Ganguli, Theory Of Plane Curves, Chapter 10

CHAPTER

X

RATIONAL TRANSFORMATIONS

198. In Chapter I, we have discussed particular methods of transformations, such as Reciprocation, Inversion, Projection, etc. and thereby from known properties of one curve deduced those of another derived from it by any of these processes. In the present chapter, however, we shall study the general principles of these methods, which consist chiefly in instituting a relation between any two points P and P' in the same plane, or in different planes lying in a common space. The case of different planes properly belongs to space-geometry, and consequently, without any reference to space, we shall regard the planes as superimposed one upon the other, so as forming a single plane. Thus we shall have to consider two figures consisting of points, lines, etc., in the same plane, instead of two figures in different planes. 199.

RATIONAL

AND BIRATIONAL

TRANSFORMATIONS:

If (or, y, z) and (:1:', y', e') be the homogeneous co-ordinates of two points P and P' in two planes (or same plane) a and {3 respecbively, then the transformation, expressed by the equations x' : y' : z'=f,(x, where

y, z) : L!», '1/, z) : fs(x,

y, z)

(1)

fll f., fs

are known functions of x, y, z, each of order a common factor, indicates that to any system of values of :1:, y, z, there corresponds a single system of values of a) ,y' ,e' ; but to a gi ven system of values of x', y', z' there will not, in general, correspond a single system but a finite number (D~l) of values of x, y, e. n (say) without

Thus, when fll f., fs are rational, there is an algebraic relation between (x, y, e) and (..(;',s'. z') expressed by the equations (1), which is called a Rational Troneformatio». 2

250

~HEORY OF PLANE CURVES

H, however, D=I,

i.e., if to a given system of values of w', y', z' there corresponds a single system of values of x, y, z, expressed byw : y : z=F1(w',

y', e') : F,(ai', y', z') : Fs("'" y', z') ...

(2)

.F i, Ft, Fs must also be rational and of order n, and when such mutual expression is possible, the relation is called a Birational TransfCYl'mation,.or Cremona Transformation, since it was first studied by Cremona. '*' 200.

LINEARTRANSFORMATIONS:

DEFINITION: Any transformation by which two figures are so related that any point and line of one correspond to one, and only one point and line respectively of the other, and conversely, is called a linear homographic transformation. Since the correspondence must be (1, 1), the required expressions cannot contain any radicals. Thus, where i..fl' fs are algebraic functions w, y, lIS, having no common factor.

and polynomials in

Since, to any straight line Zx'+my' +nz' =0 corresponds

which must be a right line for all values of l, m, n, the functions fllf.,fs must be linear in x, y, z ,

whence x, y, z can be expressed linearly in terms of ;v', y', z', Hence, assuming proper triangles of reference and the ratios of the implicit constants, we may write, without loss of generality, w' : y' : z'=:11 : y : z . • Cramona-Bologna Mam (2) Vol. 2, (1863) p. 621 and Vol. 5 (1864), p. 3, or, Giorn. di mat. Vol. 1 (1863), p. 305, Vol. 3 (1865), pp. 263 and 363.

RATIONAL

TRANSlIOR1.UTIONS

2in

The above transformation contains eight independent constants, and consequently, any four points (or lines) of one figure can be made to correspond to four points (or lines) in the other. Therefore this transformation leaves the cross- ratio of any four elements unaltered, so also the order or class of a curve remains unchanged. The method of projection explained in §. 12 is a particular case of linear homographic transformation, which involves only five constants, the vertex and axis of projection reducing the constants by three. 201.

COLLINEATION:

The linear transformation admits of a double interpretation. It may be regarded as a transformation of co-ordinates, or as a relation between the points of two different (or superimposed) planes. Let us imagine that the planes are infinitely near one another (or superimposed) and suppose that the points are referred to the same triangle of reference. If then P(x, y, z) and P'(u', y', z') represent points in the two planes respectively, the linear equationsx'=ax+by+cz y'=a'x+b'y+c'z z'=a"x+b"y+d'z

"I I

J

1)

establish between the points of the two planes an (1, 1) correspondence, which is called " linear affinity" or collineation,'*' such that to each point P of one plane corresponds a point P' (point-image of P) in the other. This relation, however, is not reciprocal, i.e., to the point P' does not, in general, correspond the same point P, but the corresponding point in the first plane is obtained by • This was first discussed by Mobius-Barycentric calculus (1827) p. 266-and afterwards by Magnus-Aufgaben and Lehrsitze aUB der ana.lytischen Geometrie, Berlin (1833).

252

THEORY OF PLANE CURVES

solving the equations (1), provided the determinant the co-efficients does not vanish. Thus,

D,.~=Av'

D,. of

+ By' + Cs'

D,.y=A'x'+B'y'+

C'z'

D,.z=A"J;' +B"y' +C"z' where A, A', A", etc. denote the minors of a, a', a", etc. 'I'hese formulee are evidently the same as for the transformation of co-ordinates, where the variables are the co-ordinates of the same point referred to two triangles of reference, while in the present case, they are the co-ordinates of two different points referred to the same triangle. Thus it will be seen that if P describes a curve in one plane, P' describes the corresponding curve of the same order in the other plane, and in particular, a straight line corresponds to a straight line, a range of points or pencil of lines corresponds to a projective range of points or pencil of lines respectively. 202.

COLLINEATION

TREATED

GEOMETRICALLY:

The geometric determination of collineation is contained in the following theorem : If to four points of one plane, no three of which are collinear, there correspond in the other four points, no three of which are collinear, the linear relation, i.e., collineation between the points of the two planes is completely determin ed. For, if we are given four pairs of corresponding points, the equations (1) of the preceding article are uniquely determined. The following geometrical construction is useful and interesting: To the line [oining

any two points

of

one

plane

II corresponds, in each case, the line joining the two corresponding points of the second plane l., and to the

RATIONAL

253

TRANSFORMATIONS

point of intersection of two lines in ~ 1 corresponds the point of intersection of the corresponding lines of ~.. Now, the four points of ~, determine six lines, which again determine three new points, namely, the diagonal points of the complete quadrilateral. To these then correspond the three points of the plane ~. obtained by a similar construction. If now the lines joining the three points of ~, are produced to meet the six sides of the complete quadrilateral, we obtain new points of ~, whose correspondents in ~. are constructed exactly in a similar manner. Thus, the entire plane will be covered over by a net-work of lines, and by the continuous crossing of the meshes of these nets, we shall obtain a point indefinitely near to a point of ~" If the corresponding nets are constructed in ~t' the respective collineation of the individual points of the two planes is established by the nets, and consequently the collin eation is completely determined."

20.3.

THE

DUALISTIC

TRANSFORMATION:

t

We have thus £ar considered only linear transformations in which a point corresponds to a point and a line to a line; but there are trans£ormations where a point corresponds to a curve, £or example, in Reciprocation a point corresponds to a. line and vice versa. Such a transformation is called "Skew Reciprocation" or " Linear Dualistic Trans£ormation." Reciprocation as described before is a special case o£ this more general linear dualistic trans£ormation and differs from it only by a linear trans£ormation.

a,

Let (x, y, z) be a system of point co-ordinates and 7], C) II. system of line co-ordinates in the same or in different planes. Then, a point in the first system corresponds to a.line in the second, if the co-ordinates of the point are proportional to «< Mobius-Bar. Cal. p. 273. For analytical treatment, the student is referred to Scott-Modern Analytioal Geometry § 223·226.

t

For a detailed account of the theory, Bee Salmon'! H. P. ourves or Scott-ibid-§ 253-256.

n. 332-342,

254

THEORY OF PLAu~ CURVES

the co-ordinates of the line, i.e., x: y : z=t : YJ: t and consequently. to any line lx+my+nz=O corresponds the point l~+mYJ+nt=O. In the general dualistic transformation, however, the co-ordinates of a line are functions of the co-ordinates of the corresponding point, and the transformation is linear when thole functions are linear. Thus,

~=a\~+bly+CIZ

"')

I

I

YJ=a.x+b.y+c.z t=asx+b.y+csz

(1)

)

where to a point (x, y, z) there corresponds the line U, Tf. ~) in the same or different planes. 1£,however, we putx': y' : z'=al;c+b1y+C1Z

: a,x+b.y+c.z

: asx+bsY+ca!:

i.e., if a point (x', y', z') is obtained corresponding to the point (:1', Y, z) by a linear transformation, there is a. correspondence between the point (:v', y', z') and the line (e, YJ,t), and we have, as stated above, the following relations: x': y': z'=~: Tf : C This shows that the systems (x', y', e') and (~, Tf, C) are reciprocal with respect to the auxiliary conic ;c. +y' +z· =0. Thus, the linear dualistic transformation differB from the interchange of point and line co-ordinates only by a collineation. 204. 1£ we solve the ' equations (1) of the preceding article for x, 3/,z, we obtain the following relations: ~x=Al~+A.Tf+Ast

I

J where AI' Ell ete., are the minors of all bll determinant 1:1 of the co-efficients in (1).

