Ganguli, Theory Of Plane Curves, Chapter 4

CHAPTBR THWHY

m'

IV.

POLE~ A~J) POLARS.

G:3. Let (.e, y, z)=O be the equation of a curve of the nth degree in any system of homogeneous co-ordinates (or in Cartesians made homogeneous by introducing requisite powi3rsof z=l). vVe shall examine the points where any line joining two given points intersect the curve by using the method of Joachimsthal. Let P (01.', s'. z') be a fixed point and Q (x, y, z) a variable point in the plane. Then the co-ordinates of any point A on PQ, dividing PQ in the ratio A : p. (where A+p.=l), are AV + fI.G',

AY + flY',

A;

+ p...;'.

The co-ordinates of points where the line PQ meets the are found by substituting these values for x, y, z in the equation of the curve, and tllen determining the ratio A : p.. from the resulting equation. curve

'1'hus,

'I' J

'0 (A.l:+ p...l;,' Ay+p..y, , AZ+p..o)=

.

This may be expanded in two ways by Taylor's theorem. We have then-

or

78

THEORY

OF' PLANE

CUltVES

where

1=/(x,

y,

J'=/(.r:,y',

z),

/::;.=( ;v' ox a +y' .Q. oy 1\ ,_( L.J.

-

, ,/

+z' ~

oz

z')

)

aax' + y aay' +_"a a;;' )

Either of these equations gives the n values of the ratio .\ : JL. Comparing the co-efflcieuts in the two equations, we obtain the following identities :-

I=!

n!

/::;.,nf .

!/::;. 'f=

r!

_1_ /::;. A-,

(11,-1) !

64.

POJ.AR

.

1_._ /::;.,n-'f (n-r)!

1=/::;.'1'

CURV!>::>:

'I'he several curves defined by the

equations /::;./=0, of the point (x', y', z') with regard to 1=0. The curve /::;.1=0 is called the first polar of the point (e', y', z') with respect to f=O ..

/::;. '/=0, etc., are called the ,. Polar

curves"

POU:S AND POL'\.R~

Similarly, the curves 6. ~j=O, etc .. arc called respectively the second, third, etc., polar curves of the point (a.', y', z') and the point (.t', y', :/j is called the pole. The equation of the kth polar curve is

or

the two equations representing the same curve In virtue of the identities of the preceding article. It follows therefore that 6.' -'j=O or 6.'f =0, which represents the (n-l)th polar curve, is the polar line, and 6. --lj=O, or L\'sf=Olwhich represent.s the (n-2)th polar curve is the polar conic, and so on. From the mode of forming the equations of polar curves it is seen that successive polars are obtained by performing the operation L\ successively on f j for instance L\"j is obtained by operating with 6. on 6.j, which is the first polar of f. Hence the second polar of (.t', y', ~') with respect to j is the first polar of the same point with respect to 6.f. Similarly, the third polar of (.t', y', :') with respect to f is the first polar of the same point with respect to L\ Sj and the second polar with respect to 6.f. In general, since 6. r( 6. 'f) = 6. r + 'I. the (r + e)th polar of a point with a respect to j is the rth polar of the same point with respect to 6. "], i.e., with respect to the sth polar of f. Thus we may state the following general theorem: The polar Cu,1'·ve of an!! rank of a point is also a polar curt'l' oj the same point with respect to all polar ourties of a rank lmoll?' than its own. It is to be further observed that 6. '(6. rf)=6. '(6. '/)=6. ""] 6. 'f=L\ '-'(6.

'I),

and

1'10On.

so

THEORY

OF PLANE

CURVE!'

This shows that the sth polar of a point with respect to the rth polar of the same point ui.r.i, the original curve is the rth polar of the point UJ.d. the sth polar of the original curve. Ex. 1. If P lies on the kth polar of Q for an n.ic, Q lies on the (n-k)th polar of P. Ex. 2. IfP is a node of the (k-l)thpolarofQ, of the (n - k )th polar of P for an n.ic.

thenQ is a node

Ex. 3. Show that the kth polars of the vertices of the triangle of reference are

a 'f =0, az' a'f =0.

ay'

Ex. 4. Prove that the polar line of a point on the curve is the tangent at the point. EJJ. 5. Prove that all polar curves of a point on the curve will touch the curve at tha t point.

65. If P(a:', y', z') and Q(.t", y", e") be any two points in the plane of a curve J=O, the points where the line PQ meets the curve are determined, as in §63, by the equation which we may write in the form ®=}..-J(x', y', z,)+}..a-1fL6,J('l;', +}}..

where

n-. fL' 6,'J(x',

6, =x"a -,

oa;'

y', z') y', e')

+ ...=0

(1)

+y" £ +z" a. Oy" -

a?'

