Conic_Equation

0. Notation

In this file I adopt the affine representation of points of the plane, by which points are represented through column
vectors x = (x,y,1)^t and directions through corresponding points at infinity: u = (u,v,0)^t.
Here X^t represents the transpose of the matrix X.
The difference of two points x-x' is a direction. The sum of a point and a direction is a point x'=x+u.
Addition of points does not make sense except when forming their barycenter (e.g. their middle):

1. Conic equation

The conic equation in cartesian coordinates (not necessary orthogonal) is given in the form:
(1)                                                    f(x,y) = ax²+2hxy+by²+2gx+2fy+c = 0.
This corresponds to a matrix (quadratic) equation for the vectors  x=(x,y,1)^t:
(2)                                                     x^tMx = 0,
where M is the symmetric matrix:

2. Degenerate conics

The conic is called degenerate or singular or reducible, when its discriminant which is the determinant |M| of the 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 Mx=0 has a non-zero solution and the symmetric
matrix M is reducible through an orthogonal matrix Q to a diagonal form:

This implies that:
x^tMx = m r(x)² -n s(x)².
Where r(x), s(x) are two linear functions of x^t=(x,y,1).
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.

3. Genuine conics, asymptotes, axis

Genuine or irreducible conics are these for which their discriminant |M| is non-zero.
They can be investigated by intersecting them with lines. In fact, assume x is a point on the conic i.e.
satisfies f(x,y) =0 and m(t)=x+tu a line in parametric form with direction vector
u^t = (u,v,0). Then a second point of this line on the conic would satisfy:
(3)                                                  (x+tu)^tM(x+tu) = 0.
Implying  by (2):
                                                     2tx^tMu+t² u^tMu=0.
If there is a second point different from x which is obtained for t=0, then the t for this point satisfies:
(4)                                                     2x^tMu+t u^tMu=0.
This equation determines exactly one second point except in the case for which the direction vector
u^t = (u,v,0) satisfies:
(5)                                                 u^tMu=0.
This is equivalent with the matrix equation:

Last submatrix or minor M₂ of M is the second after M important element of the equation. Its determinant
J₂ = ab-h² characterizes the three categories of conics:  J₂ < 0 characterizes hyperbolas, J₂ = 0 characterizes
parabolas and J₂ > 0 characterizes ellipses.
Comming back to equation (4), if both x^tMu and u^tMu are non-zero, then
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  x^tMu = 0, then u^tMu is non-zero and t=0 is the only solution of (4).
     This means that line m(t) is a tangent line to the conic.
ii) if u^tMu=0 and x^tMu is non-zero then the condition can be interpreted
     as defining a second intersection point of the conic at infinity (writing  t=1/s and letting s converge to 0).
Thus hyperbolas characterized by J₂=ab-h²<0 have two directions such that lines parallel to these
directions intersect the conic at only one point. Parabolas with J₂=0 have only one such direction and
ellipses with J₂>0 have no such directions.
The two directions determined by (5) in the case of hyperbolas are called asymptotes.
Their bisectors are two orthogonal directions called the axes of the conic.
The one direction determined by (5) in the case of parabolas is called axis of the parabola (see section-11 below).
In all cases the axes are symmetry-axes of the conic.
The direction of the axes are parallel to the directions of the eigen-vectors of the matrix M₂. This follows from an
easy computation.

4. Tangents at points of the conic

By the previous section line m(t) passing through a point x of the conic is a tangent precisely when:
                                   u^tMu is non-zero but  x^tMu=0.
The general point of the line y=m(t)= x+tu satisfies also:
                                   x^tMy=x^tM(x+tu)=0.
Thus at point x of the conic the equation of the tangent is given (as a function of y) by:
(6)                               x^tMy=0.

5. Polar of a point

The tangent line of a conic at a point of it is a particular case of a more general line associated to a point
with respect to a conic. In fact, given the point x, not necessarily on the conic, equation (6) makes
sense and defines a line (w.r. to variable y) of course depending on x and the conic.
This more general line p_X defined through (6) is called the polar of x with respect to the conic.