(2)

etc.,

III

the

RATIONAL

255

TRANSFORMATIONS

The system (2) is said to be "dual"

of the system (1)

Now consider the point (m', y', z') in the first system. Its corresponding line in the second system is then ::c'(alx+b1y+C1Z)

+y'(a.x+blly+VllZ) +z'(asx+bsY+c

or

m(al:!:'

+ a,y'

+nsz') +y(b1x'+b,y'

sz)=O

+ bsz')

+z(c1:r'+CSY'+vsz')=0

(3)

The equation (3) expresses the relation between any point (x', y', z') of the first system and any point (x, y, z) 0:1 a.corresponding line of the dual system. If now (x, y, z) is considered fixed and (x' y' z') variable, we have for th", line of the first system, corresponding to any point of the second, x(a1x' +bly' +v1z') +y(allx' +b.y' + c1z') +z(asx'+bsY'+csz')=O

(4)

The lines (3) aud (4) do not, in general, coincide j hence, in the general dualistic transformation, every point has a different corresponding line, according as the point is regarded as belonging to the first or to the second system. The conditions that the lines (3) and (4) should coincide give three values of (x', y', z'). Hence there are three points in the plane associated with their corresponding lines in a definite way, regardless of the system to which they belong. One of these points, however, is given byx' : y' : z'=c. -bs

:

as -VI: b1

-all

and the other two are real or imaginarj . The two lines (3) and (4) will coincide for all points of the plane, if for all values of x', y', z', we have al:c'+b1y'+C1Z'

: allx'+b.y'+caz':

=a1x'+aty'+asz': which requires

as~,'+bsY'+vsz'

blX'+b2y'+bs~':

c1x'+c,y'+osre'

256

THEORY

Hence, forms-

the

OF PLANE CURVES

transformation

formuloo reduce

to the

~=ax+hy+gz 'YJ=hx+by+fz '=gx+fy+cz This shows that the point and the line are associated with each other as pole and polar with regard to the general conic I'.tXa

+bya +cz!l +2fyz+2gzm+2hxy=0

Thus it is seen that in case of reciprocals with regard to a conic, the same line corresponds to a point, whether that point be considered as belonging to the first or the second system.

205.

POLE

AND POLAR CONICS :

The case of a point lying on its corresponding interesting and deserves consideration.

line

Since a point (x, y, z) lies on its corresponding ~1;+17Y+'z=O, the locus of such points is obviously(atx+b1y+C1Z),C+

IS

line

(a!lx+bsY+c2z)y + (asx+bsY+csz)z=O

(1) and this is th s same conic, whether the point be considered as belonging to the first or to the second system, and is caned the "Pole Gonic." On the other hand, the envelope of lines which pass through their corresponding points is a conic called the Polar Gonic. the

The co-ordinates of the point are expressed in terms of co-ordinates of their corresponding lines by the

'!"'"

257

RATION AL l'RANSFOIDIA'fIOXS

equations (2) of § 204.

Therefore

the required

envelope is

(A.t+.A 217+A s,)t+ (B.t+ B, 17+Bs')17 + (C .t+ C, 17+Cs')'=O

where Au B 11 C 1> etc., have the significance as in § 204. Conversely, the same pole and polar conics will be obtained, if the points of the second system correspond to the lines of the first system. The pole and polar conics have double contact, the intersection of the common tangents being the point Cbs-c.), (c 1 -as), (a. -b. ). The chord of contact is found to be the line (Bs-C.), (C. -As), (A.-B,), It will identical, if 206.

be seen that b1 bg

=a.,

QUDRIC

the pole and polar and C1 =as. '*'

conics are

=c.

I~VERsION

:

The process of circular inversion has already been described in § 15 ; but in this section will be described a more general process in which a point corresponds to a. point, while a line, in general, corresponds to a conic. This transformation can easily be effected by a geometrical construction and was given by Dr. Hirst.t In this process a fixed point is taken as origin and a fixed fundamental conic as "base." Points collinear with the origin and conjugate with respect to the base are said to be inverse. If the base is a circle and the origin its centre, the points are ordinary inverse points with regard to the circle. It' is, in fact, the circular inversion generalised and is called Quadric Inversion, Scott-loco eit., § 256. t Hirst-" all the Quad"ie [HL'e1'sio" of Plane R. 80c. of London, Vol. 14 (1865), pp. 91·106. JI

33

o
Proc. of the

258

THEORY

OF

PLANE

CURVES

Let C be the fixed origin and S the base-conic, C draw a transversal R cutting the base in the points Q, R. Then, if P, \ P' are points on the transversal, such that (PP"QR) -is harmonic, then P and P' are inverse points,

Through

\

Thus, to determine the inverse of a point P, we have to find the point P', where CP intersects the polar line of P with regard to S, It follows hence that to any position of P corresponds a single definite position of P', and vice 1'PJ'sa. If P traces out a locus ~, P' will trace out a locus ~', and ~' is said to be derived from ~ by quadric inversion. 207.

ANALYTICAL

TREATMENT:

Let CA and CB be the tangents to the base, and choose 4BC ,a.'lthe triangle of reference. Then the equation of the base-conic may be written as(1) Let Cr, y, z) and (,e', y', z') be the co-ordinates of P and P' respectively. Now, the polar line of P' isxy'+y,l;'-2zz'=0

(2)

,/'y'-fJa'y=O

(3)

and the line CP' is whence

;t; :

y : z=x' : y' : x'y'/z' =x'z': y'z': a;'y'=

.!...:.!..: ~.. y'

(4)

;);' z'

• These formulas are deduced on the supposition that the base is 1\ proper conic and the points A, B, C are distinct. Modifications are necesaary when the base is a degenerate conic, and two or more of A, B, C are coincident.

11"'"""

RATIONAL

259

TRANSFORMATIONS

Iff (.r, y, z)=o is the locus of P, the locus of P' by the equation-

1( y1" x~"

IS

given

;z )=0

Applying the linear transformation x': y' : a'=y' : x' : z' i.e., interchanging the vertices A and B of the triangle, we may express the result (4) in a more symmetrical form, and the locus of P' is now given by-·

1( .!., -.!., .!.)=0 y'

:1/

Z'

The formuloo of transformation can, however, be written under the form of bilinear relations=xie'=l, 208.

QUADRIC

z.:;'=1.

yy'=l,

h;VERSION

AS

RA'l'IOKAL

Let the formulee of transformation the formx': y': .:;'=1, :1. :/3 w here I" in e y, e.

I 13 II'

are rational functions

TRA."NSFORMAl'ION.:

in § 199 be put

into

of the second degree

To the lines x'=O, y'=O, z'=O will then correspond the three conics 1, =0,1.=0,13 and in general, to a curve of erder n corresponds one of order 2n, obtained by putting IIIIi> Is respectively for x, y, z in the equation of the n-ic.