Now, if one of the points should coincide with (.~', y', z'), it is evident that one of the roots of this equation should be fL=O. This requires that f'=J(.I;', y', .')=0, which if' otherwise evident, since, when a point lies on the curve, its co-ordinates satisfy the equation of the curve. If two of the points where

meets the curve should coincide with (,,,', y', z'), then the above equation should give two values of p.=O, i.c. fL" should he a beto!' of (r<), PQ

POLES AND

81

POLARS

This requires that both (=0 and f:::..j'=0. Now then PQ touches the curve at (,(;',y', z'), and .(x", y", z") is a point on that tangent (or tangents, if more than one). Hence, if l;~",y", ;,") be made current, f:::.. becomes

.c

a-,a.v

sey ::

+v a +" a-" a z

,

and the point (,v", y", ,11) lies on the locus

aj' _-0, ,',,aaf' -'a: +Yaaf' -,Y +Z a-:' ~

(2)

which is the tangent to the curve at the point (e', y', z'), But this is evidently the polar line of the point ev', y', z'). Hence, the polar line of a point on the curce is the tangent at that point. It follows from the preceding article that the polar line of a point with respect to a curve will also be the polar line with respect to each of the polar curves, aud since the point (.v', y', z') lies on the curve, it is seen that it lies on all the polar curves. Hence the polar curves will have the same tangent at the point (,/, y', ;;'). If, however.

af' _ af' _ af' _ a-'y - a-:T a--;·" , ~O, "j"

cs:

= a1' + y" aaj'' + - ,t a ' .II

;c

y.

z"

(3)

at az'

f

vanishes identically, whatever x", y", z" may be. Hence, in this case the line PQ meets the curve al ways in two points at (,v', y', z') for all values of ,e" y" z", i.e., the point C,t', y', :') is a double point, and every line drawn through it meets the curve in two coincident points at (e', y', z').

11

82

TliIWRY

os PLAl'
The equations (3) will not, in genp.ral, have a common solution, unless a certain condition is satisfied, which of course, is obtained by eliminating ;e',y', .' between the three equations (3). (See §47.) Aga.in, the equation 0=0 will have three roots JJ.=O, i.e., PQ will meet the curve in three points at (x', y', /), if we have f::J.j'=o, 6.'f=o (4) f=o, These show t.hat in this case the line PQ coincides with the tangent at (.v', y', s'), and that every point on it is a point on the polar conic 6. if'=O of (,e', y', z'), which must therefore reduce to two right lines. Hence, 6. If contains 6.f as a factor, and the point (e', y', z') is a point of inflexion. (See §42.) We have thus indirectly obtained the theorem that the polar conic of a point of infle:eion breaks up into two "ight lines. If, however, 6.1' and

t::. If vanish identically, whatever

.v" y" z" may be, i.e., if the first and second differential co-efficients of f vanish at (.c' y' z'), the line PQ always meets the curve in three coincident points at (.e', y', z'), which is then a triple point. (§44.) Generally, if for any pointj.e', y', z') 6. k-1f is identica.lly zero, the curve has at this point a multiple point of order k. From the mode of formation of 6.1"s, it follows that, if 6.k-'1'=O, then 6.k-<Jf=O, 6.k-sf=O, ... , M=O. The equation 6.kf=O gives then the product of the k tangents at the point in question. For, in the first place, the curve represented by this equation has also at the point a multiple point of order k, since its (k-l)th polar, the point being the pole, is no other than 6. and in consequence, vanishes independently of (.v", y", i'). But a curve of order k having a k-ple point is necessarily composed of k .lines. Hence thes-e lines touch the k different branches of the original curve which pass through the multiple point.

=r,

L

POUlS

66.

:MIXED

8~

AND POLARS

POLAR~:

The symbolic identities discussed above also hold, if two different poles are taken with respect to the 1·thand the ,th polars respectively. Let P,(.l:u Yu z,) and p.(X., Y., ~.) be any two points in the plane of a curve f=O. Then, if

and

=

we have f:l' .f:l/' /1" ,f:l'; whence it follows that the sth polar of P 1 with respect to the 11h polar of p. for any curve is the 1,th polar of P, with respect to the sth polar of P, for the same curve, The curve obtained by this polar process '*' is called a " mixed polar" curve of the two points. For a cubic curve, following theorem:

in

particular,

we

obtain

the

If S, and S. are the polar conics of two points P, and P, with respect to a cubic, and if tangents are drawn from P I and P, to S. and S, respectively, then the four points of contact will lie on a right line, which is the" mixed polar" line of P, and P, with respect to the given cubic. and P, are f:lf and f:l'f

For, the polar conics of PI respecti vely, where

f:l = "\ ~.

+ ..,..,

Now, the polar line of PI PI us. 1'. t. S., i.e., /1·/1:f=O • Elliot-Algebra

=

an d f:l/

V,('

,j'.

82.. +

.

.C

t. S. is the first polar of which is a right line passing

W. T.

of Quantiol, 163.

84

TH~ORY OF PLAN~ CURVES

through the points of contact of tangents drawn from p 1 to S •. Since b.. b. 'J= b..' b.j, the truth of the theorem follows. The polar curves

Ea.!.

,l'

Ex, 2. vertices

The

+y'=a3

conic

10. ,'.

t. the cubic

are :-

m' +y'=a"

equations

of the triangle

(a., a.)

" + y =a

Polar line Polar

of the point

of the

of reference.

pairs of tangents

drawn

from

the

to the conic

a;c' + by' + cz' + 2fyz + 2gzx + 2hxy=O are

Cy·-2Fyz+Bz·=0, Bx· -2Hroy

and

+ Ay' =0 respectively,

where A, B, C, etc., are the co- factors of a, b, c, etc., in the determinant (a b c). It follows then that

"y' +by.+c;·=O,

the three

pairs of lines->

a'.' +b'.v+c'.v·-O

and a"x' +b"xy+c"y'=O

will touch a conic, if aa!a"=cc'c". Ex. 3.

