection of corresponding planes is a cubic surface.

Of these we consider only the first two cases.

§ 94. If two pencils are reciprocal, then to a plane in either corresponds a line in the other, to a flat pencil an axial pencil, and so on. Every line cuts its corresponding plane in a point. If S1 and S2 be the centres of the two pencils, and P be a point where a line a1 in the first cuts its corresponding plane 2, then the line b2 in the pencil S2 which passes through P will meet its corresponding plane 1 in P. For b2 is a line in the plane 2. The corresponding plane 1 must therefore pass through the line a1, hence through P.

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 1 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 1 in a second flat pencil. These two flat pencils in 1 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 1 will be accordingly a line-pair or a single line, or else the plane 1 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 1 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.

§ 95. It remains to be proved that every point S on the surface may be taken as centre of one of the pencils which generate the surface. Let S be any point on the surface ′ generated by the reciprocal pencils S1 and S2. We have to establish a reciprocal correspondence between the pencils S and S1, so that the surface generated by them is identical with. To do this we draw two planes 1 and 1 through S1, cutting the surface in two conics which we also denote by 1 and 1. These conics meet at S1, and at some other point T where the line of intersection of 1 and 1 cuts the surface.

In the pencil S we draw some plane which passes through T, but not through S1 or S2. It will cut the two conics first at T, and therefore each at some other point which we call A and B respectively. These we join to S by lines a and b, and now establish the required correspondence between the pencils S1 and S as follows:—To S1T shall correspond the plane, to the plane 1 the line a, and to 1 the line b, hence to the flat pencil in 1 the axial pencil a. These pencils are made projective by aid of the conic in 1.

In the same manner the flat pencil in 1 is made projective to the axial pencil b by aid of the conic in 1, corresponding elements being those which meet on the conic. This determines the correspondence, for we know for more than four rays in S1 the corresponding planes in S. The two pencils S and S1 thus made reciprocal generate a quadric surface ′, which passes through the point S and through the two conics 1 and 1.

The two surfaces and ′ have therefore the points S and S1 and the conics 1 and 1 in common. To show that they are identical, we draw a plane through S and S2, cutting each of the conics 1 and 1 in two points, which will always be possible. This plane cuts and ′ in two conics which have the point S and the points where it cuts 1 and 1 in common, that is five points in all. The conics therefore coincide.

This proves that all those points P on ′ lie on which have the property that the plane SS2P cuts the conics 1, 1 in two points each. If the plane SS2P has not this property, then we draw a plane SS1P. This cuts each surface in a conic, and these conics have in common the points S, S1, one point on each of the conics 1, 1, and one point on one of the conics through S and S2 which lie on both surfaces, hence five points. They are therefore coincident, and our theorem is proved.

§ 96. The following propositions follow:—

A quadric surface has at every point a tangent plane.

Every plane section of a quadric surface is a conic or a line-pair.

Every line which has three points in common with a quadric surface lies on the surface.

Every conic which has five points in common with a quadric surface lies on the surface.

Through two conics which lie in different planes, but have two points in common, and through one external point always one quadric surface may be drawn.

§ 97. Every plane which cuts a quadric surface in a line-pair is a tangent plane. For every line in this plane through the centre of the line-pair (the point of intersection of the two lines) cuts the surface in two coincident points and is therefore a tangent to the surface, the centre of the line-pair being the point of contact.

If a quadric surface contains a line, then every plane through this line cuts the surface in a line-pair (or in two coincident lines). For this plane cannot cut the surface in a conic. Hence:—

If a quadric surface contains one line p then it contains an infinite number of lines, and through every point Q on the surface, one line q can be drawn which cuts p. For the plane through the point Q and the line p cuts the surface in a line-pair which must pass through Q and of which p is one line.

No two such lines q on the surface can meet. For as both meet p their plane would contain p and therefore cut the surface in a triangle.

Every line which cuts three lines q will be on the surface; for it has three points in common with it.

Hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in §§ 89-93, but with one important exception. In the last investigation we have left out of consideration the possibility of a plane having only one line (two coincident lines) in common with a quadric surface.

§ 98. To investigate this case we suppose first that there is one point A on the surface through which two different lines a, b can be drawn, which lie altogether on the surface.

If P is any other point on the surface which lies neither on a nor b, then the plane through P and a will cut the surface in a second line a′ which passes through P and which cuts a. Similarly there is a line b′ through P which cuts b. These two lines a′ and b′ may coincide, but then they must coincide with PA.

If this happens for one point P, it happens for every other point Q. For if two different lines could be drawn through Q, then by the same reasoning the line PQ would be altogether on the surface, hence two lines would be drawn through P against the assumption. From this follows:—

If there is one point on a quadric surface through which one, but only one, line can be drawn on the surface, then through every point one line