=0 ;

The simplest case presents itself in the formx': y' : z'=x' To

the

line

: y' : ~'.

l.c+ lI~y+nz=O

corresponds

the

conic

l.v~+1n .vY+n .v~=0 inscribed in the triangle xyz. Similarly, to a conic there corresponds a curve of the fourth order, and so on. It is to be noticed, however, that this transformation is not birational in general. For, although x', y', z' are

260

1'HI!lOltY

o~· PLA.NE

CURVES

expressed rationally in terms of (x, y, z), the latter are given in terms of :-e', y', z' by the equations-

i, _ i. _ t, re' -

11 - ;;-

which are 110t rational and represent comes having four common points; and consequently, corresponding to any position of (.:1)', y', z') there are foul' positions of (x, y, z) But if illI..f., have one common point, since it is independent of the position of (.-e', y', z'), it may be ignored, and to any position of (x', y', z') there will correspond only three points (x, y, z). Similacly, if t.. f., f. have two points common, to any position of (e', y', z') will correspond only two positions of (,r, y, z). Finally, if i.. f., fs have three common points, the conics have, besides the three common points, only one other common point, and to any position of (x', y', z') there corresponds only a single position of (.ll, y, z), and vice versa, and the transformation is birational. Since it is perfectly legitimate to take three conics of the form li, +mf. +vi, instead of t.. t.. fs' the three line-pairs joining each of the three common points to the other two may be taken for L. f., fs' and the formulro become: ,c:y:z=y'z':z',JJ': x'y' and

x'; y': z'=yz

: zx : xy.

Hence, the quadric inversion is only a particular case of the general birational transformation. Other special cases may arise from the coincidence of two or more of the common points. Thus, when two points coincide, we may take the COIll' mon tangent as the side y=O and the point (z, x) as the third common point. 'I'he equations of i., f., fs will be of the form ax' +2fy.;+2hxy=O. Taking x', yz, ~;y as the three conics, the formulee become x' : y' : z' =.cy : x' : yz

and

x :y :

Z=,1;'

y' : x" : 'J'z'

RATIONAL

26t

TIUNSFOR~L\'TIONS

Similarly, when the three points coincide, the equabious of ill t; is will be of the formby' +2h,fy+2i(yz-m~' Hence, taking y', iVY,Y:-lnx' formulae become:

for

)=0

i" t..I, respectively, the

:/: y': z'=,ry : y' : yz-m,v' and

il~:

y : z=:e'y' : y"

:

y'zl-m;l;'2

209. It has been stated ill § 206 that the inverse of a point P is a sillgle definite point P'. But there are excep, tional positions of P for which the inverse point P' is not in general determinate. The inverse P' is iudeterminate, if (i)

P is at C, P' is any point on AB,

(ii)

P is at B, P' is any point on BC,

(iii)

P is at A, P' is any point on CA,

(iv)

P is any point on AB, P' is at C,

(v) P is any point on BC, P' is at B, (vi)

P is any point on CA, pi is at A.

Hence it appears that if P is at any vertex or on any side of ABC, the ordinary laws of correspondence do not apply. 210.

THE

bHRSE

OF A STl(AI({UT

LI:iE

:

'I'he inverse of the straight line l,v+my+nz=O i" the locus defined by the equation (§ 207)

(1)

l/y+m/x+n/z=O which evidently triangle ABC.

represents

It

(2)

conic circurnsci-ibing

the

The following special case" are to ue noted : (i) If the line (1) passes through C, n=O and it own inverse,

IS

its

(ii) If the line passes through A, l::::::O and the inverse is the line 'mz+nx=O, which passes through B.

262

THEORY OF PLANE CURVES

(iii)

If

the

line passes

through

B, its equation

is

l.c+nz=O, and the inverse is the line h+ny=O through

A. Thus, it is seen that the inverse of a right line is, in general, a conic through A, B, 0; but in special cases it is a right line.

211.

PROPER

INVERSE:

Let 0 be the pole of a line meeting the base-conic in Q and R. Then the inverse of the line is the conic ABOOQR. But if the line passes through C, then the pole 0 lies on AB, and the conic has three points on AB, i.e., the conic consists of AB and the given line OQR. The line AB presents here as a part of the inverse simply because the inverse of 0 is indeterminate, being any point on AB. When the line passes through A or B and meets the baseconic in another point K, the inverse is a degenerate conic composed of OA, BK or OB, AK; for the pole 0 is now on CA or OB. Hence the points 0, A, 0 on the conic are accounted for by the line CA or OB and the remaining points B or A and K give the other line. Similarly, if a curve passes through A (or B), the line OA (or OB) presents itself as part of the inverse. These factors, however, occurring in the inverse are not regarded as forming the proper inverse and are rejected. The remaining factor gives the proper inverse. E»,

Consider the conic fyz + gz.;; + hilly =0

The inverse of this, by the formnlre of § 208, isf""'y'z'

l.e.,

+gw'y"z'

./y'z'U~'

=0

+gy' +hz')=O

i.e., the sides AB, BC, CA and another x"y'z',

+ hw'y'z"

the propel' inverse is the line

line. fill

Hence, rejecting

+ g-y + h~O.

the factor

•...

212.

THF.

I:;"EllsE

Of'

THE

LTXE

AT

hrIXITY:

The equation of the line at infinity being a:l+by+cz=O, its inverse, by the formulte of § 207, is the conic(1)

az.c+byz+wy=O which is evidently triangle.

a conic circumseribiug

the fundamental

The pole 0 now becomes the centre of the base-conic, the points Q, R (§ 211) are the points at infinity on tha same. The polar of the point 0 at infinity on AB passe;;; through 0 and the line 00 is parallel to AB. 'I'herefore the inverse of 0 is consecutive to 0 on the line 00, or in other words, the tangent to the inverse (1) at 0 is parallel to AB. Thus 00 is the diameter conjugate to AB and the tangent at 0 is parallel to AB. It may be noticed further that if the line drawn through A parallel to OB meets the base-conic III K, BK is the tangent at B.

Similarly, the tangent at A may be constructed. The inverse to the line at infinity is represented in the figure by the dotted Iine. 213.

INVERSIOX

OF

Let ~ be the curve

SPECIAL

POIXTS

OX

and ~' its Inverse.

A CCRYE

The following

special points are to be noticed: If ~ meets AB in P, ~' touches OP at O. For, the inverse ofP is on OP by definition, and as the polar of P passes through C, the inverse is indefinitely near to C on CPo

THEORY

01<' PLANE

OURVES

Hence CP is the tangent at C, Similarly, if ~ Cuts AB at points, other than A, B, the inverse has an n-ple point at. C. For instance, if the curve cuts AB at two points P 1 and p" there is a node at C with CP 1 and CP. as tangents. When F 1 and P, become consecutive points on ~, i.e. AB is a tangent to ~ at P " the two tangents CPI' CP. coincitls. and there is a cusp on ~' at C, with CP 1 as the cnspidal tangent.

1'1

To the tangent to ~ at P corresponds the conic osculating ~' !JotC and passing through A, and B. For, if Q is a point on ~ consecutive to P, the inverse Q'is consecutive to C on S' and the limit,ing position of CP'Q' (i,e" CP) is the tangent to S' at C, Now the inverse of PQ is a conic through A, C, B, Q', touching ~' at C (Q' being the point on ~' inverse to Q), If then Q approaches P, PQ becomes the tangent to ~ at P, and the conic ABCQ' becomes the osculating conic of ~' at C, If, however, CP be the tangent to ~ at P, the inverse hail CP as an inflexional tangent, Again, if P and P' are inverse points, as CP gradually turns about C and ultimately coincides with CA, P' approaches and ultimately coincides with A, Renee AP' becomes the tangent to the inverse of the locus of p. But AP' and BP intersect at a point H on the curve, Thus, if ~ meets CA at P, the inverse ~' touches at A the line AH, i,e. the line through A corresponding to BP, Similarly, if ~ meets CB, Hence, to c every intersection of ~ with CA (or CB) there corresponds a branch of ~' through A (or B), and conversely.

265

RA'l'01NAL TRANSFOIOrATIONS

Thus, if an n-ic ~ cuts OA (or OB) in n points, there is an n-ple point on ~' at A (or B). When ~ touches OA (or OB), the two tangents to :S' at A (or B) coincide, and consequently, A (or B) is a cusp on ~'. 214.