Any polar curve (x/a)"

of the curve

+(yfb)" +(zJc)"=O,

is of the same form. [This curve is called the triaugulnr-aymmetric Ex. 4. conjugate

If the triangle

polar

conic

of P

10. ,'.

curve.]

t. a given

n.ic

which is inscribed in '" g-iven conic, the

has

a self.

locus

of

P

is an (n- 2).ic.

67. If two roots of either of the equations (1) and (2) of §63 be equal, the line intersects the curve in two coincident points and is therefore a tangent. Hence the discriminant of the above equation, regarded as a binary quantic in A, p., equated to zero, will gi ve the equation of the tangents drawn from any poir..t (x', y', z') to the curve. The weight, i,e., degree in the roots of this discriminant is

POLES

85

AND POLARS

n(n-1),* and therefore n(n-1) from any point to the curve,

tangents

can be drawn

The calculation of the tangents by this method is not so simple, in general; but the method may conveniently be used in particular cases. Thus, for example, in the case of a cubic curve, we obtain the following equations for determining' the values of the ratio ,\ : p.-

(1) (2) Writing, for brevity, 6, and 6,' for 6,f and 6,'1' respectively, the discriminant of (1) or 1,2) becomes:

This equation is symmetrical in the two sets (:1', y, z), (.r.', y', z'), and of degree 6 in each set. Hence 6 tangents can be drawn from any point (.v', s', ~') to the cubic. The form of the equation (3) shows that it represents a locus touching f at the points where the latter meets 6,f. The other points where f meets the locus lies on the curve (6,') 2 -46,.f' =0. Hence the geometrical interpretation of the equation is:If from any point six tangents are draum to a curve of the third order, their six points of contact lie on a conic 6, =0, and the si , remaining points, in which these tangents meet the cubic again, lie on another ctniic (6,')' -46,.f' =0, having double contact with the firet at the points ioliere 6,' =0 meets 6,=0. If however (./, y', z') is a point on the cubic, 1'=0. The discriminant equated to zero gives 6,' =46,'}, which is of the fourth degree in (.v, y, z). Hence, only fOUT tangents can be drawn to a cubic from any point on the same. jfo

Elliot-Algebra of Quantics, § 77.

86

THEORY

OF PLANE CURVES

It has already been saidthat the roots of the equation (1) of §63 gives the points where the line meets the curve. Hence, from the conditions that the equation has one or more pairs of double roots, or has two or more roots coincident, it will be possible to obtain bitangents or other multiple tangents of the curve; but the investigation by this method is by no means simple, and we shall see later that these can be obtained by other simpler methods. 68. POLAR

GEOMETRICAL

INTERPRBTATION

OF

THE

EQUATIONS

OF

CURVES:

If ll.'f=O, it follows from the equation (2) of §63 that the Rum of the roots vanishes, i.e. where, Au At, As, ..•..• Aft are the n points the line PQ intersects the curve.

in which

If now we put

Then,

~ R-r =0,

i,e.

l'

or,

nil -=R r1

1 1 +-++...+-. r. rs r.

The geometrical interpretation of this property has been given by Cotes (1722) in his Harmonia Mensumrmn as follows :-

87

POLES AND POLARS

If on each radius vector through a fixed point P, there be taken a point Q such that 1

nIl

PQ =PA 1.+PA +· .. ·.. +PA'

ft

where Au At, are the points of intersection with the curve, then the locus of Q will be a right line; or, in other words,Aft

The locus of the harmonic mean Q of the points of intersection with a curve of all lines of a pencil, drawn th1'Ough a firlJedpoint P, is the polar line of P. Similarly for the polar conic. The point Q is called the harmonic mean, In general, we call a point Q the "harmonic order k," which satisfies the equation

mean of

~QA.l QA. QAh -0 . "PA 'PA ..·..·PA - , l'

k

thus we can say that the harmonic mean of the kth order lies on the kth polar, and therefore it geometrically signifies the vanishing ofthe (k + 1)th term in the equation (2) of §63. If the pole P is at infinity, then Q is called the centre- of mean distances, and the several polars are called the diametral curves. E», 1. If the given curve be a conic, it has only one polar line, and this is the locus of a point Q which is the harmonic conjugate of P w. r. t. A, and A•. EtIJ. 2. If the point P lies on a cubic curve, then the two points :.&. I and A2I where any line through P meets the cubic are determined by the two values of ,,/,.,.given by

"'f + ",.,.t::.f +,.,.' t::.'f

=0.

If now the line meets the polar conic of Pin Q, t::.f=O; and consequently we have-

88

THEORY PQ-PA,

i.c.,

pLANE

CURVES

+ PQ-PA.

=0

Q}'

~~ 211 - + -PQ PAl PA.

i.e.,

-

which

shows

that

P, A"

Q, A.

form a harmonic range,

in other

01'

worda.s-The P'Jints 10hae any line ilvrouqh: a point on a cubic

meets

the

cubic

and the polar conic of the point [orm. with the point a ha'Y'monic range. E». 3.

