nets of conics in the real domain

Nets of Conics in the Real DomainAuthor(s): Alan D. CampbellSource: American Journal of Mathematics, Vol. 50, No. 3 (Jul., 1928), pp. 459-466Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370813 .

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Nets of Conics in the Real Domain. BY ALAN D. CAMPBELL.

The nets of conics in the complex domain were reduced to canonical types by C. Jordan,* and for the Galois Fields GF(p4) p > 2 by A. IH. Wilson.t The purpose of this paper is to solve the corresponding problem for the real domain. The nets are reduced to twenty-six independent types.

Denoting the net by (a) AC1 +VJC2 + VC3 0 where

Ci = ajX2 + b y2 + C,Z2 + 2fiyz + 2gizx + 2hixy

(i 1, 2, 3) we apply linear transformations to x, y, z and A, c, v to reduce the net to a typical form. The reduction to types is aided by considering the cubic curve A = 0 in the plane (A, ,u, v) whose equation is obtained by setting the discriminant of (a) equal to zero. To a line in the (A, u, v) plane corresponds a pencil of the net (a) whose intersections with A = 0 give values of A, ,u, v for which the corresponding conic in (a) is composite. Nets of conics may therefore be divided into two main categories according as they do or do not possess a pencil of composite conics. In the former case A - 0 is also composite. In the latter case the net may be reduced to a standard form by taking for A 0 the tangent at a real inflection on A = 0, for ja - 0 the harmonic polar of the point of contact of A = 0, and for v 0 a line through the inflection and tangent to the cubic (at a point on u = 0) or through a node or cusp (on ,u= 0) if the cubic have a node or cusp.

The types of pencils of conics in the real domain are known.t The types or pencils with A 0 are

(b) Ax2 +?y2 0 (C) Ax2 + 2Ixy 0 (d) 2Axy + 2juxz =O (e) 2Axy + #(X2 _ y2) = 0.

* "Reduction d'uii Reseau de Formes Quadratiques," Journal de Mathematiques, Vol. 6, Series 2 (1906), pp. 403-438.

t "The Canonical Types of Nets of Modular Conics," Americaun Journal of Mathe- m'atics, Vol. 36 (1914), pp. 187-210.

t L. E. Dickson, " On Families of Quadratic Forms in a General Field," Quarterly Journal of Mathematics, Vol. 39, pp. 316-333.

10 459



460 CAMPBELL: Nets of Conics in the Real Domain.

For a net containing the pencil (b) we have the general form

AX2 + 'ty2 + V( CZ2 + 2fyz + 2gzx + 2hxy) = 0.

If c 7y 0 we can put

T: x x', y=y', z=-(g/c)x' (f/c)y'+z'; A = Al + (g2/C)VI, fL = t,J + (f2/C)V', V =v,

and get (after dropping the primes on the variables) the form

AX2 + ?y2 + V(CZ2 + 2h'xy) =0

from which we derive the typical nets

(1 ) AX2 + ?Ly2 + VZ2 =0

(2) Ax2 + ,(y2 + 2vxy= O (3) Ax2 +,uy2 +V(z2 + 2xy) 0 If c = 0 and f and g are not both zero, we can choose a new z and v so as to get the form

AX2 + ?y2 + 2v(fyz + gzx) =0

from which we derive the types

(4) Ax2 + y2 + 2vyz 0 (5) AX2 + ?y2 + 2v(yz + zx) 0.

For a net containing the pencil (c) we have the general form

AX2 + 2yxy + v(by2 + cz2 + 2fyz + 2gzx) - 0.

If c #7 0 we can subject this net to a transformation similar to the above transformation T and get the form

AX2 + 21%xy + v(b'y2 + cz2) = 0

and derive the new types

(6) AX2 + 2,Lxy + V(Z2 y2) =O (7) AX2 + 21%xy + V(Z2 + y2) =O.