EFE'ECTR

OF INVEHRTON

O:-! SINGULAIlITIES

:

Prom what has been said above, it follows that, in general, an ordinary point inverts into an ordinary point. But if three consecutive points at the given point and the three fundamental points A, B, 0 lie on a conic, their inverses are collinear on the inverse curve, and there is, therefore, an inflexion on the mverse. Thus the inverse of an ordinary point may be either an ordinary point or an inflexion. Similarly, an inflexion is inverted into an ordinary point, unless the inflexional tangent passes through any of the fundamental points. Again, a double point, in genoral, inverts into a double point of the same nature ; and consecutive double points invert into consecutive double points, but the appearance may be slightly altered. Thus, a tacnode inverts into a tacnode, an oscnode inverts into an oscnode, but if the three nodes are initially collinear, the oscnode on inversion loses this property, unless the tangent passes through a fundamental point. Similarly, a curved oscnode may be straight on inversion. In the case of a bitangent, the inverse becomes a COIlIC having double contact with the inverse, unless the bitangent passes through a fundamental point, and then it inverts into a bitangent. Conversely, a bitangent may be gained on mveraion. Thus it follows that as regards points and lines, not belonging to the fundamental triangle, the point singularities of a curve and its inverse are the same, but line singulari ties are changed. Hence inversion can conveniently be used for analysing singularities on curves. 34

14

266

THEORY

215.

E~'FECTS

OJ,' PLANE

OF INVERSION

CURVES

01'1 A CURVE:*

Let the curve ~ be an n-ic having a q-plepoint at A, an r-pie point at B and an s-ple point at C. Then, ~ meets AB, BC, CA respectively at n-q-1', n-j'-s, and n-q-s other points. The inverse ~' has therefore an (n-q-r)-ple, (n-1'-s)-ple and (n-q-s)-ple point respectively at

C, A, B. Again, the q intersections of the tangents at A to ~ with BC are points on ~'. Similarly,~' meets CA and AB in r and s points respectively other than A, B, C. Since ~' meets AB in {(n-r-s)+(n-q-s)+s} i.e., (2n-q-1'-S) points, ~' is of order (2n-q-r-s). Thus, the inverse of an n-ic ~ with a q-ple point at A, an r-ple point at B and an s-plepoint at C is a curve ~', of order (2n -q-r-s), with an (n-r-s)-ple point at A, an (n-q-s)-ple point at B and an (n-q-1')-ple point at C. n'='2n-q-1'-S,

Putting

q'=n-1'-s, s'=n-q-1',

we may establish a reciprocal relation between ~ and ~'. Thus,

n=2n'-q'-r'-s'

q=n'-1,1-s'

r=n'-g'-8'

s=n'-g'-r'

We shall now show that, in general, the deficiencies of the two curves ~ and ~' are the same. Since a q-ple point is equivalent to %q(q-l) the deficiency P of the first curve ~ is given byp=H (n-l)(n-2)

• Effects discuased

of

inversion

in Chap. XIII.

-q(I}-I)

on

higher

-;{r-l)

singular

nodes,

-s(s-I)}

points

will be fnlly

RATIONAL

and

267

TRANSFORMATIONS

p'=H(n'-I)(1t'-2)-q'(q'-I)-1"(T'-1) =H (2n-'l- r-s-l)(21l-'1-r-s-2) - (n-r-s)(n-

-s'(s'-I)}

r-s-l)-(n-'1-s)(n-q-s-l) -(n-'1-r)(n-'1-r-l)

=H (n-l)(n-2)

-'1(q-1) -r(r-l)

}

-8(8-1)}

=ri.e., the deficiency of a curve transformation. 216.

ApPLICATION

OP

unaltered

lS

QuADRIC

by quadric

INVERSION:

'I'ha process of quadric inversion affords a very convenient method of investigating the properties of one curve from known properties of another. The following examples will illustrate the method. E1).

triangle

1.

Oonsider a

in three

conic

cutting

the

sides

of

the

fundamental

pairs of points.

a"" + by' + cz' + 2Jyz + 2gz;!\ + 2h",y=O

Let

be the equation of the conic cutting

(1)

the sides BO, OA and AB in the

pairs of points A" A, ; Bll B, and 0,. 0, respectively. The inverse of (1) is the quartic curveal»? + bjy'

+ c/z' + 2f/yz + 2g!o>; + 2h/?'!7J=O

The points A, B, 0 are evidently

nodes (§ 213) on the

the lines All.." A A, ; BB" BB, and 00" But these lines all touch one and inverted

into themselves.

The nod"l

tangents

the

00, same

conic,

Hence we have the theorem

oJ

It

irinodal. quavtic

curve

with

as nodal tangents. and

they

are

:-

touch. on~ and

the same

conic, Again, the pairs of tangents inverted

into themselves,

drawn from A, B, 0 to the conic are also

and their inverses are tangents

quartic. :. Hence, the s;?! tangents drmuH [ron: the three

to the trinodal

nodes to a trinodal

quartic touch one and the sume e011ic. Finally,

the' four bitangents

of a trinodal qnartic are obtained

sa-me process from the fact that through

three

given points,

by the

there

call

268

THEORY

be drawn four conics

having

the inflexional tangents

OF l'L.-\NE doable

are obtained

CURVES

contact with a given One, while from thc fact

that

through

given points can he drawn six conics, having three-pointic

three

contact

with

a given conic. E», 2

Show

that the

three

cuspidal

tangents

of a

bricuspidal

quartic are concurrent. If in

Ell!.

coincident,

I, the pairs

of

points

Au A,;

then the lines joining the vertices

of the inscribed conic with

the opposite

sides

inverse of the conic is evidently a triouspidal at the

vertices,

and the

joining

lines

which are again the cuspidal tangents,

B"

E,;

C" C,

are

to the points of contact are

concurrent.

quartic,

are

having

inverted

whence the

The

the cuaps

into themselves,

truth of the theorem

follows. E», 3.

Through

any

point

call be drawn two

trinodal quartic and passing through This follows two tangents

immediately

from

can be drawn to

becomes the trinodal

quartic

the fact that from

any

and

lines

touching a

its nodes.

couic.

On

the two

any point only

inversion

tangents

invert

the

conic

into two

conics through the nodes touchiug the quartic, and these evidently

pass

through the inverse of the given point.

217.

CIRCULAR

hiVEHSION

:

*

A particular case of quadric inversion IS the transformation by reciprocal radii the principles of which have been explained in § 15. If we take k=1, the relations between the rectangular co-ordinates of P and P' are-

X'=

y'=

_!J_.

a;2+y"

::1"

and

~-u=---- , ,e'2 +y"

1

whence

a/ +iy'=

Writing

X:Y

and

X' : Y' : 7,' =.l"+iy' : a!-iy':

«<

Moutard-Sn,.

;lJ-iy'

7,=.c-iy

Z" tru usfornuuion.

Nouv. Ann. t; 3(2) (1864), pp. 306-30'J.

P""

; x+iy

: 1 1

myuns vee/en!"s "eeiproques-

RATiOS

we obtain the relations

Y' : ZI

X':

or in other words, the transformation The

geometrical

will be best

269

A L TItAXSFORMATIONS

significnnce

understood

from

= YZ

: ZX : XY,

is a quadric of these

the

inversion.

transformations of § 206, if we

figure

consider that the points A and B are circular points at infinity, so that the base-conic

S now becomes a circle with centre

and P, pI are in verse

points

with

respect to the circle.

fact, we have taken the circular lines through the

line

the origin and

at infinity as the sidos of the triangle

Hence, circular

inversion

transformation,

and

generalisation

by projection

vVe may established

is a particular quardric

deduce

C In

of reference.

case of quadric

transformation

is

a

of the process of inversion

a number

in the preceding

of theorems

articles.

circle is a circle, that of u straight

from the results

Thus, the inverse of a line is a circle

through

C, and so on. The

inverse

of a conic, in general,

the nodes being the origin infinity.

and

is a trinodal

the circular

quartic, points

at

If the origin be the focus of the conic, the inverse

is a limacon

, if the origin be on the conic, the

nodal circular An

cubic, the origin

osculating

osculating through

circle

circle of the

to

Spr;CIAL

QUADRIC

inverse

is a

being the node. a curve

inverse,

thc origin, the inverse

218. In

(C)

will

invert

into

an

but when the circle passes is an inflexional

tangent.