The polar line of a point at infinity is the

diameter

of the

system of parallel chords directed to that point. E», 4.

The polar curves of any point W.1·.t. a curve

into the polar curves of the projected

are

point tv .r,t. the projection

projected of

the

given curve.

E». 5.

A couic touches a cubic at 0 and cuts it in four other points

P, Q, R, S.

Show that OP, OQ, OR, OS meet the cubic

again

in four

points lying on a conic, which also touches the cubic at O.

69.

CE~TRB

OF A CURVE:

In a conic the pole of the line at infinity is defined as the centre. But in the case of a general curve of order n, a line has (n-l)' poles, and there is therefore no unique point for such a curve corresponding to the centre of a conic. If, however, the curve be regarded as an envelope, every line has a pole, a polar curve of the second, third and higher class, and finally a polar curve of the (n-l)th class, which is touched by the n tangents a.t the points where the line meets the curve. Thus, we may obtain a unique point-the pole of the line at infinity,-when the curve is defined by its tangential equation.

a,

n

Let '1, '), (f, "I', be the co-ordinates of any two lines. Then the co-ordinates of U,llY line through their intersection, dividing the angle between them in two parts whose sines are in the ratio A : p-may be taken as (A~+P-~',

+

ATI P-'1', At+p-O·

If we substitute these values for ~,'1,'in the tangential equation of the curve, the equation corresponding to (1) of

I" (

POLES AND POLARS

89

§63 now determines the ratios of the sines of the parts into which the angle between the two lines is divided by each of the tangents drawn to the CUrTethrough their intersection. If now P is a variable point, and 0 a fixed point on any given line whose pole is to be determined, and Au A~,...A~ the points of contact of tangents drawn from P, the pole Q of the line has the property (1) For the conic, regarded as an envelope of the second class, this relation becomes sin QPA1 sin A,PO

+ sin

QPA. =0

sin AiPO

In the language of geometry follows :-

(2)

this may be stated as

If from any point P, on a fixed line 0 P, tangents PAl' PA2 are drawn to a conic, and a line PQ such that {P·OA1QAs} is harmonic, then the line OQ passes through a fixed point." The relation (1) may be written in the form (3) where M, is the foot of the perpendicular drawn from A, on the line PQ, and N 1 is the foot of the perpendicular from the same point on the line OP. If the line OP now moves off to infinity, then all denominators in (3) tend to equality, and we have lM,A, =0, i.e., the sum of the perpendiculars drawn from the points of contact of any system of parallel tangents on a parallel line through Q is zero.

the

•• Salmon, Conics, §57.

12

90

THEORY OF PLANE CURVES

Hence we have the following definition for the centre of a curve, given b1 Chasles * ;The centre of mean distances of the points of contact of any system of parallel tangents to a given curve is a fixed point, which may be regarded as a centre of the curve. Thus, the middle point of the line joining the points of contact of parallel tangents to a curve of the second class (a conic) is a fixed point. Similarly, in a curve of the third class the centre of gravity of the triangle, formed by the points of contact of three parallel tagents, is a fixed point, and so on. lt is to be noticed, however, that in the system of Cartesian point co-ordinates, when the equation of the curve contains only terms of odd, or only terms of even degree, the curve is symmetrical about the origin, and it may be brought to self-coincidence by rotation through 1800 about the origin. The origin may, in this case, be called a centre of the curve. E», 1. If P is a point on a curve and PP 1P •... a secant cutting the curve in P" P 2 •.•• P 0-1> and the polar conic of P in Q, then

n-I

PQ =

1

lpp,

Ew.2. IfPbeapoint of inflexiou on a curve and PP,p•...p.he s secant cutting the harmonic polar in Q, prove that

1

n-I =l ~. PQ PP, E», 3. If a curve has a centre, the polar curves of the centre have this point for a centre. E». 4. The polar curves of any point at infinity, in Ex. 3, have the centre of the curve as a centre. Ew. 5. If an n-ic f(w, y) =0 has a centre, all the partial derivatives of f uur.t. wand y vanish at that point.

[When the origin is taken at the centre (w', y'), the transformed equation becomes f(w +w', y+y')=O, and the terms of orders (n-I), (n-3), (n-5) ... must be absent from this new equation.]

* Quetelet, Correspondence Math6matiques et Physique, VI, 8.

POLES AND POLARS

70.

MACLALRIN'S

91

THEOREM:

If through any point P a line be drawn meeting the curve in n points, and at these points tangents be drawn, and if any other line through P cut the curve in Au A" As, ... A., and the system of n tangents in BlI B" B", then 1 PA

~-

1 PB

=~ - .

Consider two Iines drawn through P which intersect two curves of order n in the same points R" R., Rs ... R. and SuS., 8s S. respectively. The polar line of P with respect to both curves must be the same, since the two harmonic means Rand S are the same for both. Now, if PR and PS coincide, the two curves touch each other at n collinear points-Ru R" Rs, R., but still the polar line of P for both is the same. The tangents at the n points Ru R., Rs,'" ... R. may be taken to constitute a curve of the nth order touching the other original curve at n collinear points, and therefore, if a line through P intersect the curve in Au AI, As A. and the tangents at Bu B., B, B., the harmonic means of the two systems are the same, and consequently we have-

Em. 1. The tangent drawn from any point on the polar line of a point P w.r.t, a conic is cut harmonically by the tangents drawn from P. Ex. 2. A radius vector drawn through II point 0 on a cubic meets the curve again in P and Q. Shew that the locus of the extremities of harmonic means between OP and OQ is a conic, which reduces to a right line when Q is II point of inflexion.

