vectors

Here X

The difference of two points

Addition of points does not make sense except when forming their

(1) f(x,y) = ax

This corresponds to a matrix (quadratic) equation for the vectors

(2)

where M is the symmetric matrix:

is zero. In this case equation (2) splits into a union of two real or imaginary lines. This is seen most easily by elementary

linear algebra. In fact, in this case equation M

matrix M is reducible through an orthogonal matrix Q to a diagonal form:

This implies that:

Where r(

Thus the equation can be written:

Of course the factors, depending on the signs of {m,n}, can be real or imaginary. Anyway the resolution into two

lines is given by this procedure when |M|=0. For another less structural but simpler procedure to split the equation

into the product of two lines see the file ConicsDegenerate.html . There is also proved the inverse statement, that if

the equation splits into two lines then |M|=0.

They can be investigated by intersecting them with lines. In fact, assume

satisfies f(x,y) =0

(3) (

Implying by (2):

2t

If there is a second point different from

(4) 2

This equation determines exactly one second point except in the case for which the direction vector

(5)

This is equivalent with the matrix equation:

Last submatrix or

J

parabolas and J

Comming back to equation (4), if both

there is a second point of the conic on the line m(t). Last quantities cannot be simultaneously zero, since in that

case the whole line m(t) will be contained in the conic and the conic will be degenerate. Thus,

i) if

This means that line m(t) is a tangent line to the conic.

ii) if

as defining a second intersection point of the conic at infinity (writing t=1/s and letting s converge to 0).

Thus

directions intersect the conic at only one point. Parabolas with J

ellipses with J

The two directions determined by (5) in the case of hyperbolas are called

Their bisectors are two orthogonal directions called the

The one direction determined by (5) in the case of parabolas is called

In all cases the axes are

The direction of the axes are parallel to the directions of the eigen-vectors of the matrix M

easy computation.

The general point of the line y=m(t)=

Thus at point

(6)

with respect to a conic. In fact, given the point

sense and defines a line (w.r. to variable

This more general line p

The polar is characterized geometrically as the locus of points Y for which the cross ratio (A,B,X,Y)=-1, i.e.

as the locus of the harmonic conjugates Y of X with respect to {A,B} which are the intersection points of

a variable line through X with the conic. Denoting the line through X as usual with m(t)=

the intersection points are the roots of the equation:

(m(t))

which amounts to the quadratic:

(7) t

If t

t

identities for product and sum of roots reduces to:

(8) t

Thus the locus of harmonic conjugates

(9)

and (6) is easily veryfied.

The polar p

the vector

hence also the equation of the tangent:

the conic at a single point i.e. it is a

(

It is then readily verified that

(

This is a quadratic equation reducible into a product of two lines which are the tangents to the conic from point

if equation (6) is satisfied. This is a symmetric relation (because of the symmetry of M). It implies that if Y is on the polar p

of X, then also X is on the polar of p

(10) F

of the projective plane P

older bibliography is referred as the

polar p

corresponds to every line p represented by its coefficients (q,r,s) the point X(

is called the

(11)

This is equivalent with the linear equation:

involving the inverse symmetric matrix M

p the line X

To every pair of points (X,Y), theline L

{p

poles (X

its pole X

All these wonderful things make the theorems of the projective plane "double". Fixing a conic and taking

the images of lines through F

theorem for points (their poles) and lines joining them.Inversely every theorem about points and lines joining them

transforms to a theorem about lines (their polars) and intersections of them.This is the much celebrated

defined by a conic.

the point defined by the equation:

In particular for parabolas (ab-h

touches the line at infinity at its center. For the other (central) conics the center is determined by its cartesian coordinates:

For central conics whose center O is an ordinary point, every line through the center intersects the conic at two

points {A,B} which are symmetric with respect to the center. This follows from the fact that the harmonic conjugate

of O with respect to (A,B) is at infinity, according to the definition of the center.

Since the equation of the conic can be multiplied by a constant often last equation is written in terms of the

adjoint matrix M

The argument can be reversed and shows the following:

with a symmetric non-degenerate matrix N, then the line envelopes a conic i.e. is tangent to a conic,

namely the conic:

(12)

Conjugate directions are defined by the middles of chords drawn in a fixed parallel direction

In fact, if

Later implies by substituting

(

(13) 2

On the other side the middles of the segments defined by

Thus taking the product and taking (13) into account we find:

This shows that the middles

(14)

Writing the line described by

Since this is identically satisfied for all t it implies both:

(15)

The first shows that the tangent at

The second proves the initial claim about the conjugacy of the direction of the chords and the direction of

the line of their middles.

In particular the eigen-vectors of M

Since the parallel chords to the one direction are bisected by the conjugate diameter it follows that

the directions of the eigen-vectors of M

First it should be noticed that the relation concerns really the submatrix:

This because of the zero third coordinate of

|M

This means that for all directions

for all possible directions

defines vectors

and it happens that the conjugates of all directions in the case of parabola coincide with the axis direction.

The case of

are orthogonal. This amounts to the condition:

a + b = 0.

the

The reduction can be studied in [Loney, I, p.323], [Eisenhart, p. 208]. The matrix equation results then easily.

Harmonic.html

[Loney] Loney, S. L.

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