TIIANSFORJIATlOIlS:

§ 206 we have discussed the general case of quadric

transform/ltioll; positions

of the

but

special

points

Base must he considered

A,

cases B,

arismg

from

special

C or the

nature

of the

for a systematic

treatment

of the

subject. Case 1:

One special case presents

A and B coincide (§ 208.)

itself when the points

270

THliORY

01" PLANE

CURVES

III this case, any line through C, the line CA, and the polar of C are taken as the A sides of the triangle of reference. The base-conic IS now a pair of lines, whose equation, by a proper choice of coordinates, can be put as ;t·-z"=O. The polar of c any point P'(x',y',o') is the line X.l/-zz'=O and CP' is ~ty'-x'y=O, whence the inverse point P is given by-

z : y : z=:J.l' : y' : f»"/z'=,lJ'Z' : y'z' : :1:"

~' : y' : z'=za; : yz : a;2=.c : y : x'iz. Hence, if l(x,

y, z)=O be the locus of P, that

of P'

IS

It is to be noticed that this transformation is equivalent to the three transformations in succession, in which the pole is the point C(O, 0, 1) and the bases are the three conics x"-xy+z'=O,

m'-y'+z2=0

and x·+xY-Z2=0.

Case II: When the three points A, B, C coincide at C, any chord through C, the tangent at C and the tangent at the other extremity of the chord are taken as the sides of the triangle of reference. 1n

The base-conic is now of the form 2yz-mx' is at our disposal.

=0, where

The polar of P'(.()', y', z') is y'z+yz'-ma:x'=O and CP' is the line xy'-x'y=O, whence pea', y, z) is given byx : y : ~= ..c'y' : y" : m,lJ"-y'z' and

x' : y' : ,.'=:cy : y2 : mx' -yz

(§ 208)

RATIONAL

219.

271

'l'RANSFORMATIONR

TRANSFORMATION:

KOETHER'S

We have so long used the same triangle of reference for the our're and its transform; but if we take CBA instead of ABC a·s the triangle of reference for the transformed curve, this amounts simply to the interchange of IV and z in the transformed equation. Hence, the curve f(x, y, z)=O is transformed into f(z, y, z·/.c)=O Writing this equation in the form f(x/z, we see that in the Cartesian system, the curve transformed into f(x, 'l'Y) =0. Hence, the £ormulre of transformation

f

ay/z2, 1)=0 (:c, y)=O IS

become-

x : y : 1=:t' : x' y' : 1, i.e., a=.c', Y='JJ'Y' and y'=yj.l',

a/=,r.

'I'his form of transformation was given by Noether * and was used by Newton and Cramer for the analysis of higher singularities. A series of successi vp transformations are at times required for complete analysis. E:D.

1. Examine the singularity

at 0 on the curve y'z=w'.

The

inverse

is x"y"=lJ",

the proper

by

inverse

the is y"

forrnuleo

and

consequently

= a;'2,

which has at 0' (x', '!I') a cusp with

JII'

=0 for tangent.

Oonsequently, the singularity at 0 on the original triple

point

apparent

form is

inflexion, but

curve is a

(§ 213) the

that

whose of an

penultimate

form is shown as in the figure.

E»:

2. Verify the following:

(i) The inverse of a line through C or B is a line through

C or B.

(i..:) The inverse of a line is a conic through 0 touching AB

at A.

E z, 3.

Examine the singnlarity

• Noether-Uber

die siugularen

Function und die singularen

Pnnktc

Ann. Bd 9 (1876), pp. 166.182.

at the origin on the curve y'

Wertsysteme

= ,,,'.

einer algebraischsn

einer algebraischen

Curve-e-Math

'-. 272

THEORY

220. As

CrmMONA

in § 199,

z'=I, :I. : 13

is

system it is not, in general, the form x : y : z=F/, rational

functions

Luigi

general

has

possible

to deduce

investigated

of

1!", are

F/2l

the conditions

under

are possible.

we are given

the corresponding

F'"

this

another

in ,r/, y', z',

expressions

If in the one system

transformntion

i.e., from

: F'. : F', where

*

Cromona

intersections

the

not birational,

(polynomials)

which such mutual

then

CURVE:;

CONDITIONS

explained

;r/: y':

01" PLAN8

;1;'

y' : zl=a : b : c,

:

points in the other

are given as t.he

of the curves-

(1) Now, smce in (x, y, z), the if

11' t., 13

will point

number

are polynomials of the nth degree of intersections will be n2. But

in p common

intersect

evidently

remaining

i.. t.,/3 pass

tbrough

points,

the

curves

these common points,

n" -p points will then

correspond

to

(1)

and the

the

given

(a, b, c).

When p =n" -1, the curves will intersect only in one variable point, that is to say, all but one intersections of the curves

(1) being known, the co-ordinates

ing point will be determinate, of (a, b, c.), i.e. of

and

we see that

tion, if the three

rational

y', z', and we shall have-

,t/,

x : y : z= F/, Hence

of the only remain-

thus

this

curves

: F'" : F/3

will be a birational

I" 1.,13

functions

,

transforma-

bave n2 -1 common points

of intersection. " Cremona

has thoroughly

theory is due to him-see dellefig'u"e piane-Mem. For applications

his

investigated

these

conditions

and the

Memoir Bulle transjorma zioue qeometriclie

di Bologna, Vol. II (1863). and Vol. V (1865), of Oremona transformations,

see a paper by A. B.

Coble in the Bull. of the Am. Math, Soc., VoL 28 (1922), pp. 329-364, to which is appended a number of important

references

on the subject.

RATIONAL

This

agam

i.. i ; i,

be cubic

is

sufficient

:t

curves

having'

cond ition. eight

common

no variable

intersection.

But

if

points,

here again,

suppose that the en hies have on e node common intersect

For,

have a ninth point common, and consequently

they certainly there

not

IS

273

TIUNSFORMATLONS

in four

other

ordinary

points,

and

given a node is equivalent

to three conditions,

intersections

and therefore,

are

known,

ditions are required

to determine

to

if we

all,

they

since to be

seven

of their

only two more con-

any curve

afl +bf2 +cfs =0

Now, the common points are equivalent

to eight intersections,

the

one

node

counting'

obtained In

as fonr.

corresponding'

fact,

the

ponds to the therefore

Hence,

variable

point is

to the giyen point.

system

of curves

system of lines

be perfectly

a,t!

general

afl+bf2+C!3=Ocorres-

+ by' + c:' =0, *

and

should

and must not be determinate

except when a, : b : c are given, which is equivalent

to two

conditions. Therefore,

the

i.. f., i, must

number

ditions determining 221. n being

From the above considerations greater

and no variable etc.,

U1

ordinary

common,

intersections less

by

to be satisfied by

than

two,

i..

of con-

points, such

U2

that

the number

have

n2-1 point

If, however, points, are

necessary

of order n, we obtain on" remaining

U3

thus

t.. f., fa

triple points, to n2-1

equivalent

of conditions

implied

to determine

variable

that

common

cannot

another

double these

it follows then

f2,f3

have

point of intersection.

and the number

2 than

number

a curve of order n.

common points, for then they have

of conditions

be at least two less than the

bit

a curve

point of inter-

section corresponding to the given point and the transformation becomes rational. Since, to be given an r-ple point on a curve is equivalent to {1' (r+l)

conditions

• See Montesano

=

and two n-/c8

intersect

in r2

points

Napoli Rendi, Vol. 11(:1) (1905), p. 25!l.

274

THEORY OF PLANE CURVES

at an I'·ple point on each, we may state the above two conditions as follow: a, and

+22.:>., +3'a. + ...

a, +3a.+6a.+

+1·2a,. =n2 -1

(1)

... +-}?·(?·+1)ar=tn(n+3)-2

(2)

Combining (1) and (2), we may state the second condition a simpler form :

III

a, +2a. +3a.

+

+ra, =3(n-1)

(2')

Positive integral values of at> a., ... satisfying the equations (1) and (2') will then determine the transfermations, provided the number of higher singularities assumed to belong to the curves does not exceed the proper limit. Cremona has tabulated all the admissible solutions, for cases up to n=10, of the above equations, which are often referred to as "Cremona conditions." For a detailed discussion of the theory, the student. referred to Cremona's Memoir and to Cayley's paper above referred to and also to his paper--" Note on the theory of the rational transformation between two planes and of special system of points, Coil. Works, Vol. VII, pp. 253-55. IS

222. THEOREM: EveTY Cremona tmnsformation may be reduced to a number of successive quad?ic tmnsfm'mation;* and converselu, each birationai transformation of a plane into another ~8 equivalent to a finite number of quadric transformations, Consider the transformation: x' : y' : «=t, :f. : f3' where f" f., f. are curves (polynomials) of order n, having a, ordinary points, a. double points, etc., in common. • Vide

Prof.