71.

POLAl~ CURVES OF 'fHB ORIGIN:

Let the Cartesian equation of a curve be-

uo+ut+u.+

+u.=O

or, this may be written in the homogeneous form:

92

THEORY OF PLANE CURVES

The co-ordinates of the origin may be taken as (0, 0, I). _

••

A

_

L..l.=

a

Oz

Therefore, the different polar curves are respectively-

The polar line of the origin becomes-

The polar conic is-

The polar cubic is-

+(n-2)

! U2 z+(n-3)

! tts

=n(n-l)(n-2)uo+3(n-I)(n-2)u1 +6(n-2)tt. +6tts =0 etc. etc. etc. From these the polar curves of any point on a curve can easily be found; the point may be taken as the origin of a system of Cartesian axes and the corresponding equation of the given curve may be obtained by the usual method.

POLES

AND POLARS

93

72. From a study of these equations we at once draw the following inferences :(1) If the origin lies on the curve, Uo =0, all the polar curves pass through the origin, and U1 =0 is the common tangent to all. Thus, the polar curves of any point on the curve pass through the point and have a common tangent there, i.e., the polar curves all touch the curve at that point. (2) If the origin is a double point on the curve, the first degree terms are absent from the equation, and it is found that they are absent also from the equations of all polars, The terms of the lowest degree are ft. in an of them. Hence we infer that all the polar curves of a double point on the curve have a double point with the same tangents at the double point on the original curve. Further, the polar conic of the double point is U1 =0, which represents the two tangents at that point. Hence the polar conic of a double point on the curve breaks up into two right lines. (3) In general, if the lowest terms in the equation of a curve are Uk, the lowest terms in all the polar curves Hence we infer that all the polar curves of any multiple point of order ,,~on the curve have a multiple point of the same order with the same tangents at that point. (4) If the origin is a point of inflexion on the curve, the linear terms are a factor of the second degr6e terms, and hence, from the equations of § 71, it follows that the origin is an inflexion on all the polar curves; and similarly in general. Thus, the polar curves of a point on the curve at which the tangent has r-pointic contact have r-poinoic contact at the point with the same tangent.

94

THEORY OF PLANE CURVES

Ex. 1. The (n-k)th polar of a k-ple point of at: n-ie consists of the ktangents at the point. Ex. 2. If P lies on a given ",-ie, the envelope of the polar line of P with respect to a given n.ic is of class m(n-l). Ex. 3. The kth polar of P with respect to an n-ic having an (n-I)-ple point at O is an (n-k).ic having an (n-k-l)-ple point at O.

73. If a point Q (x", y", z") lies on the kth polar of a point P (.u', y', z'), then P lies on the (n-k)th polar of Q with respect to a given ourve. This is only a geometrical statement of the §63. For, the kth polar of P is-

(x'..2ox or,

(

+s' ~+z,

oy

ill

Q. )kf=o

oz

0 0 0 Oot,+y ail + Z oz'

X

equations

)"-k 1'=0.

1£ Q.lies on this, we must have

o x' +y" 0~ y' +z" A0 z'

( a/'~'

)"-kl'=O_ .

Therefore, the locus of P (x', y', s') is-

which is the (n-k)th

polar of Q.

74. 'I'h« locus of all points w hose polar lines pass through a {i.codpoint is the first polar of that point. The polar line of any point (x', y', z') is

,

(

:c

0

0 x' +y

0

0 ) _

cfi/+ Z 0 z'

f' -0.

95

:POLES AND POLARS

If this passes through a fixed point Cr", y", ZU), we must ha:ve .~"of' +y" of x' y'

a

,0

which

a

shows that

+_"

aof'

=0

Zl

0

the locus of (x', y',

I:::.f=O, which is the first polar of (a/', y",

Zl)

IS

the curve

z"),

In a like manner, it can be shown that the locus of points whose polar conics pass through a given point is the second polar of the. point; and in general, the locus of points whose kth polars pass through a fixed point is the (n-k)th polar of the point. 75.

Every point has only one polar line.

For, the polar line of the point (x', y', of

:r~1

va;

of' +y ~I+Z vy

Zl)

1S-

of' _ ~I-O. o»

The co-efficients in this equation are known, determinate functions of (x', y', z'), and therefore the line is determinate and unique. Consequently, there is only one polar line of a given pole. 76. The first polar of every point on a 1ight line passes through the pole of that line. 'I'be polar line of any point (,r', y', Zl) is,c of' +y of' x' oy'

a

+z

~'

oz'

=0.

