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```PROJECTIVE PROPERTIES OF CONICS

by

GLEN HERBERT STARK

B. S. , Bethany College, 1962

A MASTER'S REPORT

submitted in partial fulfillment of the
requirements for the degree

MASTER OF SCIENCE

Department of Mathematics

KANSAS STATE UNIVERSITY

Manhattan, Kansas

1964

Approved by:

Major Professor

11

S 7<M~

INTRODUCTION 1

PROJECTIVE COORDINATES 2

CONJUGATE POINTS AND LINES 9

POLES AND POLARS 13

SELF POLAR TRIANGLES , 19

ACKNOWLEDGMENT 24

BIBLIOGRAPHY 25

INTRODUCTION

This report contains a development of some of the projective
properties of the conies. The particular properties investigated are
those pertaining first to conjugate points and lines, second to poles
and polar s, and third to self polar triangles. Conjugate points have
the following properties: Two points, the first of which lies on the
conic, are conjugate if -and only if the second point lies on the tangent
at the first, and two points neither of which lies on the conic are con-
jugate if and only if they separate harmonically the points of inter-
section of their line with the conic. Conjugate lines have properties
corresponding to those of conjugate points.

Poles and polars, which are developed from conjugate points and
lines, exhibit the following property: The pole of the line joining two
points is the point of intersection of the polars of the two points, and
the polar of the point of intersection of two lines is the line joining the
poles of the two lines. The concept of poles and polars leads to the
notion of self polar triangles.

A triangle is self polar relative to a conic if and only if each pair of
sides are conjugate lines relative to the conic, and each pair of vertices
are conjugate points relative to the conic. The report concludes with
the result that there are only two types of conies in the projective plane,
those with real traces and those without real traces.

PROJECTIVE COORDINATES

For a system of projective coordinates in a plane consider the
four points P , P , P , Q, no three of which are collinear. Let p ,
p , p be the sides of the triangle p p 2 p, and q , q , q, be the lines
joining the point Q to the vertices of the triangle. Let A be an arbitrary-
point and let a. , a , a be the lines which join the point A to P , P ,
P respectively.

\

p.

Consider the cross ratios:

(!) <*! = (P2P3' q i a i>'

<*2 = (p 3 P l' Vz h

0^3 " (P 1 P 2 » q 3 a 3^

where (L L , L L ) means the ratio that L divides L and L_ divided

by the ratio that L. divides L. and L. with L n , L_, L ot L„ distinct

4 1 2 12 3 4

concurrent lines.

"William C. Graustein, Introduction to Higher Geometry, p. 72.

The numbers <* , «* , ^ could be used as coordinates for the point A
except that if A were on one of the sides of the triangle, then at least
one of the three cross ratios would be undefined. This difficulty is
overcome by using for the coordinates of A any three numbers
(x , X-, x,) such that

(2) o< i =x 3 /x 2 , * 2 = Xl /x 3 , «* 3 =x 2 / Xl . •

■

Then there are no exceptional points which have to be defined in some
special manner. The coordinates (x , x , x ) are called homogeneous
projective coordinates of the point A.

Consider the general homogeneous equation of second degree in
projective coordinates x , x , x :
3

\ p..x.x. = 0,

1,J = 1

p..x.x. = 0, p = p..

If the coefficients p.. of equation (3) are all zero, then the locus of

ij

points whose coordinates satisfy the equation is the set of all points on

the plane. If the coefficients are real and not all zero, then the locus of

points whose coordinates satisfy equation (3) is called a point conic.
3

Suppose _> p..x.x. can be factored into two linear factors as
i.J=l

follows:

4 ZZ p..x.x. = (J! a.x.) (T" b.x. .
i,j = l i=l i=l

Then the locus of equation (3) is the locus of

3

(5)

J a.x. = 0,

ti xl

together with that of

(6)

J~~ b.x. = 0.
iTi 1 1

Equations (5) and (6) represent two straight lines which may be
distinct and parallel, coincident, or intersect in one point. In any of
these cases the point conic is called degenerate. If this is not the case,
that is, if equation (3) cannot be factored into linear factors, then the

point conic is called nonde generate . A necessary and sufficient condition

2

that a point conic be nondegenerate is that lp..| \$ 0. Since this report

deals with nondegenerate conies, the added condition |p..| £ is
assumed hereafter and a nondegenerate point conic is called simply a
point conic.

