McCarthy, J. P.

Simonds, Eugene Francis.

University of Queensland eSpace

VOLUME 1.
D:BJPAR,TrMENT OF MATHEMATIC'S.
1940. NUMBER 2.
I
I
I>RO.JE(I11IVI<~ DIFFEREN1~IAL INVARIANrrS OF
rrvvo CTJRVI1JS.
By
E. P. SIMONDS "\ND .J. P. JHcCARrrHY.
Department of Mathe-matics, Uld1;fr"ity of (JueMtslanil.
[Published as an original paper by the University of Queensland.]
DATE OJ~ PUBLICA'rJON:
28TH OCTOBl<jR., 1940.
•

PROJECTIVE DIFFERENTIAL INVARIANTS
OF TWO CURVES.
By E. F. SIMONDS AND ~r. P. MOOARTHY.
Department of Mathematics, University of Queensland.
INnWDUOTION.
The projective group of transformations in the plane
has eight essential parameters. According to the general theory,* in addition to the
absolute invariants of a single curve, there are eight of " mixed" type, i.e., functions
of both curves. Those of the first type have already been treated by G. H. Halphen
in his celebrated thesis,§ where it is shown that the simplest are of orders 7 and 8.
It follows t,hat the eight mixed invariants are of orders not exceeding 6.
The objeot of the present paper is to determine them in algebraio form, leaving
their geometric interpretation to be dealt with bter.
1. THE AFFINE INVARIANTS.
In the first paper of this seriest the six mixed invariants of the affine group
were found, viz.,
'They were the solutions of a complete set of equations involving derivatives of the
fourth order. The affine invariant,s of a single curve are
.15 2.14 _ 3 , .16.14 _ 2
where
.15 =' 9y"2yv - 45y"y'"yIV + 40y"'3
.16 = 9y"3yVI ._- 63y"2y'''yV _. 45y"Zylv2 + 255y"y"'2y lV .__. 160y"'4
* E. F. Simonds, Trans. ArneI'. Math. Soc. Vol. XIX, No.2, pp. 223-250.
§ G. H. Halphen, Oeuvres, Vol. II, pp. 197-2:33.
t Affine Differential Invariants of 'l'wo Analytic Arcs.
P110JECl'IVfiJ DIFFERENTIAL INVARIANTS OF TWO CURVES.
be proved in a paper shortly to be published in this series, entitled" Affine
invariant :families of cuspidal cubics." In the meantime, their invariance can be
readily verified. J 4 = 0 is the equation of all parabolas; J 5 c= 0 is the equation
of all conics; J 6 = 0 has no algebraic general solution, the attempt to integrate it.
leading to a Riccati equation of the so-called insoluble type.
The affine group being a sub-group of the projective, the invariants of the-
latter for two curves will be functions of
(I)
jJ![32
the last four being invariants of the first type, that is, belonging to a single curve,
indicated by the additional suffix 1 or 2.
2. EXTEND,ED INFINITESIMAL TRANSFORMATIONS OF THE PRO,mOTIVE GROUP.
For the projective group we have, in addition to the extended infinitesima]
transformations of the affine group, the follo"",ing two which are non-affine;
1'J j'- Y'. 2 af " af "( . ') af 3"'" af
- 7 = ...,.r 'a-- + ~xy -a-- + ~ y -- .ry -a'·, - ~.ry ~a"
x y y y
--}; (5xy'" + 3y") aya~, --}; (7xylV + 8y"') oaf
aylV
alf' alf alf alf
Usf'= };xy - + };y2 __ + };y' (y - xy')-, - 3};;:cy'y" ~"
ax ay dy ay
- }; (4.xy'y'" + :3xy"2 + 3y'y" + yy"') aa{~,
y
- }; (5xy'ylV + lO:J;'y"y'" + 8y'y'" + 6y"2 +2yy,v).aL
• aylV
._}; (6xy'yv + 15xy"ylv + 15y'ylV + 1O,xy"'2 + 30y"y'" + 3YYV)'!'[
ayV
- }; (7::cy'yvl + 21xy"yv + 24y'yV + 35::cy"'ylV + 60y"ylV
+ 40y'''2 + 4yyVl) af
ayVI
PBO,JECTIVE IJIFFBRENTIAL INVARIANTS OF TWO CURVES. 3
The projeetive mixed invariants will be the solutions of
'The method of solution will be-in prineiple-to assume the 10 solutions (i) of the
first six equations, and to use them as new variables in the last two. We shall thus