If c = 0 and f # 0 in the above general form of net we can put

x=x', y=-(g/f)x'+y', z=z'; A = A' + 2(g/f)l + b(g2/f2)v', ,u = u' + b(g/f)v', v

and we get the form

AX2 + 2juxy + v(by2 + 2fyz) 0



CAMPBELL: Nets of Conics in the Real Domain. 461

and then a new z gives the type

(8) Ax2 + 21xxy +2vyz = 0.

If c = f = 0 we have the form

AX2 + 21uxy + v(by2 + 2gzx) = 0

and derive the new types

(9) Ax2 + 2/xy + 2vzx 0 (10) Ax2 + 21jixy + v(y2 + 2xz) = 0.

For a net containing the pencil (d) we have the general form

2Axy + 2p-xz + v(ax2 + by2 + cz2 + 2fyz) = 0.

A binary transformation on y and z and an obvious transformation on A, ,u, v gives if a =4 0 the new types

(11) 2Axy + 2,xz + V(X2 + y2 + Z2) =0

(12) 2Axy + 2pxz + v(x2 _ y2_Z2) o

(13) 2Axy + 2ptxz + v(X2 + 2yz) = 0

(14) 2Axy + 2xz + v(x2 y2) 0

and if a =0

(15) 2Axy + 2uxz + v(y2 + Z2) 0

(16) 2Axy + 2uxz + 2vyz = 0.

For a net containing the pencil (e) we have the general form

2Axy + 1(X2 _ y2) + v(ax2 + CZ2 + 2fyz + 2gzx) 0.

If c ,7 0 a new z and v give the form

2Axy + (X2 _ y2) + v(a'X2 + CZ2) = 0.

Hence we have the new types

(17) 2Axy + ?(x2_y2) + V(z2 + X2) =0

(18) 2Axy + (x2 _ y2) +V(Z2_X2) 0

(19) 2Axy + (x2 _ y2) +VZ2 o0.




If c = 0 we have the form

2Axy + (x2- y2) + v(ax2 + 2fyz + 2gzx) =-0.

We can put (if g70)*

x=x'-(f/g)y', y=y', z=Z'; A = A', = (2fg/f2 g2)A' + ,u', v-v',

and we get the form

X(2,xxy' + a'x'2) + 14x'2 + b'y'2 + 2h'x'y') + v(a"x'2 + b"y'2 + 2gz'x' + 2h"x-y') = 0.

If we now put x'=x", y' (a'/a)x" + y", z' =z", follow this by a transformation in A, ja, v that will rid the new C2 and C3 of x"y" and rid C. of ybt2, we arrive (after an obvious transformation of the variables) at the form

2Axy + p4X2 - y2) + v(a"'x2 + 2zx) = 0

which reduces to (14) by choosing a new z. When the cubic A = 0 is non-composite it has at least one real inflec-

tional tangent say A = 0. The harmonic polar of the point of contact of A 0 may be taken to be ,u 0. If one of the tangents from the point of contact, or the line joining the point of contact to a singular point, is v= 0 the equation of the cubic is now of the form

A == 2X -v(aA2 + b,v + CV2) = 0.

There is no Xuv term in A since every line A= -av cuts the cubic and ,u 0 in a harmonic set. If v =- 0, then A =_ 2, hence the pencil AC1 + ,UC02 0 must have AX2 as discriminant. From Dickson (loc. cit., p. 318) we see that all such pencils belong to classes with typical forms reducible to

AX2 + 2pAyz = O, Ax2 + /U(y2 + Z2) 0, or A(x2 +z2) + 2,uxy= 0

On the other hand A = 0 in A gives A- =- Cv3 and there are just two classes of pencils with such a discriminant, namely those with typical forms reducible to

J_1V2+v(y2+2zx) =0 or 2pgxy_v(y2+2zx) =0.

* If g = 0, f ; 0, we change X, Iu, v so as to replace $2 by y2 in C8, put X =y', y =', z=iz, and we get the above result. If g-f =0, a, ; 0 we have a pencil 2Xxy + Vx' = 0, contrary to hypothesis.