Let a and b, represented by (a , a , a,) and (b , b , b ), be two
distinct points on a straight line L. An arbitrary point on the straight
line L is given by

(7) x = o^a + fih, o^, A arbitrary constants not both zero.

This arbitrary point lies on the conic given by equation (3) if and only if

Graustein, op. cit. , p. 190,

H Py( Ka.+^b.Jt^a.+^b.;

(8) > p..( o<a. + 3b.) ( <*a. + /3b.) = 0.

Since p.. = p.. equation (8) implies
ij ji

(9) <* YL P.,a.a. + 2«/3 5Z

p. .a. a. + 2©cg > p..a.b.

.2 3 -

+

/* II Pyb.b=0 (

i,J=l J J

which is a homogeneous quadratic equation in e<, A with constant
coefficients. Equation (9) determines the coordinates of the two
points ©<- a + l3,b, and'c( a + -Q b which are common to the line and

the conic provided the coefficients are not all zero. If the coefficients
are all zero, then the lines (7) and the conic coincide.

THEOREM: A straight line always intersects a nondegenerate

3
point conic in two points, distinct or coincident.

A tangent line to a conic is a line which intersects the conic in

two coincident points. Since the two points coincide, ^ a + /3,b and

e^.,a 4 fib are the same.

A secant line to a conic is a line which intersects the conic in two
distinct points. Then the two points 6L a + Q b and «^ ? a + fib are
not the same.

3
Graustein, op. cit. , p. 192.

If the equation, of a conic is equation (3) and a is a point on the
conic, then the equation of the tangent to the conic at the point a is

3

(10)

• p..a.x. = 0, p.. |

jt o.

For a system of projective line coordinates in a plane consider
the four real lines r , r , r , s, no three of which are concurrent.

JL £» -J

Let r , r , r be the sides of a triangle and R , R , R the vertices
opposite r , r , r respectively. Let S , S , S be the pointsof
intersection of s with r , r , r,. Choose an arbitrary line b and

X Lt 3

let B , B 7 , B be the points of intersection of b with r , r , r .

Ri

Form the cross ratios

(11) A =(R 2 R 3' S lV

/S 2 = (R 3 R 1 , S 2 B 2 ),

/ 3 3 = (R 1 R 2 , s 3 B 3 ),

Graustein, op. cit. , p. 193.

where (P P„, P,P,) means the ratio that P_ divides the line segment
v 1 2 3 4 3

P P divided by the ratio that P divides the line segment P P,, with

5
P , P , P , P distinct collinear points. Represent these cross

ratios in terms of v , v , v as follows:

(iz) /VVV /3 2 = v 1 /v 3 , /3 3 = V v i-

The coordinates (v., v ? , v ) are called homogeneous line coordinates
of the line b.

Consider the equation of the second degree with v , v , v pro-
jective line coordinates

3

7_ q..v.v.

2_ ^. v . v ; = °. ^ = ^i

If the coefficients are all real and not all zero, then the totality
of lines whose coordinates satisfy equation (13) is called a line conic .

A line conic may consist of two points which are distinct or coincident.
If the line conic is two points, distinct or coincident, then the line conic
is called degenerate; otherwise it is called nonde gene rate. A line of a
line conic is a singular line if every point determined by it and a line of
the conic belongs to the line conic. A nonde gene rate line conic, then,
has no singular line. A necessary and sufficient condition that a line
conic be degenerate is that lq..J = 0. Since this report deals with non-
degenerate conies, hereafter it is assumed that jq..j fi 0. A

5
Graustein, op. cit. , p. 74.

nonde generate line conic is called simply a line conic.