Dbtain the projeetive invariants as funetions of the affine. It will be seen that the
work is not nearly as tedious as the equations might lead us to expeet; but this is a
common experience in group-theory.
a. SOLUTIONS OF Ud = 0 '1'0 THE FOURTH ORD}]R.
We begin by finding the results of applying the operator U 7 to the relativo
:invariants of the affine group. We find that
U7 (RI ) = x2R I U 7 (R~) = x lR 2 U 7 (R3 ) = R I - x 2R3
U 7 (R4 ) = - axl R4
U7 (Ra) = (X2 - 5x l ) Ro -- aR IR4
U7 (RIO) = - lOxlRIO + 6R4!h'"
U 7 (R5 ) c= - ax2R5
U 7 (R7 ) = (Xl - 5x2 ) R 7 - aR2R5
U 7 (Rll ) = ._- lOx2Rll -+ 6R5Y2'"
From these we proceed to the affine absolute invariants;
,similarly,
6RI yt'U 7 (M41 ) = M 41 (x2 - Xl) -+ -R-2-
4
6R Y ",U7 (M42 ) ccece M 42 (Xl - X 2) + ---i+-
5
(2)
(3)
PIWJECTIVE DIlJ'FEREN1'IAL INVARIAN1'!:3 01/ TWO CURVES,
From these we find at once that U7 MM~ = 0, and we have
. 22
as the first solution of U 7 = 0. It is of order 2.
Again, we find that
U 1 vr Il2 -.HI7 og 1,.1 21 = ---R--
3
1
(4)
Combining (2) and (3),
2.H2 (. 3)U 7 log (M 21 • M 31) ,~-" .H
3
1 - ~]lZl-: Ji.la~
Putting f ~ \,/M21 • 1ll;, (4) becomes
(5)
Similarly, we find that
(6)
where
.. ~. ~ 3 U7 (f --- 3) =~; .
U 1 A. .Hz7 og ('f> - 3) = .H-..
'3
Combining (5) and (6), we get
U 1 A. • ,f .Hz - R1 U 1 1I,r7 og ('f> - 3) (~J - 3) = -R- = 7 og IL Z1
. 3
PllOJECTIVE DIPFERENTfAL INVARIANT/i OF TWO UURVBS. 5
Our secOI~d solution of [17 0 is t,herefore
(7)
It is of order 3.
(JM 21 • M:n -3) (JM~;~~-M~ - 3)
-,~-,~"'--~---------,---~
M 21
Again, we find
[17 (M31 + M 41 ) = (X2 - Xl) (M31 +M41 ) +; 2 (R l Y1'" - Ro)
4
(8)
But from (2)
(9)
and from (5)
(10)
(ll)
Combining (8), (9), and (10) we get that
U
7
logM21 ~!l1 +M~L+ 18) = 0(1) -- 3)2
M 21 (M31 + M 41 + 18) . .( 1M M __ 3)2 IS a solutIOn of Ud = o.\ 21' 31
Similarly it can be shown that
(12)
is a solution.
M 22 (M32 +- M 42 + 18)
(JM 22 • M~2 ~~- 3)2
4. SOLUTIONS OF U 8 ~-~ 0 TO THE ]'OUR'l'H ORDER.
We have now obtained four solutions of U 7 = 0, and we proceed to show that
these very four are al~o solutions of Us! = o.
PROJ.ECTIVE DIF'FEREN'l'IAI, INVARIANTS Oil'TWO CURVES.
The work follows very similar lines to that for lId = O. We find easily that
II8 (RI) c= (Yl -- XIYI' + Y2) R I ; lIs {R2) ,= {Y2 ._- X 2;1h' + YI) R2
Us (Rs) = Y/ (YI -- X1YI') - Y'/ (Y2 .-- X2-Y2') ;
U 8 (R6) = (Y2 --- 5xly/) R6 - 3RIR,IYl' ; US {R7J =, (YI -- 5X2Y2') R 7 - 3R2R5Y2'
Us (RIO) = - 2 (Yl + 4x1y/) RIo -+- 6yt" (YI'yt''' - 3y/,2)
from which we derive
(13)
(14)
(15)
(16)
(17)
(18)
From (13) and (14) we have obviously
(19)
From (13) and (15) it follows that
Us log (M21 ,MSI) =, ~YR"l!2 - 6yl'
. :1
1
(19a)
hence
V 8 log (rp -- 3) Z = ?_llZY1'
.Ra
Combining (20), (21), and (22), remembering that Yz- Yl = YZ'!ll..+ Yl'Rz ,
.Ra
5. PROJECTIVE INVARIANTS. UP '1'0 THR }1'OUR'rn ORDl1;R.
whieh are known t,o slttisfy the six affine equations
We have now obtained four solutions of
Vd 0 , Vsf=O
UJ 0 , (i == 1, 2, 8, 4, 5, B)
These four expressions (1), (7), (I I), and (12) are therefore inva,riants of the projective
group.
(rp - 3) (If; -- 3)U8 log--·-----·--··-· = O.M Z1
PRO,lJi!CTJVE DIF'7i'RRRN.7'JA {, INV A RJANT8 OF TWO CURVES. 7
(20)
we get
But
(22)
(21 )
V 8 log ~Zl (~~-~~~~tl-~Ll~ = O.
Thus (11) and its symmetrical counterpart (12) a,re fourth-order solutions of V sf ,=, O.
a,nd from (I9a)
Finally, from (15) and (17)
V 8 (M:n +- M 41 ) = (yz -- ?h) (M:n + M 41 + ]8)
Similarly
S PIWJEC1'IVlC DIFFERENTIAL, INVARIANTS OJ? TWO CURVES.