If (x", y", z") is a point on this line, we must have-

[1;"

of' +y" of' +z" of' =0 ;v' oy' oz'

a

~\

~. which shows that the first polar of (x", y", z") passes through ~. the point (,f, y,

,'), which

ia 'he pole,

96

THEORY

PLA.NE CURVES

Q]'

77, There are 2(n-2) points on a right line, of which the fi1'st polars are touched by the line. The polar conics of these 2(n-2) points of contact are also touched by the same line. Let, l.c+my+nz=O point on it.

(1) be any line, and (x', y', z') any

The first polar of (x', y', z') is-

+ y'

F=a:' of ox

f=o

of

Oy

+ z'

=0'

of

0Z

(2)

being the given curve.

The tangent to this curve at a point (.v", y", Zll) is-

x

i.e., x (x'f"

11

+y'f"

o}i'

-0 x II+Y +z'f"

1~

of ~o II+Z y 1 S)

of _

-0 z ,,_0,

+y(x'f'

1~

+s'I"

2g

+Z'f"

2 S)

=0

(3)

wheref"=f(x",y", Zll), and flu f12' f13' second differential co-efficients of f.

etc., denote the

+z(.v'f"

13

+y'f"

2S

+z'f"

3 3)

If this is to be the same as the line (1), we must have x'f" 11 +y'f"

also

+z'f"

1S

=kl

x'f" 19

+s'F»« +z'f"

23

=km

x'f'~

+y'f"

3S

=kn

3

11

28

+z'f"

lx"+my"+nz"=O

'I

I

rI

(4)

)

Eliminating k ana (x", y", z"] between the equations (4), we shall obtain the locus of (x', yr, z'). Now the eliminant" will be of degree 2(n-2) in the variables x', y', z', which therefore represents a 2(n-2)-ic, and the points * Clebsoh, Leeons sur la Geometrie, Tom II, p. 13.

I

i

1

97

POI.ES AND POLARS

where l,e+1ny+nz=O intersects the locus are, the required points. Hence the truth of the theorem follows. If, however, we eliminate k and (e', y', e') between the first three equations in (4) and Z,t' + my' + nz' =0, we obtain the locus of (.1;", y", z") in the form of a determinant equation, namely,(5)

l =0 m

n n

o

Eqnation (5) is of degree 2(n - 2) in the variables, and nberefore the given line intersects this curve in 2(n-2) points which are the points of contact of the first polars with the given line. Again, the polar conic of (a/', y", z") is-

f' 11:1l"+I", ,y' +I" s sZ'

+2".

,yz+2J"

If this touches the line lc+nty+nz=O,

r.,

r I'

r

f' ••

f' ••

"Sl

r.. r

r

1 1

m

S 1:,11+2"

we must have

=0

1 ,

8.

n

which is satisfied, since (z:', y", z") curve (5).

1.;ry=O.

m n 0 is a point

on tlte

From what has been said above, we ma.y deduce the following theorem: Through any point we may draw two lines, on each of which there is a point whose first polar touches the line at the given point. These are the two tangents wln'ck can be drawn fmw/' the given point to its polar conic.

l~

98

THEORY OF PLANE CURVES

These two tangents may coincide either when the point lies on the polar conic, in wh ich case the point must lie on the original curve, or when the polar conic breaks up into two right lines, and the tangents coincide with the line joining the point to the intersection of those two lines. In this case, as we shall show later on, the point lies on the Hessian. From the theorems just stated it will be seen that there is somewhat of a reciprocal relation between the first polar and the polar conic of a point, as will appear in the sequel. 78.

Every straight line has, in general (n -1)' poles.

For, take any two points A and B on the line. The first polar of each of these points passes through the pole of the line (§ 76). Therefore the points of intersection of these two first polaI'Sare the poles of the line. But each of these curves is of the (n-I)th degree, and they intersect, in geI!eral, in (n-I)' points, which are the poles of the line. Cor. The first polars of all points on the line pass through these (n-I)' poles. For, if P and Q be the first polars of A and B, then the first polar of any point on AB is P+AQ=O, which evidently passes, for all values of A. through all the intersections of P and Q. 79. If, however, the curve has a node, the first polar of every point passes through it (§ 65), and therefore the two first polars intersect in only (n-l)' -1 other points, which are the poles of the line. Therefore, if the curve has 8nodes, the first polars intersect in only (n-I)'-8 other points, which are the poles of the line. H the curve has a cusp, the first polar of any point not only passes through it, but also touches the cuspidal tangent." Therefore the cusp counts as two a.mong the intersections of the two first polsrs, which then intersect only in (n-2)S -2 other points, and these are the • To be proved hereafter in § 85.

J

1

99

POLBS A.ND POURS

poles of the line. If the curve has K cusps, the two first polars intersect only in (11 -1).- 2K other points, which are the poles of the line, Hence, combining all these the following theorem :-

results,

we may enunciate

Every 'right line has, in qeuerul (11 -1) ~ poles with reqard. to a non-singular Cttrve, but if the curve has 15nodes and K cnsp8, the number of poles is reduced to (n-l)' -15-2K. Ex.

The pole

degenerate

of

cubic ;uyz=O

[The point and

the

the

line

lx + my + nz = 0

with

regard

to the

is the point (1//., 11m, 1/,,), line are

called

the pole and

polar

I/};r,t·.

the

triangle.]