The dual of a point conic and its tangent lines is the line conic and
its contact points. If equation (13) is the equation of a line conic and r
is a line on the conic, then the equation of the contact point of the line
conic on the line r is

( 14 ) Z- <^ r 4 V ; = °. hJ * °-

I*" q..r.v. = 0, Jq..l t

There is a correspondence between point conies and line conies
which is indicated in the following theorem.

THEOREM: The set of all tangents to a point conic is a line conic,

7
and the set of all contact points of a line conic is a point conic.

One can speak of a conic then in terms of either line coordinates or

point coordinates. If the equation of a conic in point coordinates is

3

(15) > p..x.x. = 0, p..=p.., |p..| t 0,

then the corresponding equation in line coordinates is

3
(16) > P..v.v. = 0, P.. = P.. , JP..I M,

with P.. the cofactor of p.. in jpf .

If the equation of the conic in line coordinates is

6^
Graustem, op. cit . , p. 197.

7
C. W. O'Hara and D. R. Ward, Projective Geometry, p. 116.

5
(17) y~ q..v.v. = 0, q..=q... jq..f

1 » J *■

t 0,

then the corresponding equation in point coordinates is

3
(18) Q..x.x. = 0, Q.. = Q.. , ' Iq..| t

i» j~ - 1

with Q.. the cofactor of q.. in |q..| .
ij iJ / yl

The point conies with their tangent lines and the line conies with

their contact points are considered to be identical, and the point conic

8
and line conic is hereafter called a conic.

CONJUGATE POINTS AND LINES

There are three major classes of properties of conies: Projective,
affine, and metric. Only the first of the three is considered in this
report. Conjugate points and lines are the first properties investigated,
since they serve as groundwork for other properties.

A conic is represented by either of the two equations

3
(19a) JZ P-.x.x. = 0, p ..=p.., j p / £ 0,

or

(19b) SI q^v.v. = 0, % - q.. . ^| * 0.

1 » J *■

8
Grau stein, op. cit, p. 199.

10

where x., x. are point coordinates and v., v. line coordinates. These
i J i J

two equations will be occasionally referred to as equation (19), and
the next two as equation (20). If equation (19a) is given first then q.. =
P , the cofactor of p... If equation (19b) is given first, then p.. = Q..,

ij y y y

the cofactor of q...

y

If the coordinates of two points a and b satisfy either

3

or

(20a) > p..a.b. = 0,

20b) > p..b.a. = 0,

i,j=l

then a and b are called conjugate points with respect to the conic
represented by equation (19a). Two conjugate points are related to
the conic in a manner stated in the following two theorems.

THEOREM 1: Two points, the first of which lies on the conic, are
conjugate if and only if the second point lies on the tangent at the first.

Proof: Let the two points be a and b. If a and b are conjugate,
then equation (20) is true. If a is assumed to be the point on the conic,
then b is on the tangent at a since equation (10) is the equation of the
tangent to the conic at the point a.

If b lies on the tangent at a then equation (20a) is true and the
points a and b are conjugate by definition.

THEOREM 2: Two points, neither of which lie on the conic, are
conjugate if and only if they separate harmonically the intersection of

11

their line with the conic.

Proof: Let the two points neither of which is on the conic be a
and b. The two points determine a line and the points where that line
crosses the conic are x and x whose coordinates are given by

(21) x = a + <* b, x^ = a + o< 2 b,

1 ■ £

- + 2<* ^> p..a.b. + * T" p..b.b. = 0.
i»J=l i»J=l

^— , ij i j
i,j=i

If a and b are separated harmonically by the points x and x , then

1 u

by definition c<, + oL = 0. Since the sum of the roots of the quadratic
equation is zero the coefficient of o( in the middle term of equation (22)
is zero, and that is the condition that a and b be conjugate.

If a and b are conjugate, then the middle term of equation {2.2) is
zero. Hence the sum of the roots of equation {22) is zero and a and b
are separated harmonically.

