PENCIL OF CONICS
§ 87. Through four points A, B, C, D in a plane, of which no three lie in a line, an infinite number of conics may be drawn, viz. through these four points and any fifth one single conic. This system of conics is called a pencil of conics. Similarly, all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic. We have here the theorems:
The pairs of points in which The pairs of tangents which any line is cut by a system of can be drawn from a point to conics through four fixed points a system of conics touching four
are in involution. fixed lines are in involution.
We prove the first theorem only. Let ABCD (fig. 36) be the fourpoint, then any line t will cut two opposite sides AC, BD in
the points E, E', the pair AD, BC in points F, F', and any conic of the system in M, N, and we have A(CD, MN)=B(CD, MN). If we cut these pencils by t we get
(EF, MN)=(F'E', MN)
or (EF, MN) = (E'F', NM).
But this is, according to § 77 (7), the condition that M, N are corresponding points in the involution determined by the point pairs E, E', F, F' in which the line t cuts pairs of opposite sides of the fourpoint ABCD. This involution is independent of the particular conic chosen.
§ 88. There follow several important theorems:
Through four points two, one, or no conics may be drawn which touch any given line, according as the involution determined by the given fourpoint on the line has real, coincident or imaginary foci.
Two, one, or no conics may be drawn which touch four given lines and pass through a given point, according as the involution determined by the given fourside at the point has real, coincident or imaginary
focal rays. For the conic through four points which touches a given line has its point of contact at a focus of the involution determined by the fourpoint on the line.
As a special case we get, by taking the line at infinity:
Through four points of which none is at infinity either two or no parabolas may be drawn.
The problem of drawing a conic through four points and touching a given line is solved by determining the points of contact on the line, that is, by determining the foci of the involution in which the line cuts the sides of the fourpoint. The corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point.
corresponding which meet the plane r in the same point or in the In the first case the point of contact is said to be hyperbolic, in the same line. In this case every plane through both centres SI 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
§. 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 SI and S2 be the centres of the two pencils, and P be a point where a line al in the first cuts its corresponding plane a2, then the line b2 in the pencil S2 which passes through P will meet its corresponding plane t31 in P. For b2 is a line in the plane a2. The corresponding plane tai must therefore pass through the line al, hence through P.
The points in which the lines in SI 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 Si.
The locus of these points is a surface which is cut by a plane in a conic or in a linepair 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 SI 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 a1 be a plane through Si. To the flat pencil in SI which it contains corresponds in S2 a projective axial pencil with axis a2 and this cuts a1 in a second flat pencil. These two flat pencils in a1 are projective, and, in general, neither concentric nor perspective. They generate therefore a conic. But if the line ¢2 passes through SI the pencils will have SI 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 al will be accordingly a linepair or a single line, or else the plane al will have only the point SI in common with the surface.
Every line lI through SI cuts the surface in two points, viz. first in SI and then at the point where it cuts its corresponding plane. If now the corresponding plane passes through SI, as in the case just considered, then the two points where lI cuts the surface coincide at Si, and the line is called a tangent to the surface with SI as point of contact. Hence if 11 be a tangent, it lies in that plane r1 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.
X". 23
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 I' generated by the reciprocal pencils Si and S2. We have to establish a reciprocal correspondence between the pencils S and SI, so that the surface generated by them is identical with 4h. To do this we draw two planes a1 and /3i through Si, cutting the surface 4, in two conics .which we also denote by al and t31. These conics meet at SI, and at some other point T where the line of intersection of al and t31 cuts the surface.
In the pencil S we draw some plane a which passes through T, but not through SI 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 SI and S as follows:—To SIT shall correspond the plane a, to the plane al the line a, and to $1 the line b, hence to the flat pencil in al the axial pencil a. These pencils are made projective by aid of the conic in al.
In the same manner the flat pencil in hi is made projective to the axial pencil b by aid of the conic in RI, corresponding elements being those which meet on the conic. This determines the correspondence, for we know for more than four rays in SI the corresponding planes in S. The two pencils S and SI thus made reciprocal generate a quadric surface which passes through the point S and through the two conics al and F31.
The two surfaces ,1, and have therefore the points S and SI and the conics at and hi in common. To show that they are identical, we draw a plane through S and S2, cutting each of the conics al and RI in two points, which will always be possible. This plane cuts 4' and 43' in two conics which have the point S and the points where it cuts al and. t31 in common, that is five points in all. The conics therefore coincide.
This proves that all those points P on lie on di which have the property that the plane SS2P cuts the conics a1, /31 in two points each. If the plane SS2P has not this property, then we draw a plane SSIP. This cuts each surface in a conic, and these conics have in common the points S, SI, one point on each of the conics al, 0'1, 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 linepair. 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 linepair is a tangent plane. For every line in this plane through the centre of the linepair (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 linepair being the point of contact.
If a quadric surface contains a line, then every plane through this line cuts the surface in a linepair (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 linepair 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 §§ 8993, 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 II
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 a 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 a and S. The locus of the points Q in a is a line a, the polar of P with regard to the conic in which a cuts the surface. Similarly the locus of points Q in $ is a line b. This cuts a, because the line of intersection of a and f 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:
I. 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. cIf 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 onjugate with regard to the surface.
too. 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.
End of Article: PENCIL OF 

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