Cayley's paper-"

On

the

Rational

Transformation

between two spaces," Coil. Works, Vol. VII, pp. 189.240. For other proofs see Noet.her,

Ueber Flachen

3( 1871), pp. 161.227, Segre, Un'osservazione Vol. 36 (1901),

pp. 645·651 and

Castelnuovo,

eto.," ibid, Vol. 36 (1901), pp.861.874.

etc.,

relativa,

Math. Ann. Bd.

etc., Torino Atti,

" Le transformazioni,

RATIONAL

275

TRANSFORMATIONS

Then, there are three of these points '* (one

q-ple,

r-ple

one

and one s-ple, say) the sum of whose orders exceeds n, so that

N ow, take those quadric

three

transformation.

points

quadric

transformation,

may be further

reduced.

ultimately

obtain

'rhus

Cremona

the

of successive

unaltered

223.

lines

by any Cremona

and

U)lAllmRIW

13YCR";.\IOXA

the line corresponding'

in k points a

k-ple

all

point.

becomes a kr-ple

no multiple the principal

n-ics.

to a number

unaltered it

remains

transformation.

F =0 be a curve of order lc and

transform

curve

we shall

to the

consequently

A in common,

If now

In

TH.AN:;r'ORMATJON :

let

us

IIII., I, to A

corresponding

to

general,

point.

apply

will

meet

A, which

any

the

have a point

of

the

the then r-ple

Hence, if the given curve has

points, the transform

will

have

none

except at

points, i.e., at the common points of 11' f. and

Thus, the degree of the ponding'

way,

is reduced

to this curve.

becomes

in this

corresponding

transformation

points

of this

that the deficiency remains

transformation,

DEFICI&NCY

Let

and by a

degree

inversions.

But, it was proved quadric

the

transformation

less than

is reduced,

Proceeding

rig'ht

quadric

points of a

which is certainly

i.e., the degree of the given curve

second

by

principal

Then the degree of the transformed

curve is, by § 215, 2n-q-,·-s, 11.,

as

maximum

Hnk-l)(nk-2).

number

is nk and

transform of double

Also the multiple

points are equivalent

~alk(k-l)+!a

points,

toa ' 2k(2k-l)+

Viele

Salmon,

corres-

as usual,

is

points at the principal

... +~a,.h(rk-l)

or

*

the

is.

H. P. Curves,

§ 356.

27G

THEORY

OF

PLANE CURVES

which, by equations (1) and (2') of § 221, is equal to ~k'(n2 -1)-~k.3(n-l)

=tk2 (n' -1) -ik(n-l).

Hence, the deficiency of the transform becomeRHnk-l)(nk-2)-

{~k' (n2 -1) -~k(n-l)}

=Hk-l)(k-2), the same as that of the original cnrve. If, however, the original curve has other multiple points, the transform will have corresponding multiple points of the same order and the deficiency will remain unaltered (§ 222). Further modification is necessary when the original curve passes through any of the principal points. Again, when the curve F=O passes through the principal points au a., ... the degree of the transform will beN=:nk-u,-2u2-3a3 224.Rn:~IANN

•••

-1'(1,.

TRANSFOIDIA'l'ION:

We have hitherto considered the Cremona transformations which are birational with regard to points of the whole plane, under certain conditions. But there are other transformations that are birational * only as regards the points of a curve of the plane, but no such conditions are uecessary ill this case. Let F=O be a given curve and apply the transformation

where t, 12,fa are homogeneous functions of the nth degree in ";, y, z, not necessarily satisfying Cremona's conditions, which have no common factor. The above equations are not by themselves sufficient to express .e,y, z rationally * For the birational

transformation

of a curve into itself, see H. A.

Schwarz, Crelle, Bd. 87 (1875), p. 189, also F. Klein, Theorie del' atgebraischeu

Functionen

(1882), p. 64.

Uber Riemann's

RATIONAL

277

TRANS1<'ORUATlONS

terms of x', y', Zl ; but when they are combined with the equation F=O of the curve, it is possible to express x, y, z rationally in terms of x', y', Zl by the following equations:

in

where cp'1> cp'., cp's are homogeneous functions of the same degree 'It, without a common factor.

x', y', z

111

In fact, when x, y, z are eliminated between the equations of transformation and F=O, we obtain an equation l!"=O, which is the condition for the co-existence of the system of equations, When this condition is satisfied, z, ?I,;; can be determined rationally in terms of x', y', ",',* DEFIN[1'ION: An algebraic transformation that IS birational as regards the points of two curves but not as regards t.he points of the whole plane is called a Riemann Transjormatiou,

EJ;. 1.

Consider

the two curves-

z (y' +w')=x' both of which formation

have

the

and

deficiency zero.

which will transform

y

Z'2 '=W'3

We can

determine

a trans-

the two curves one into the other.

Any point on the first can be expressed asill

:

y : z=(1 +A')

: A(1 +A')

and any point on the second is given by~' : y' : z'=l\.'z : J\'3 : 1.

If now

we associate

same parameter, 1L

y'

and

W =;1

whence and also If, again,

x : 'lJ : ill'

the

points of the two curves which hare the

i,e., A=A', then

:

Z=

.:.= ••

_1_ 1 + A"

z

,

~/(a.:'+ z') : y'(.v' + z'} : z'x',

u' : z'=x(JJ-z)

: Y(JJ-z)

: z».

A and A' are connected by a bilinear

relation

of the form

AAA'+ BA + CA' + D=O, we may, in a similar manner, express x' : Y' : z' in terms of x, y, z. Ii

Salmon's

Higher Algebra, Lesson X.

278

THEORY OF PLANE CURVES

Ex. 2.

Consider the two curvesy : z= t' : t: 1 + (2

IV :

and

y' : z'

X' :

Associating show that :Il, y, vice

the Z

=

points

I(t 2 -1) : (t 2 - 1)' : t.

which

have

t'ersa.

x

In tho first curve,

:v' z' - =t y' X

.,

In the second curve,

y'z'

z=1

y

W'2

=x"(.,'

+2/)

,"

x'

';1 x z

and

2',

and

(1 )

1 + t'

z

and

we shall

in terms of :1)', s',

.,

and

- =t

Y

whence

the same parameter,

can be expressed rationally

(2)

=t'-1

F ~' +z' 1 + t' - .v' + 2z '

a/ +2z' .1:' +:,' : .v',(.,' + 2~').

: y'z'(x' +Z')

(A)

II' z' ~ =t.y' QJ'

and

:.-al': y': z'=I:

y" =xy(x'-y') .V'_y2

002_y1

----xy

'I'hus, by Riemann

: --.

transformation

: (x'-y')':

xy'.

(B)

the two given curves can he trans.

formed one into tbe other. Ex. 3.

Apply the transformation Now

X'+y3+Z'=O

Consider the curve

w': y' : z' =x'

: y' :

z'.

e : y: Z=2X+y3z3 : 2x3Y'Z3 : 2Z3y3z'

=x+(x" +2y3z3_X6)

:

Y'(Y" +2Z3X3_y6) : Z+(Z6

=z'{x.

+2y3z3_(y3

+Z3)"} : y'fy"

+2X3y3_Z6) +2Z3X3_(Z'

: zOfz" + 2x3Y"_(X3 =x+{2Z"-(x"

= z"

{2.C'3 -k} where

+a;3)'}

+y3)'}

+y" +z")} : y+{2y6_(X· +y. +z") 0 6 : z'{2z -{x +y. +Z6)} : y" {2y'3 -k}

:

Zl.