Two lines r and s which satisfy

3

(23a) T"' q..r.s. = 0,

i,J = l

or

(23b)

q..s.r. = 0,
i,J=l

12

are called conjugate lines with respect to the conic represented by
equation (19b). Theorems 1 and 2 have two corresponding theorems
which are now given in terms of conjugate lines.

THEOREM 3: Two lines, the first of which is a tangent to a conic,
are conjugate if and only if the second passes through the point of
contact of the first with the conic.

Proof: Let the two lines be r and s, and let r be the tangent to the
conic.

If r and s are conjugate then equations (23a) and (23b) are true.
The line s then passes through the point of contact of r with the conic
because equation (14) is the equation of the point of contact of the line r.

If s passes through the point of contact of r with the conic, then
equation (23a) is true and r and s are conjugate lines by definition.

THEOREM 4: Two lines, neither of which is a tangent to the conic,
are conjugate if and only if they separate harmonically the tangents from
their point of intersection.

Proof: Suppose that neither r nor s is a tangent to the conic. Let
v and v ? be the tangents to the conic from the'point of intersection of r

J. w

and s. The lines v and v are given by

(24) v l =r + /3l S ' V 2 = r+ ^2 S '

with R and R roots of the quadratic equation

13

3 3 3

(25) 5~~ q..r.r. + 2/3 V" q..r.s. +/Q T" q..s.s. = 0.

If r and s separate harmonically the tangents from the point of
intersection, then Q + /3 = 0. Since the sum of the roots of equation
(25) is zero, the coefficient of the middle term is zero and that is the
condition that r and s be conjugate.

If r and s are conjugate the middle term of equation (25) is zero,
hence the sum of the roots of the equation is zero and r and s are
separated harmonically.

POLES AND POLARS

For the discussion of poles and polars, the conic is considered to
be given by equation (19). Poles and polars are developed from two
viewpoints: First, they are developed from conjugate points, and second
they are developed from conjugate lines.

Given a point a, then a point x is conjugate to a with respect to the
conic in equation (19) if and only if

3

(26) > p..a.x. = 0.

f\ -, lj ij
i,J=l J J

Consider the locus of all points x which are conjugate to a given
point a with respect to the conic in equation (19). The locus is a line v.
The line v is called the polar of the point a with respect to the conic
and the polar of a is given by equation (26).

14

THEOREM 5: If a point is on the conic, then the polar of the point
is the tangent at the point. If a point is not on the conic, then the polar
of the point is the secant joining the points of contact of the tangents from
the point to the conic.

Proof: Let the point a be on the conic, then the polar of the point a
is given by equation (26). Equation (26) also represents the tangent to
the conic at the point a. Thus the polar of a point a on the conic is the
tangent at the point a.

If the point a is not on the conic, then there are two points which
are the points of contact of the tangents from a. The two points of con-
tact define a line which is a secant of the conic. Each point of contact
is conjugate to the point a. Hence the secant determined by the point
of contact is the polar of the point a with respect to the conic.

If a point a has the line v as its polar, then a is called the pole of v.

THEOREM 6: If a line is tangent to a conic, then the pole of the
line is the point of tangency to the conic. If a line is a secant of a conic,
then the pole of the line is the point of intersection of the tangents to
the points in which the secant meets the conic.

Proof: If the line v is tangent to the conic, then v is represented by
equation (10) which is the same as equation (26). Hence the line v is the
polar of point a and the point a is the point of tangency.

If the line is a secant of the conic then the points of intersection
with the conic are conjugate to the point of intersection of the tangents

15

from these points. All points conjugate to a given point are on the
polar of the point which in this case is the secant. Hence the given
point is the pole of the secant.

Poles and polar s are now developed from the viewpoint of conjugate
lines. Given a fixed line v, a line y is conjugate to v if and only if

3

27 > q..v.y. =

i,J = l

The totality of lines y which are conjugate to a given line v with
respect to the conic in equation (19) pass through a point a. The point
a is called the pole of the line v with respect to the conic, and the pole
a is given by equation (27).