Since the equations extended to the fourth order form a complete system,
tlw number of independent solutions is 12 8 4. so that we have them all.
G. PnOJW)TIVE INVARJAN1'S OF OnDER G.
Thn mixed in variants of orders greater than 4 will involve the affine relati\'e
invariants .14 , .15 , .1(; for nach curve: and wn ;;hall di;;tingui;;h the two curves by
adding the subscript 1 or 2. Thus, for example,
Applying the opemtor;; U7 and Us we find
(2G)
(2G)
(27) Us (J(il)
and the others are obtained from the;;e by changing the subscript 1 to 2, and R4 to R 5 •
We find from (23) and (25) that
(J 2)t• '1U 7 log J:
1
3
But we have already
U 7 10gMu =~ (a;2
U7100" ('~5!£
o .1
41
,vhencc
(28) \ .1511 M41 /U ----------- -----------
7 (JH (M:ll-I- M'l1 -1- 18) j o.
and the same expression i;; easily shown to satisfy Usf= o.
Thus we have an invariant of order 5, f1ndwe can obtain another by changing
the subscript 1 to 2.
(~-3) + ..... , .
therefore there is an invariant of the type
dJ[ clI] dx]
Zlx~ d:r; dX-~
PROJECTIIlI lilFF'ElIFNTIAL lNI'A,RLiNTS Of!' nvo CURf'lIS. !l
=-lr
Before obtaining Lhe last Lwo lnixed invariantsw!> prove t,h8 following lemma;
dII
dX]
~iT;
([XL
dl z dI2 .d,I:1
dXI dx] riX,
For if I j is the transform of 1 L' on differentiating the equation I] =0 I with
regard to Xl' we get
7. PIW.J EOTI VJ£ JN VAKIAWI'S OF ORDJm Ii.
which proves the IClTlma.
and we have already
Now if we multiply the two invariants (11) and (28), invert the produet, and
extract the square root, we get an invariant
I] ""•. J5]-~ . R3R<] (~ - 3)Rz
'if I] and I" are two absolut;e invariants, then Cdill / dd 1z is an itbsoluLe
" ,Xj x]
invariant."
SimHarly
10 I'IW,JE'CTlF'E' f)JT!'FIUUXNTIAL INVAlililNT8 OF TWO CURVI£8.
where A [lttS still to he determined. 'rhe two expressions
.161 .I41 !
.J~I* .fhi l
ttre affino invariants, therefore A is an a,ffine invaria,llt.
Now from (2;1) and (20)
(2U) U7 (.~~l, ) :~ytJ 5lt
(30)
But
U7 (~::: ) = ~ytYj2f~I-:~
U7 (;;,If + A -;::~) == 0
which in virtue of (2D) and (:~O) is equivalent to
also
:~y/,
I n !
3Yl"Zh" f
J 4J
Com,paring (:31) and (32) we see tImt
A~=- V~~
it remains to verify tha.t this expression is also a solution of Usf = o.
U /111;]1
S '\M 41
3yt'y/'
.f4 ,t
3y/' (Yl'yt' - :3y/' 2)
---- --------~...._..
.141
IM),
'\ M 41
3y/y/,
J 5ll
3.111--! 211" (Y/llt'" ._. 3YI" 2)
.1511
PRO.J]J)Gl'IVIC ])IFF'ERENTTAT~ INVARIANTS OF TWO CURVP:S. H
whence it easily follows that
U 8 ({~;- - V~~
With a similar expression obtained by intercha,nging the two curves, we complete our
list of invaria,nts.
8. SUMMAHY.
To SUIll up, the mixed invariants of the projective group are as follows ;--
(a)
(b)
(c)
One of order two, viz., ~21M22
One of order three, viz.,(~.lI:f21~~=_~L_(_v'_X!=~,---M_""'-",~=-'"_-_3-'.)
M21
Two of order four, viz.,
M 21 (Mal -+ M 41 + 18)
-(~M~~M:Jl ... 3) 2
M 22 (M32 + M 42 -I- 18)
(JM22M~; ..- 3)2 .
(d) Two of order five, viz. ,
(e) Two of order six, viz.,
J4!'\
J5~J,
No matter how varied the configuration of ana.Iytic arcs, all the projective
invariants are functions of these types, and of the types found by Halphen for a,
single analytic arc.
The geometrical meaning of the mixed invaria,nts will be considered in another
paper.
A. ,H. TUCKER, (lOft. Printer, llri,bane.
o


Projective differential invariants of two curves

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Projective differential invariants of two curves

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