{2Z'3 -k}

ItATIONAL z, y, z have

Thus,

been

279

TRANS}'ORMATIONS

expressed

rationally

/ J: ,

in terms of

1/', z' with

the help of the equation of the given curve, Now applying the equation

which

gives

known,

there

the

225.

transformation

reciprocal

to the given curve, we have for

after rejecting

polar

cnrve

is (1, 1) correspondence

of two reciprocal

CURVE:

this

of the transformed,

a factor,

of x3

between

+ y3 + Z3

= 0, and as is

the points

and lines

figures,

REDUCTION

OF

THE

ORDF.R

OF

THE

'l'RANSFORMF.ll

'*'

From what has been said before, it follows that if we apply the transformation of § 221 to the n-ic F, the order of the transformed curve will be N snk-a, -2a •... etc., where all a., etc., denote the number of single, double, etc., points common to the k-ice III 121i, lying on F. We shall now consider how this transformation can be applied so as to reduce the order of the transformed curve as low as possible, i.e., to make N a minimum. Now, the curves 11'I., Is can be made to satisfy, as has been seen in § 221, ik(l,+3)-2 conditions. Hence, N will be a minimum, if ill i21is be assumed to pass through as many as possible of the double points of the given curve F. If then the deficiency of F be denoted by P, the of its double points isHn-l)(n-2)-p, (i)

Suppose

Then,

number

i.e., 1n(n-:3\-p+1.

k=n-l.

t.. I•.i, may be made to pass through 1k(k+3)-2=-Hn-l)(n+2)-2 =tn(n+l)-3

• ot.

points only.

Salmon, H. P. Curves, § 365.

:!80.

THEOlty

01-' PLANE

CURVES

Therefore, besides the double points, the curves can be made to pass only through

f

l'

t.. t,

{}n(n+ 1)-3} - {tn(n-3)-p+l} i.e., 2n+p-4 ordinary points on F, so that we may take a,=2n+p-4

and

a.=tn(n-3)-p+l.

Therefore, the order of the transformed curve is

=n(n-l)-(2n+p-4)-2{{n(n-!3)-p+

I}

=p+2. Put k=n-2,

U'i) Ail

(n>2).

before, we may take al

a.=t12(n-3)-p+l;

=B-k(k+3)-2}-Hn(n

RO

that

-3)-p+l}

=H(n-2)(12+ 1)-2}-Hn(n-3)-p+

I}

=n+p-4 N=n(n-2)-a,

-2a.

=12(n-2) -(n+;o-4)

-2{tn( n-3) -p+ I}

=p+2. (iii)

Put

k=n-3.

We take a. =tn(n+3)-p+l, and consequently, al =p-3 as before, so that p is to be taken always greater than 2. Hence,

N=p+l.

Since the transform has the same deficiency as the given curve, we may summarise the above results in the following theorem: A curve of order n with deficiency p may be transformed into a curve of order p+2 with deficiency p or with tp(p-l) double points. If P >2, the order of the transform may be p+ 1 with deficiency p, or with -}p(l-' -3) double points.

28]

RATIONAL l'R.\NSFOR~L\TIONS

If, however, p=O, the curve may be transformed into conic, which however can be further transformed into a straight line. If p=1, the transform is a cubi

so on.

For a detailed discussion, the student IS referred t.o Brill-Noethers paper-s-" Ue11e1'die algebraichen Fnncktionen and ihre Anwcndung in del' Geometrie "-l\IatJl. Ann. Rd. 7, pp. 297-398, and also to Cayley's paper-" On tlle trnnsforrna.tion of plane curves," ColI. Works, Vol. G, pp. 1-9. 22G.

REOUCTIOX

The following

OF

A CURVE

formal

WITIl

proof

for

~IULTIPr.F.

the

PO[!(TS:

general

case of a

with

CIlj'\'C

mnlt.iple points was given by Scott.' Let F have mult.iplo points of orders points P"

let p"

POl

the

curves

of k;

12' t,

curve

:::".P;

to determine

N=nk-~r~p~

enc., and at

multiple

points

these

of orders

I.;and p's, so that the order of the may

be a rnin imnm,

trans.

i.c'lfora~ivcn\'all1f\

is to be made a maximum.

The cnrvesf"f,,/,

can be made to satisfy

Oldy, but if a p.point of the's nnmber

,." "",."

have

.. etc. (whero nny of the ,.'s or p's may be zero or unity).

It is required Iormnd

jll

of conditions

11.(k+:3)-2

is placed at an ordinary

imposed is ~p(p+

1), while the

conditions

point of F, the point

counts

as p

intersections. ';;p(p + 1)~p,

Evidently,

on F will count

j'», i.e., if

according

as most

as p~l.

intersections,

Hence, all ordinary

p= 1.

Again, if a p-point is placed at an r.point, the number is -Isp(p + I), while the number of intersections

is "p, and

certainly

a

p=l,

difference

betwe.on the

conditions

is to be made a maximnm.

Since

positive

the

I" I" Is

• Scott,

quantity,

multiple

of other multiple npon

if ,.>1 and

number

+ 1) is

and generally,

of intersection .• and

points are snpposed

of conditions I'P-~p(p

independent,

the number

the of

the existence

points will not affect the number of conditions imposed by supposing

"Note

the p.point at the ,..plc point of F. Hence

on Adjoint

Vol. XXVIII, pp. 377·381.

36

point

if it be an ordinary point of

Curves,"

Quarterly

Jonrnal

of Math.,

-

':'--_ ...•.

282 Itt

THOORY

every r-ple

OF

PLANE

CURVJ<:S

point of F we have to make the difference 1'p--}p(p+1)

i,e"tomake

or, -}p(21'-p-1)

a maximum

a maximum.

Now the sum of the two factors heing given, t he product maximum

when

the

two factors

are

as

nearly

equal

will be a as possible,

and ,.-1. Thus we may take p=r or p=,'-1,

i,e. when the factors are,'

and in either case the above expression=-}r(r-1), Hence, at every r-ple point of F we may take p=!' then take other

ordinary

(1'

points

or p= "-1,

on F sufficient

to make

number of conditions necessary for the transformation. But since an r-ple point with r of the the case P="-l. The

('It""e.'

of

the

()'

sha.U have

(,·-I).p7c

011

n u.mbc» oJ condition»

paint

o1'(lina1'Y

IT

requ.ireii

DE~'INITION

An" adjoined"

:

an (1'-1)'1'10

cnl've as Loioe«!

in ",hieh

or "adjoint"

point at every r-ple point of an

adjoints

of I. which

of

will

order

k=n-3,

minimise

on P, ond 'll!

words,

the

of adjoint. en!'ves. curve is one which has II

.ic.

We shall next show that best results can be obtained, using

jJo .«sibl e

to nuil:e

or, in other

net;

(18

the tranejormat ion.

F', sufficient

011

the

micst be e,(fecled by werm"

il'rlll.-jol'mntion

as an ('·-I).pie

may be included in

at cve,'?! r-pl.e point.

pnillf....:

for

r

the result as follows :-.

genel'a.l

a

Ii!! mca.ns of a tmn.-jo!'lHfltion

shall pass throu.o]: other the

o/'der

net,

may be regarded

Hence we m:1y summarise

rc.luct ion

mnst be effected

point on J's

other conditions, t he case p =

0-

and

lip the

i.c.

we

N=nk-'2,I'iP,-o-,

in general, by

shal l

find the

where

p=1'-I,

value and

0-

gives the number of ordinary points on F that may be chosen arbitrarily of f

for the determination But

there

1l

3'

is a limit to the value of

on an n-ic of deficiency curve

i-, J

l'

which

For the number

0-.

can

be chosen

of lower order lcis nk- p. and the remaining

determined,

if k~n-2.

But

if k~1'I-3,

Hence, for the net of adjoints, if k~n-2, used in imposing

conditions

falls

which shows that

'2,,(r- 1) + 0- = nk: - 1'- 2

i.e.

k~,,-::l, we have 0-=

a

points are thereby

there is no such limitation. the number

short of the

when

:'2,1'(r-1) + tT= ~k(k + 3)-2 ik(k + 3) - i( It -1)(n-:!)

of inbersect.ions

total number by p + 2.

N=nle-'2,1'(r-l)-o-=p+2, If

of points

to determine

+ p-2

which implies that the expression must not be negative.

k~!t-2.

Now,

writing

whieh shows that upon

lL\TION:\L

THANSFORMATION~

k=II-3+1,

we have

.'. t must
ordinary

for the

1, subject

Thus, in general,

i.c., the

value

of

depends

II

is a special case and is not to be considered.

be zero

and N=lJ+

is obtained

2
2p~6-t(2n-3+/),

which therefore

II,

2S:S

general

case,

to the condition

and

hence

k=n-3,

p~3.

the lowest possible order of the transformed

by means of adjoints

of order 1/.-3, passing

curve

through

certain

points of the given curve.