From these considerations we see that a net of conics with the above A can be put in one of the three following forms:

Ax2 + 2,uyz + vC3 = 0 A(X2 _Z2) + 21,xy + VC3 = 0

A(X2 + Z2) + 2,uxy + vC3 ? 0.

In the first case we have the general form

,X2 + 2,uyz +v(ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy) 0. Here

A--j a, b, c V3 2(gh-af)2v2-au2v + (bc-f2)Av2-_Ap2-2fAuv =O,

so that a = f = 0 and either g or h is zero and we have the form

Ax2 +2,xyz+v(by2+cz2+ 2hxy) =0.

We therefore have the new types

(20) Ax2 + 2,ttyz + v(y2 + Z2 + 2xy) =0 (21) AX2 + 2uyz + V(y2 Z2 + 2xy) = 0 (22) Ax2 + 2juyx + v(z2 + 2xy) = 0.

In the second case we have the general form

X(X2 -Z2) + 21jxy + v(ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy) - 0

whose discriminant is

A_{ a, b,c V8 +2(fg-ch)puv2-c/_L2v +(-ab + bc- f2+ h2)Av2 -bA2v + AtL2 + 2hApuv = 0.

Hence h-c =0 and either f or g is zero. If f0 we have the form

X(X2 - Z2) + 2p2y + v(ax2 + by2 + 2gzx) = 0

and therefore the new type

(23) X(x2_ Z2) +2pxy+v(2ax2+y2 + 2zx)-0, a 0,

where 2c is used instead of a to avoid fractions when solving A = 0 with &=0. If g = 0 we have similarly the new types

(24) X(x2 _ Z2) + 2xy + V(/X2 + y2 + 2yz) =0, # O, and (25) X(X2 - z2) + 2sxy + v(X2 + 2yz) = 0,

according as b is not or is zero.




In the third case above we have the general form for the net

X(X2 + Z2) + 2Ztxy + v(ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy) 0

whose discriminant is

A |a, b, c V3v + 2(fg- ch)IwV2 _cI 2v

+ (ab + bc -h2 f2)v2 +bX2v X,2 -2hklv=0.

Hence h=-c =0 and either f or g is zero. If f =0 we have the form

A(X2 + Z2) + 21.xy + v(2ax2y + y2 + 2ZX) =0

with discriminant A = -, v(X2 + 2 av - v2). The pencil AC1 + VC3 = 0 has three distinct degenerate conics. From Dickson we see that every such pencil belongs to one of three classes whose typical pencils are readily reducible to:

Ak (X 2 _y2) + IA(X2 _ Z2) O 0, or k(X2 +y2) + (X2 Z2) = O,

or 2AXy + tk(X2 - y2 + Z2) = 0, respectively. Each one of these pencils con- tains at least one conic that is a pair of real lines. If we take this conic for C,= 0 we can transform the net into (23) or (24). If g 0 and b 0 we have the form

X(x2 + z2) + 21Lxy + v(/3x2 + y2 + 2yz) 0

with discriminant A _ A-,u2 + v(X - v) (X + /3v). Here again the pencil AC1 + V03 = 0 has one pair of real lines, and so the net can be transformed into (23) or (24). Finally if g = b 0 we have the new type

(26) (X2 + Z2) + 2PXy + V(X2 + 2yz) = 0.

We shall now discuss the equivalence or non-equivalence of these twenty- six typical nets. Nets with non-equivalent cubics are not equivalent, but some non-equivalent nets have the same (or equivalent) cubics. The nets (1) to (19) inclusive all have composite cubics and so are not equivalent to the nets (20) through (26). These nets with composite cubics fall into four distinct classes as follows: Class A of nets (1) to (5) inclusive where each net has a pencil (b) in it; Class B of nets (6) to (10) inclusive where each net has no pencil (b) but has a pencil (c) ; Class C of nets (11) to (16) inclusive where each net has no pencil (b) or (c) but has a pencil (d) ; Class D of nets (17), (18), (19) where each net has no pencil (b) or (c) or (d) but has a pencil (e). The cubics A = 0 of the nets in Class A serve to' prove the non- equivalence of these nets to one another, except the two nets (4) and (5);




but the pencil A = 0 in (4) cannot be transformed into the pencil A + u = 0 in (5), so these two nets are non-equivalent. In Class B the nets (6) and (7) have equivalent cubics but there is no real transformation that will send (7) into (6) for any such transformation must send the pencil Ax2+2-2Xy Q 0

of (7) into the same pencil of (6) and so must be of the form:

T: x = a1x', y =a2x' + b2y', z = a3x' + b3y' + C3Z',

where a1b2c3 #' 0. But T cannot send the conic Z2 + y2 = 0 of (7) into a conic whose y2 and Z2 terms have opposite signs, such a conic as occurs in (6). The other nets of Class B are distinguished by their cubics. In Class C the nets all have non-equivalent cubics. In Class D the only nets with equivalent cubics are (17) and (18); but in (18) the sum of the coefficients of X2, y2, Z2

is identically zero in A, ,u, v (hence we must have this sum of coefficients in any equivalent net identically zero in A, ,u, v) also any transformation must send the pencil 2Axy + _(X2 y2) = 0 of (17) into the same pencil of (18). There is no real non-singular transformation that will satisfy all these conditions, hence these nets are non-equivalent.

The nets (20), (21), and (22) have each a double line and so are not equivalent to the nets (23) to (26) inclusive; the cubics of these three nets with double lines have respectively a crunode, an acnode, and a cusp and so distinguish these nets. The cubic of (25) has an acnode and the cubic of (26) a crunode, so these two nets are not equivalent to each other or to the nets (23) and (24). The cubic of (23) for a = ? 1 has a node. If = + 1 we can reduce (23) to (26) by the transformation

X=-A' + 2v', p= ,u, v=XA'-v'; x=-y', y=-x', z=y'-z'

By a similar transformation we can reduce (23) for o = - 1 to (25). If I a, < 1 in (23) the harmonic polar u = 0 of (0, 1, 0) cuts the cubic in one real and two conjugate imaginary points, hence for these values of a the net is not equivalent to (24). If , = 1 in (24) this net has a double line and is transformable to (21) by

A =- ', /u ', v= A' -v'; x= z', y =y', z-x' -y'.

If ,B < 1 in (24) we put

A w A'gt -(2 ) fo/r],u', v =-(1 1/,)A, X [ _/(1 1)4] y" y= [,8/(l f)71A]XI Z ]X' [/ -) ]z+

I

and we get (23) for 2ac B 2;(-B%




To show that for 8 > 1 the net (24) is not equivalent to (23) we use a projectively invariant relation connecting a line conic Au2 + BV2 + Cw2

+ 2Fvw + 2Gwu + 2Huv = 0 and a point conic a'x2 + b'y2 + c'z2 + 2f'yz + 2g'zx + 2h'xy = 0 and given by the equation

R: a'A+b'B+c'C +2fF+2g'G+2h'H=-.*

To a net of line conics dual to (24) for 8 > 1, namely

(24') X(u2 - w2) + 21,uv + v(/3U2 + V2 + 2vw) = 0,

there corresponds by R a net of point conics

N: X( y2 +yz) +XZ +V[(1/P)x2 + (1//)z2 y2] = O.

We can reduce N to (23) for I acc < 1 namely c=- (2 -3)1/, by the transformation

x=y', y x' + z', z = 2x'.

But the cubic A- 0 of (23) for a Ic| < 1 cuts the harmonic polar ju== 0 in one real and two conjugate imaginary points and therefore A = 0 shows the non-equivalence of (23) for I a I < 1 to (23) for ac <- 1 or c > 1. On the other hand it is easy to show that a net of line conics (23') dual to (23) for ac <- 1 or > 1 is apolar to a net of point conics (23) with c > 1 and ac < 1 respectively. Hence (24) for 8 > 1 cannot be reduced to (23) for c > 1 or ac <- 1, and its cubic A = 0 shows it is not equivalent to (23) for Ic c < 1.

* H. J. S. Smith, Collected Works, Vol. II, pp. 524-540.



nets of conics in the real domain

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