THEOREM 7: If a line is tangent to a conic, then the pole of the
line is the point of tangency to the conic. If a line is a secant of a
conic, then the pole of the line is the point of intersection of the tangents
to the points in which the secant meets the conic.

Proof: If a line v is a tangent to the conic, then the point of contact
is represented by equation (14) which is the same as equation (27), and
equation (27) gives the pole of the line.

If a line is a secant, the tangents to the conic at the point of inter-
section of the secant are conjugate to the secant: The two tangents
meet at a point which is the pole of the secant.

16

If a point a is given, there is a line v which has as its pole the
point a. The line v is called the polar of a with respect to the conic.

THEOREM 8: If a point is on the conic, then the polar of the point
is the tangent at the point. If a point is not on the conic, then the polar
of the point is the secant joining the points of contact of the tangents
from the point to the conic.

Proof: If the point a is on the conic, then the polar of a is repre-
sented by equation (26) which is the same as equation (10), hence the
polar of the point is the tangent at the point.

If the point is not on the conic, then there are two lines through
the point and tangent to the conic. Their points of contact determine a
secant line intersecting the conic at the points of tangency. The point
not on the conic is the pole of this secant, hence the secant is the polar
of the point.

It is to be noted that Theorem 5 and Theorem 8 are identical and
Theorem 6 and Theorem 7 are identical even though they are developed
differently. Hence one can speak of poles and polar s without regard to
whether the development is based on conjugate points or conjugate lines.

THEOREM 9: A point is on a conic if and only if it lies on its polar.
A line is tangent to a conic if and only if it contains its pole.

Proof: If a point is on a conic, by Theorem 5 its polar is the tangent
at the point. Hence the point lies on its polar. If a point lies on its

17

polar, by definition the polar is tangent to the conic at the point and
the point is on the conic.

If a line is tangent to a conic, by Theorem 6 its pole is the point of
tangency and hence the line contains its pole. If the line contains its
pole, by definition the line is tangent to the conic. Hence the point of
tangency is the pole of the line.

THEOREM 10: If a first point lies on the polar of a second point,
then the second point lies on the polar of the first point. If a first
point passes through the pole of a second line, the second line passes
through the pole of the first line.

Proof: If one point lies on the polar of a second point, then by the
definition of conjugate points, the first point is conjugate to the second
pant. By the same definition of conjugate points, the second point lies
on the polar of the first point since the polar contains all conjugate
points.

If one fixed line passes through the pole of a second line, the
fixed line is conjugate to the second line by the definition of conjugate
lines. Then the second line is conjugate to the first line and from the
definition of a pole, the second line passes through the pole of the
first line.

THEOREM 11: The pole of the line joining two points is the point of
intersection of the polar s of the two points. The polar of the point of
intersection of two lines is the line joining the poles of the two lines.

18

Proof: Let p be the pole of the line 1 which is determined by
the two points a and a,. The line 1 is the polar of p and since the
two points a and a lie on 1, the pole p lies on the polar of a and a .
Since p lies on both of the polars, it is the point of intersection of
the polars of a and a_.

Let 1 be the polar of the point a which is the point of intersection
of two lines k and k . The point a is the pole of 1 and since the two
lines k and k intersect at a, the polar of a lies on the pole of k and

i. w i.

k . Since 1 meets the pole of both k and k , 1 must be the line joining
the two poles.

The following theorem is an extension of Theorem 11.

THEOREM 12: The polars of any number of points which lie on a
line all go through a point which is the pole of the line. The poles of
any number of lines which all pass through a point, lie on a line which
is the polar of the point.

Proof: Consider the polars of two distinct points P , P on a line
L. By Theorem 11 the polars of P , P ? pass through a point P which
is the pole of the line. Consider then one of the first points, say P ,
and any other point P 1 on the line L. By Theorem 11 the polars of P
and P 1 pass through a point Q which is the pole of the line L. The
pole of a line is unique so P = Q.