80. If the polar line (or £Lny other polar curve) of passes through the point, the point lies on the curve.

It

point

For. if we put (ol, y, z) for Ce', y', z") in the equation of the polar, it becomes identical with the equation of the curve, since the polar curves are obtained by operation with 6., which becomes in this case

the effect of which, by Euler's Theorem, on a homogeneous function is only to multiply the function by a numerical factor. It is to be noted that the polar line of every point on a curve is the tangent at that point. 81. The converse theorem is also true (§72),i.e., every polar ctt1'Ve of a point on the curve touches the curve at fhat point.

~

Let .I!\ be the rth polar of any point Ce', y', its equation is gi ven by-

:

-.;

:..I!'r=( ~' Q +

ax

y' §..

oy

+

.1

oza )'1=0

Zl),

80

that

100

TItEORY 0]'

PLANE

CURVES

which evidently passes through the point (.e/, y/, z/), since the point lies on the curve. Now, the tangent at (
.§ O,l)/

+y 0 +z 2 oy/ oz/

)F:=O

...

t2)

But F', is of degree n-r, and therefore, by the identities of § 63, equation (2) may be written as-

( a/ -~ 8;c

+

y/ ~

oy

+

z/

0

OZ

r:

F,=O

i.e., as

or, which is the equation of the tangent curvef=O.

at (.v/, y', z') to the

82. The points of contact of tangents drawn to a curve from any point li« on the first polar of that point. at

Let (x', y/, z/) be a point on the curve. y', z/) is-

The tangent

(oll/,

or / +y of' ~/+ v
01l~

of' /-0. _

Z -~-

VZ

If this passes through any point (,v", y", have-

a;"or +Z"Of' -0 o ,c/ +y"Or 0 y/ 0 z/ -

Zll),

,

which shows that the locus of (:l, y/, z/) is:v"of o'{:

+y"of +z"of oy oz

=0.

'

and this is the first polar of the point (",", y", ~" .

we must

, i l

POLES

AND POLAR~

101

From this it fclk-ws that the points of contact of the tangents drawn from a given point to the curve are the points of intersection of the curve with the first polar of the point. Now, the first polar of any point is ('Ii degree (n-I), and consequently it intersects the curve of the nth degree in n(n-l) points. Thus we see tbatfro?n a given point there can be drawn, in general, n(n-I) tangents to a curve of the nth degree. DE~'INITION ; The Class of a curve is determined by the number of tangents which can be drawn from a gi-ven point to the curve, and will usually be denoted by ?no

The above theorem can therefore be stated as follows

r-e-

The Class of a curve is, in general, n(n-l), or as we shall see later on, the degreeof the reciprocal. polar curve is n (71-1). 83. Let us examine ~he case when the given point lies on the curve. We have seen (§72) that the first polar touches thq curve at the given poin t.· Hence that point counts as two of the intersections of the first polar with the curve. 'I'herefore, the number of tangents (different from the tangent at the poinc] which can be drawn from the point to touch the curve elsewhere is n(n-l)-2, or, (n+l)(n-2). Hence we obtain the tneor em i-sFrom any point on a curve. not 1nare than (n+l)(n-2), 01', m-2 tangents (excluding the tangent at the point) can be drawn to the CU1·ve. If the tangent at a point on a curve has a contact of the second order with the curve, i.e., if it be a point of inflexion on the curve, the tangent is to be regarded once as the caugent at the point and once as one of the tangents which can be drawn fr nn the point to the curve. Therefore, t~ere can be drawn only (n+l)(n-2)-I, or, 71(71-1)-3 other tangents to the curve. Hence we obtain the theorem

1l! -

r->

From a point of inflexion on a cm:ve, only n(n-l)-3, 3 tangents can be drawn to toucli the curve eleeuihere.

01',

lO~

THEORY OF PLANE CURVES

84. If, however, the point is a double point on the curve, we shall have to distinguish between the cases when (I) it is a node, (2) it is a cusp. Suppose the point is a node on the curve. We have seen that the first polar of a uode has a node with the same nodal ta-igents at the point. Therefore, the double point connts as SIX among the intersections of the first p-ilar with the curve ~§51). Of these, however, two belong to the two nodal tangents. 'I'hus the number oj tanqenis, exciueioe of the nodal tangents, which can be drawn from a node to toueli a curve elsewhere is n(n-l)-4, 01', m-4. Next, suppose that the point is a cusp 011 the curve. The ficst polar has also :L cusp at. the point with the common cuspidal ta.ngent. Hence the point counts as three among' the intersections, besides the two which belong to the cuspidal tangent. Therefore the number of tangents, exclusive of the cuspidal tangent, which I;an be draum. front a cusp to touch. a curve elsewhere is n(n-I)-3, or, nt-3. In general, if the point is a multiple point of order k, there are k tangents at the point, each of which counts as two among the tangents which can be drawn from the point to the curve. Thus, the k tangents count as 2k tangents drawn from the point, and ntn-I)-2k or m-2k other tangents can je drawn, distinct from the tangents at the multiple point, to touch the curve elsewhere. Ex. 1.

Prove that the points of contact of tangents

drawn from the

point (h, k) to the curve x' + y' =3 axy lie on a conic through the origin. Discuss the case when the point is at the origiu.

E», 2.