The question

of further

of the ordinary

227.

what

has

been

curves,

it is seen that

whieh

is so familiar'

no unimportant curves.

said

the

above about

special

class,

to the student

a part

As

ill

the

geometry

we have

seen,

the

adjoints

curve

are

important

that

they

always

curves.

ive adjoi nts as a means

to the

and so

The

of investigation

lower by

from

the

fact

adjoints

adjoints

Oil,

plays algebraic

of order

into corresponding

Adjoints

Second Adjuint6,

"adjoints"

plano

of

the original

transform

transformation

called

than

are called

choice

of function-theory,

three

the transform

by a propel'

separately.

CURVES:

AOJOI:'iT

From

reduct.lon of the order,

points, should be discussed

of

of

a curve

use of success-

is due to S. Kantor

and G. Castelnuovo.* Now, given

the

fact

point

is

Consequently,

a curve

must

extending

over

be all

therefore

has (r-l)-ple

to -tl'(r-l)

the co-efficients

(n-:3)-ic its equation

that equivalent

ill the equatiou

by

connected

the

multiple

of the

~{1'(7·-1)

points

of the

~n(n -:3) - ~-}l'(r-l)

contains

point

conditions

at a

(§ 50). adjoint relations n-ic, and arbitrary

co-efficients. Hut

}ll(n-:n

Thus

we

more

than

equation

= {t(n-l)(1l-2) -~tr(i'-l)}-l

-~Jr(l'-I)

may

state

the

number

=p-l

(§ 5:3).

the

deficiency

that of

of the most general

*

acbitcary adjoint

of

all

co-efficients

(n-3)-ic.

Math. Ann. Bd. H (18!J4), pp, 127.

lHC IS

In

one the

Z84

THJ;;ORY at' PLANE

CURVES

It IS to be noticed, however, that for n=1 or 2, p=O; and for n=3, 1'=0 or 1 according fLS the n-ic (cubic) has or has not a double point, 228.

IN'l'fo:RsEcT!o~s

Ob' A CUBVE WITH

lTS ADJOINT:

Since at every r-ple point the adjoint has an (1'-1 )-ple point, the point counts as 1'(1'-1) intersections, and the fact that the adjoint has an (1'-1)-ple point is equivalent to ~1'(r-1) relations between its co-efficients. Hence we obtain the theorem: '1'he number of intersections of an n-'Ie and an arl;j()'int at the multiple points of the u-z'c is double the number of relations between the co-efJicients of the adjoint curt:e. If the adjoint be an (n-3)-ic, since there are p-1 arbitrary co-efficients, the Dum bel' of relations between its co-efficients is -~n(n-3)-p+1=}(n-1)(n-2)-p, t».

Show that

are adjoints

229.

to C /I

the identity

(1) of §:38 holds,

if

CIII,

em"

C/,' C/,'

•

IK'l'ERSECTlO~S

'YlTH

A

PEKC/1.

OF ADJOIN'l'S

:

Let l: be the order of a curve adjoint to the n-ic, with multiple points of orders '1'1,1'" 1'" .. , Then the multiple poin ts count as l'r{1'-1) intersections and the co-efficients of the adjoint 7,-ic aie connected by r~'I'(1'-1) relations. Therefore the lr-ic req uires tk(l, +2)-ilT(-r-1) ether conditions to be uniquely determined, i.e., we may take i"( k+ 3)-t~r( 1'-1) other ordinary points on the u-ic besides the mul tiple points, so as to completely determine the adjoint. Now, the two curves intersect in nk points. Hence the num bel' of remaining intersections =nk-~I'(1'-l)=nk-t~r()'-l) =nk+p-H

Hk(k+;3)-i~I'(I'-1)} -~"k(k+3)

n-1)(n-2)-tk(k+:3)

=H2nk-n2 -Ii' +8n-3k)+p-1 =Hn-Z)(k-n

+3) +p-l.

RATIONAL

285

TRANSFOR1IATIONS

This result shows that if we describe a pencil through the multiple points and through

of k-ics

{t7,(k+3) -I} -~:SI'(r-1) other ordinary points on the n-ic, then this pencil will meet the n-ic in -}(n-k)(k-n+3) +p variable points. Hence, we may state the theorem : Any curve of a pencil of adjoint k-ics, through the 1Iwltiple points and other ordinary fixed points on the n-ic, will meet the n-ic in tin-k)( k-n+8) +p variable points. If k=n-1 or n-2, it is equal to p.

this number is p+ 1;

if k=n-3,

Thus, any adjoint (n-3)-ic through the multiple points and through -in(n-8)-1--i-(n-1)(n-2)+p, i.e., p-2 ordinary points on the u-ic will meet the n-ic in pother variable points. Ere.

A poncil of adjoint

that (n-k)(k-n

k-ic« has its base points on an n-ic,

+ 3) + 4]1-2 curves

of the pencil

touch

the

Show n-ic at

points other than a base-point. 230.

TRAXS,'ORMATIOX

BY

Let there be ", double points

on

the

given

AD.JOIKTS

points,

u-ic

",

F = 0,

:

triple

so

that

common a, single points, ", double points,

and a,. (J' + 1).ple

points, the

adjoint

k-ics

have

... « • "-ple points on F.

If p' denotes the deficiency of the adjoints, by § 63. p =}(n-l)(n-2)-}:J:I'(J' p' =}(k-l

Now,

1>' -p=

(2)

(§ 221).

-}tr(r-l)a,.

-3(k-l)}

{,l-(k-l)(k-2)-}tr(r-l)a,} - H(n-l)(n-2)

p' =~(k-l)(k-2)

k= n- 2,

p' =

+ 3(k-l)

-&(n-l)(n-2) -}( n-l

Hence, if 1.=11-1, If

(1)

)",

-3(7,-1).

-}tr(r-l)a,

=}(k-l)(k-2) i,e.

+ 1)",.

+2a, + ... +1'a,.)

p=-}(n-l)(n-2)-}tl'(I'-I)",-(a, =}(n-l)(n-2)

••

)(1.--2) -~tl'("-1

)(n-2)

p'=2n+p-4=number Ii

+ p-4=number

+ 3(k-l)

+ p.

of ordinary of ordinary points.

points.

286

TUJ<;ORY

If k=II-3,

OJ<' PLAN1<;

l"=p-3=.r=llumuer

CUltVES

of ordinary intersections.

Since

it is a (1,1) correspondence, thero should be no ordinary intersection on F.
-r~:"(r+ 1la, =i( ?I-I

As before,

",ne!

)(1/-2)-1'

= ;(k-l)(k-2)-p'

J~r(r-1)",.

and if '1 be the number of free intersections

besides

the

multiple and

other points on the given curve, then q=k2-~"'a,

U) If

k='II.-3,


~~"(j'-l )a;. From (I) and (6) ']=

(ii)

_(I"

(n-3)'

con be written

as (6)

~r2a,.=n'-6n+l1-p-p' - 6" + II-p-1")- (p-3)=p'

we have

this value of
(5)

(4)

= -;(n-.1,)(n-5)-).'

we have

If k>,,-3,

-
~"'a,

obtained

+ 1.

l)a,.

IT=llk-p-2-~r(-i'+

Substituting

by addition of (1)

and

(2)

in equation (5), we obtain'I =k' -"k-,\-(,,-I)(n-2)

+1" + 2

=Hn-k)(n-k-3)

-2) + 1" + 1.

= i(n-k-l)(n-k

if

k= n-l

or

".- 2,

- 2) + p' + 2

- Hk-l)(k

'I

=s/

+ 1,

Hence, by means of adjoints of order

the same as above.

L n, but {:n-3,

and deficiency

p', the plane is subjected to a (q, 1), i.e., (p' + 1, 1) transformation. :.

It will be a (1, 1) Cremona transformation,

if p' = 0, i.e., if the

adjoints are uuicursal. Thus the necessary and sufficient condition that the by adjoints is a (1, 1) transformation

transformation

is that the adjoints

be unicursal.

It is to be noticed, however, that the number of free intersections a net of curves

(not necessarily

a specified manner through

adjoints)

of deficiency

p',

passing

of in

fixed points is p' + 1, and this is a particular

case of a theorem due to Sogre." ,. Segre, Rendiconti (1887),

p. 217.

del Circolo

Matematico

di

Palermo,

Vol. 1