can be drawn, and all these lines meet in a point. The surface is a cone of the second order.
If through one point on a quadric surface, two, and only two, lines can be drawn on the surface, then through every point two lines may be drawn, and the surface is ruled quadric surface.
If through one point on a quadric surface no line on the surface can be drawn, then the surface contains no lines.
Using the definitions at the end of § 95, we may also say:—
On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic.
As an example of a quadric surface with elliptical points, we mention the sphere which may be generated by two reciprocal pencils, where to each line in one corresponds the plane perpendicular to it in the other.
§ 99. Poles and Polar Planes.—The theory of poles and polars with regard to a conic is easily extended to quadric surfaces.
Let P be a point in space not on the surface, which we suppose not to be a cone. On every line through P which cuts the surface in two points we determine the harmonic conjugate Q of P with regard to the points of intersection. Through one of these lines we draw two planes α and β. The locus of the points Q in α is a line a, the polar of P with regard to the conic in which α cuts the surface. Similarly the locus of points Q in β is a line b. This cuts a, because the line of intersection of α and β contains but one point Q. The locus of all points Q therefore is a plane. This plane is called the polar plane of the point P, with regard to the quadric surface. If P lies on the surface we take the tangent plane of P as its polar.
The following propositions hold:—
1. Every point has a polar plane, which is constructed by drawing the polars of the point with regard to the conics in which two planes through the point cut the surface.
2. If Q is a point in the polar of P, then P is a point in the polar of Q, because this is true with regard to the conic in which a plane through PQ cuts the surface.
3. Every plane is the polar plane of one point, which is called the Pole of the plane.
The pole to a plane is found by constructing the polar planes of three points in the plane. Their intersection will be the pole.
4. The points in which the polar plane of P cuts the surface are points of contact of tangents drawn from P to the surface, as is easily seen. Hence:—
5. The tangents drawn from a point P to a quadric surface form a cone of the second order, for the polar plane of P cuts it in a conic.
6. If the pole describes a line a, its polar plane will turn about another line a′, as follows from 2. These lines a and a′ are said to be conjugate with regard to the surface.
§ 100. The pole of the line at infinity is called the centre of the surface. If it lies at the infinity, the plane at infinity is a tangent plane, and the surface is called a paraboloid.
The polar plane to any point at infinity passes through the centre, and is called a diametrical plane.
A line through the centre is called a diameter. It is bisected at the centre. The line conjugate to it lies at infinity.
If a point moves along a diameter its polar plane turns about the conjugate line at infinity; that is, it moves parallel to itself, its centre moving on the first line.
The middle points of parallel chords lie in a plane, viz. in the polar plane of the point at infinity through which the chords are drawn.
The centres of parallel sections lie in a diameter which is a line conjugate to the line at infinity in which the planes meet.
Twisted Cubics
§ 101. If two pencils with centres S1 and S2 are made projective, then to a ray in one corresponds a ray in the other, to a plane a plane, to a flat or axial pencil a projective flat or axial pencil, and so on.
There is a double infinite number of lines in a pencil. We shall see that a single infinite number of lines in one pencil meets its corresponding ray, and that the points of intersection form a curve in space.
Of the double infinite number of planes in the pencils each will meet its corresponding plane. This gives a system of a double infinite number of lines in space. We know (§ 5) that there is a quadruple infinite number of lines in space. From among these we may select those which satisfy one or more given conditions. The systems of lines thus obtained were first systematically investigated and classified by Plücker, in his Geometrie des Raumes. He uses the following names:—
A treble infinite number of lines, that is, all lines which satisfy one condition, are said to form a complex of lines; e.g. all lines cutting a given line, or all lines touching a surface.
A double infinite number of lines, that is, all lines which satisfy two conditions, or which are common to two complexes, are said to form a congruence of lines; e.g. all lines in a plane, or all lines cutting two curves, or all lines cutting a given curve twice.
A single infinite number of lines, that is, all lines which satisfy three conditions, or which belong to three complexes, form a ruled surface; e.g. one set of lines on a ruled quadric surface, or developable surfaces which are formed by the tangents to a curve.