Show that the conic in Ex. 1 will bisect the

the nodal tangents, Ex. 3.

angle

between

if h + k = U.

If the tangent

at an) point (XI' !I,) on the cubic x' +.y3+a'

meets it again in (.v", !/:), show that

'''.ixi

+-Y:/YI +1=0_

)0:1

POl,ES AND POLARS

The first polar of a po£nt passes throuqh: every double point and its tangent at that point is harmonic conjugate of the line joining the double point with the pole with respect to the nodal tangents of the curve. 85.

The equation of a curve having the origin for a node, with the axes of ,I: and y as tangents, may be written as

or, in the homogeneous form-

The first polar of any point (.r', y', z') isF(x, y, z)==( x'y+y',r)z·-·

+ lower powers

of z=O

which shows that the first polar passes through the origin (double point), and that its tangent at that point is(I) Now, the line joining the point (,t', y',

Z')

to the origin is

;v'y-y'z=O

(2)

The lines (I~ and (2) are evidently harmonic conjugates with respect to the two axes, which are the nodal tangentR.'*' If, however, the origin is a cusp, with the axis of y as

the cuspidal tangent, the first polar becomesF(x, y, a) =( :li'+y')xz'-'

+ lower powers of z=O .

.~. The axis of y (J~=O) is also a tangent to the first polar, i.e., the first polar of any point touches the cuspidal tan~erlt and meets the curve three times at a cusp. From what has been said in §82, it, f'.)llows that, the RrBL polar of a point passes through all the double points, etc., and the points of contact of tangent:; drawn from the ~iven point to the curve.

~\··. i * Salmon, Conic SectionB, § 57.

•

A_.

i,,..

104

THEO&Y OF PLANE CU&VE8

Ex. 1. Show that the points of contact of tangents drawn from the point (a, b) to the cnrve bx' +ay'=llie on a circle. E». 2. The locns of points of oontact of tangents drawn from any point to the system of curves y=~Qj' is a rectangular hyperbola.

Ex. 3. Show that all the polar conios of the ourve y=ao +a,'" + a,:I)' + + anx' are parabolas. Ex. 4. Show that tbe points of contact of tangents drawn from the origin to the curve ilJy=ailJ' + bx'Y + cz'1/' + dtey' + ey' are collinear.

86. If the curve has a multiple point of order k, that paint will be a multiple point of order k-l on the first polar, of order k-2 on the second polar, and 80 on. Let ~he origin be a multiple point of order k on the curve. The equation of the curve is therefore of the form-

where u i~ a binary r-ic in form-e-

.l'!

and y, or, in the homogeneous

The first polar of any point (:1:', y', z') isof ox + y' oy

.FaoV' of

The lowest terms in

+z'

of oz

=0

(1)

of and ~fy are of degree (k-I) ooV v

and those in ~~ are of degree k in .cand y.

Therefore, the

lowest terms in J) and y beiug of degree k-I in (I), origin is a multiple point of order k -Ion the curve F.

the

Similarly, the lowest terms in the equation of the second polar are of degree k-2 in I; nnd y, and FlO on, which proves the proposition,

POLES

105

AND POLARS

87. It appears from the above that if u ; has a square faccor U17 i.e., if ttk =u, 'Uk_I' then this factor occurs also in the lowest terms in the equation of the first polar.

For,

Similarly,

Also the lowest terms in

U

contains u,

3

as a factor.

Thus the lowest terms in the equation of the first polar contain 'It, as II. factor, and therefore u, =0 is a tangent to the first polar. Hence we obtain the following theorem :If two tangents at a multiple point coincide, the coincident tangent touches the ji'rst polar of every point. In particular, the first polar of; any point touches the cuspidal tangent, as was proved in § 85. has any factcr u, in the lth degree, i.e., if (l-1)th degree in the lowest terms of the equation of the first polar, in degree (l-2) in those of the second polar, and so on. Again, if

Uk=U,I.t'k_l,

Uk

u, will occur in the

Hence we conclude that, in general, if l tangents at the k-ple point on a curve coincide, the coincident tangent will appear as (l-l) coincident tangents at the multiple point of the first polar, as (l-2) coincident, tangents at the multiple point of the second polar, and so on.

14

106

THEORY OF PLANE CURVES

Em. 1. Show that the points of contact of tangents drawn from the cubic x3 + y' +z' +611i
point (0, -1,1) to the the line y-z=O. Ex. 2.

The

points

of contact of parallel

the 11th degree lie on a curve of the (n-l)th

tangents degree.

to a curve

of

(Serret.)

(They lie on the first polar of a point at infinity.) Ex. 3.

The

polar

curve are rectangular poles, any

conics of two points A, B with regard to a cubic hyperbolas.

Prove that

one of which is the orthocentre

the

of the

line

AB has four

triangle

formed

by

the other three. Ex. 4.

Prove that the properties of polar curves

are unaltered

by

projection. Ex. 5.

Show

that, the envelope of the polar lines of points

ou a

given line w. ,'. t. an n.ic is of class (n-l). Ex. 6.

Show that

the

tangents

to the curve

from the point (a, 13) touch the same at poiuts lying

(m, + n)xy=n13x + may.

x"'y'=am+"

drawn

on the hyperbola