Consider two lines which pass through a point. By Theorem 11
the poles of the two lines lie on a line which is the polar of the point

19

of intersection of the two lines. Consider one of the two original lines
and any other line which passes through the point of intersection of the
first two lines. By Theorem 11 the poles of the two lines lie on a line
which is the polar of the point of intersection of the two lines. But the
polar of a point is unique so the polar in each case is the same.

SELF POLAR TRIANGLES

If p is a point of the plane, and q is any point on its polar with
respect to a given conic, and r is the intersection of the polar s of p
and q, then the triangle pqr is such that each vertex is the pole of the
opposite side and each side is the polar of the opposite vertex. Such
a triangle is called self polar relative to the given conic.

THEOREM 13: A triangle is self polar relative to a conic if and
only if each pair of sides are conjugate lines relative to the conic, and
each pair of vertices are conjugate points relative to the conic.

Proof: If a triangle is self polar relative to a conic, then each side
is the polar of the vertex opposite it. Each vertex is conjugate to the
other two vertices since they lie on its polar. Hence any two vertices
are conjugate points. The pole of each side is the vertex opposite it.
Thus the intersection of two sides is the pole of the third side. Hence
the polar is conjugate to each of the other sides, and each pair of sides are
conjugate lines.

20

If a pair of vertices in a triangle are conjugate relative to a conic
then the remaining vertex is the pole of the side opposite it. Thus each
vertex is the pole of the side opposite it. If a pair of sides in a triangle
are conjugate relative to a conic, then the remaining side is the polar
of the opposite vertex. So each side is the polar of the vertex opposite
it. Hence by definition the triangle is self polar.

Self polar triangles are used as triangles of reference for the
reduction of the equation of a conic to an equation in standard form. Let
the equation of a conic be

3

(28) 2— P-.x.x. = 0, p..=p... /p.. I * 0.

1 » J i

Consider a triangle pqr that is a self polar triangle relative to the conic
represented by equation (28). Introduce new projective coordinates
(x ', x ' x ') with pqr the triangle of reference. In the new coordinate
system equation (28) becomes
3

(29) > p'..x'.x'., p\.=p'.., p'..j * 0.

The pairs of vertices of triangle pqr are conjugate points. Also the

vertices of triangle pqr have in the new coordinate system the coordinates

9
(1, 0, 0), (0, 1, 0), (0, 0, 1). Two distinct points a and b are conjugate

with respect to the conic represented by equation {29) if and only if

9
Graustein, op. cit. , p. 157.

21

(30) > p*..a.b. = 0.

Since (1, 0, 0) and (0, 1, 0) are conjugate points with respect to the conic
represented by equation (29), p' = 0, and by definition p'^ = p' 21 -
In a similar manner p' = p 1 = 0, and p' = p' = 0. So equation
(29) reduces to

(31) p^x'j 2 +P' 22 x' 2 2 +P'3 3 x'3 2 = 0, *

p, ll p, 22 P '33 *°-

Equation (31) is now reduced to an equation in standard form by
another change of coordinates. There are two cases to be considered.
If all of the coefficients of equation (31) are of the same sign they are
assumed to be all positive. If they were all negative, multiplying
equation (31) by -1 would make them all positive. So the linear
tr an sf or mation

(32) o-x^' = \p' u Xl ', <Tx 2 " = N/p"^ x,',

22 A 2

rx 3 " = ^ X 3'«

reduces equation (31) to

(33) Xl " 2 + x 2 " 2 + x 3 " 2 = 0.