It follows that all lines in which corresponding planes in two projective pencils meet form a congruence. We shall see this congruence consists of all lines which cut a twisted cubic twice, or of all secants to a twisted cubic.
§ 102. Let l1 be the line S1S2 as a line in the pencil S1. To it corresponds a line l2 in S2. At each of the centres two corresponding lines meet. The two axial pencils with l1 and l2 as axes are projective, and, as, their axes meet at S2, the intersections of corresponding planes form a cone of the second order (§ 58), with S2 as centre. If π1 and π2 be corresponding planes, then their intersection will be a line p2 which passes through S2. Corresponding to it in S1 will be a line p1 which lies in the plane π1, and which therefore meets p2 at some point P. Conversely, if p2 be any line in S2 which meets its corresponding line p1 at a point P, then to the plane l2p2 will correspond the plane l1p1, that is, the plane S1S2P. These planes intersect in p2, so that p2 is a line on the quadric cone generated by the axial pencils l1 and l2. Hence:—
All lines in one pencil which meet their corresponding lines in the other form a cone of the second order which has its centre at the centre of the first pencil, and passes through the centre of the second.
From this follows that the points in which corresponding rays meet lie on two cones of the second order which have the ray joining their centres in common, and form therefore, together with the line S1S2 or l1, the intersection of these cones. Any plane cuts each of the cones in a conic. These two conics have necessarily that point in common in which it cuts the line l1, and therefore besides either one or three other points. It follows that the curve is of the third order as a plane may cut it in three, but not in more than three, points. Hence:—
The locus of points in which corresponding lines on two projective pencils meet is a curve of the third order or a “twisted cubic” k, which passes through the centres of the pencils, and which appears as the intersection of two cones of the second order, which have one line in common.
A line belonging to the congruence determined by the pencils is a secant of the cubic; it has two, or one, or no points in common with this cubic, and is called accordingly a secant proper, a tangent, or a secant improper of the cubic. A secant improper may be considered, to use the language of coordinate geometry, as a secant with imaginary points of intersection.
§ 103. If a1 and a2 be any two corresponding lines in the two pencils, then corresponding planes in the axial pencils having a1 and a2 as axes generate a ruled quadric surface. If P be any point on the cubic k, and if p1, p2 be the corresponding rays in S1 and S2 which meet at P, then to the plane a1p1 in S1 corresponds a2p2 in S2. These therefore meet in a line through P.
This may be stated thus:—
Those secants of the cubic which cut a ray a1, drawn through the centre S1 of one pencil, form a ruled quadric surface which passes through both centres, and which contains the twisted cubic k. Of such surfaces an infinite number exists. Every ray through S1 or S2 which is not a secant determines one of them.
If, however, the rays a1 and a2 are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having A as centre. Or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order. In other words, the projection of a twisted cubic from any point in the curve on to any plane is a conic.
If a1 is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by a1 will pass through Q. There will therefore be one line of the congruence passing through Q, and only one. For if two such lines pass through Q, then the lines S1Q and S2Q will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it. Hence:—
Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn.
§ 104. The fact that all the secants through a point on the cubic form a quadric cone shows that the centres of the projective pencils generating the cubic are not distinguished from any other points on the cubic. If we take any two points S, S′ on the cubic, and draw the secants through each of them, we obtain two quadric cones, which have the line SS′ in common, and which intersect besides along the cubic. If we make these two pencils having S and S′ as centres projective by taking four rays on the one cone as corresponding to the four rays on the other which meet the first on the cubic, the correspondence is determined. These two pencils will generate a cubic, and the two cones of secants having S and S′ as centres will be identical with the above cones, for each has five rays in common with one of the first, viz. the line SS′ and the four lines determined for the correspondence; therefore these two cones intersect in the original cubic. This gives the theorem:—
On a twisted cubic any two points may be taken as centres of projective pencils which generate the cubic, corresponding planes being those which meet on the same secant.
Of the two projective pencils at S and S′ we may keep the first fixed, and move the centre of the other along the curve. The pencils will hereby remain projective, and a plane α in S will be cut by its corresponding plane α′ always in the same secant a. Whilst S′ moves along the curve the plane α′ will turn about a, describing an axial pencil.