22

If not all of the coefficients are of the same sign, it is assumed two
are positive and one is negative. A preassigned coefficient is made
negative by either multiplying equation (31) by -1 or renaming the
vertices of triangle pqr, or both. So assume p'- and p' are positive
and p 1 ,, is negative. Then the linear transformation

(34) rx"=^ Xj', <Tx 2 "='vp T ,

22 A 2'

2 ..2

,?.

x '■ + x '' •

- x;

i

1 2

3

<Tx 3 " =^ 3 x 3 \

reduces equation (31) to
(35)

Equation (33) represents an equation in point coordinates of a conic
which has no real points, and equation (35) represents an equation in
point coordinates of a conic which has real points. This results in the
following theorem:

THEOREM 14: The equation of a conic is reducible by a change of
projective coordinates to

2 2 2

(36a) x + x + x = 0,

if the conic has not a real trace, or

23

(36b) X;L + x z - x 3 = 0,

if the conic has a real trace.

The equations in line coordinates of a conic which correspond
respectively to equations (36a) and (36b) are

2 2 2

(37a) v x + v 2 + v 3 = 0,

and

z z z

(37b) v + v - v = 0.

Equation (36a) represents a conic which is determined by a change
of coordinates from equation (28). The aforementioned properties of
the conic are preserved in the transformation so all conies without a
real trace can be represented as one type of conic. Also all conies
with a real trace can be represented as one type of a conic. So there
are only two types of conies in the projective plane, those with real
traces and those without real traces.

24
ACKNOWLEDGME NT

The author wishes to express his sincere thanks and appreciation
to Dr. Neal E. Foland for his helpful suggestions and assistance with
the preparation of this report.

25

BIBLIOGRAPHY

1. Adler, Claire Fisher. Modern Geometry . New York:

McGraw-Hill Book Company, Inc., 1958.

2. Grau stein, William C. Introduction to Higher Geometry .

New York: The Macmillan Company, 1930.

3. Holgate, Thomas F. Projective Pure Geometry. New York:

The Macmillan Company, 1930.

4. O'Hara, C. W. , and D. R. Ward. An Introduction to

Projective Geometry. London: Oxford University Press,
1937.

5. Young, John Wesley. Projective Geometry . Chicago:

The Open Court Publishing Company, 1930.

PROJECTIVE PROPERTIES OF CONICS

by

GLEN HERBERT STARK

B. S. , Bethany College, 1962

AN ABSTRACT OF A MASTER'S REPORT

submitted in partial fulfillment of the
requirements for the degree

MASTER OF SCIENCE

Department of Mathematics

KANSAS STATE UNIVERSITY

Manhattan, Kansas

1964

This report contains a development of some of the projective
properties of conies. As background material for the projective
properties, the definition of a point conic and a line conic is given
in terms of projective coordinates. The particular properties which
are discussed are those pertaining to conjugate points and lines,
poles and polar s, and self polar triangles.

Conjugate points have the following properties: Two points,
the first of which lies on the conic, are conjugate if and only if the
second point lies on the tangent at the first, and two points neither
of which lies on the conic are conjugate if and only if they separate
harmonically the points of intersection of their line with the conic.
Conjugate lines have the following properties which correspond to
those of conjugate points: Two lines, the first of which is a tangent
to a conic, are conjugate if and only if the second passes through
the point of contact of the first with the conic, and two lines, neither
of which is a tangent to the conic, are conjugate if and only if they
separate harmonically the tangents from their point of intersection.

Poles and polar s are developed from conjugate points and lines.
Poles and polar s exhibit the following properties: A point is on a conic
if and only if it lies on its polar, and a line is tangent to a conic
if and only if it contains its pole. In addition, the following

is true: The pole of the line joining two points is the point of
intersection of the polar s of the two points, and the polar of the
point of intersection of two lines is the line joining the poles of
the two lines. The concept of poles and polar s leads to the develop-
ment of self polar triangles.

A triangle is self polar relative to a conic if and only if each
pair of sides are conjugate lines relative to the conic, and each pair
of vertices are conjugate points relative to the conic. The report
concludes with a self polar triangle being used as a triangle of
reference to reduce the equation of a conic to standard form. This
yields the result that there are only two types of conies in the
projective plane, those with real traces and those without real
traces.

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