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)=x+tu
the intersection points are the roots of the equation:
                                                      (m(t))^tM(m(t)) = 0,
which amounts to the quadratic:
(7)                                           t² (u^tMu) +2t (x^tMu) + (x^tMx) = 0.
If t₁, t₂ are the roots of this equation, then the harmonic conjugate of X is given by the parameter
t₃ = 2t₁t₂/(t₁+t₂) (follows from Harmonic.html 2.1 by setting h=0), which by the well known
identities for product and sum of roots reduces to:
(8)                                                t₃ = -(x^tMx)/(x^tMu).
Thus the locus of harmonic conjugates y to x is described by:
(9)                                                y = x -[(x^tMx)/(x^tMu)] u,
and (6) is easily veryfied.
The polar p_X at its intersections {D,E} with the conic has tangents passing through X. This is seen by considering
the vector y₀ representing such an intersection. It satisfies both (6) and the equation of the conic y₀^tMy₀=0,
hence also the equation of the tangent:
                                                    y₀^tM(y₀+t(x-y₀))=0.

6. Tangents from a point

The existence of a double root of equation (7) implies that the corresponding line   y_t=x+tu   intersects
the conic at a single point i.e. it is a tangent to the conic. The condition on the coefficients is:
                                                   (x^tMu)² - (u^tMu)(x^tMx)=0.
It is then readily verified that y_t satisfies a similar equation:
                                                   (y_t^tMx)² - (y_t^tMy_t)(x^tMx)=0.
This is a quadratic equation reducible into a product of two lines which are the tangents to the conic from point x.

7. Conjugation, pole of a line

Two points {X,Y} represented by x and y respectively are called conjugate with respect to the conic
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_X
of X, then also X is on the polar of p_Y. In fact the conic defines through its matrix a bijective transformation:
(10) F_M : P² -----> P²^*, x |------> x^tM,
of the projective plane P² onto its dual P²^* consisting of all lines of P². This is an example of a correlation and in
older bibliography is referred as the polar reciprocation with respect to a conic. To each point X this map corresponds the
polar p_X. It turns out that this map is a projectivity between the two projective spaces. Its inverse G = (F_M)^-1
corresponds to every line p represented by its coefficients (q,r,s) the point X(x) having for polar the given line p. This point
is called the pole of p and is found by solving the equation (with k non-zero constant):
(11) x^tM = k(q,r,s).
This is equivalent with the linear equation:

involving the inverse symmetric matrix M^-1. The projective nature of F_M makes it correspond to each line
p the line X^* of P²^*consisting of all lines through the pole X of p (we say X^* is the pencil of lines through X).
To every pair of points (X,Y), theline L_XY is the polar of the intersection point Z of the polars
{p_X, p_Y}. To every pair of lines (p,q), their intersection point X_pqis the pole of the line defined by the
poles (X_p, X_q) of (p,q). Besides the cross-ratio along a line p and the corresponding pencilof lines through
its pole X_p is preserved by F_M.
All these wonderful things make the theorems of the projective plane "double". Fixing a conic and taking
the images of lines through F_M every theorem about lines and coincidences on them transforms to a corresponding
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 duality
defined by a conic.

8. Center of the conic

The center of the conic can be defined as the pole of the line at infinity (z=0). Thus, by the previous section it is
the point defined by the equation:

In particular for parabolas (ab-h²=0) the center is a point at infinity, thus lying on its polar, consequently the parabola
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.

9. Dual conic

The tangents to the conic are given by (11) for points x on the conic. It follows that their coefficients
y = (q,r,s)^t satisfy the equation resulting by substituting x into the conic equation:

Since the equation of the conic can be multiplied by a constant often last equation is written in terms of the
adjoint matrix M^# of M which is M^-1 multiplied with the determinant |M| of M.
The argument can be reversed and shows the following:

Corollary If the coefficients L^t = (q,r,s) of a line satisfy a quadratic equation:
L^t N L = 0,
with a symmetric non-degenerate matrix N, then the line envelopes a conic i.e. is tangent to a conic,
namely the conic:
x^t N^#x = 0.

10. Conjugate directions, conic-axes

Conjugate are directions (i.e. vectors of the form (s,t,0)) which are conjugate with respect to M:
(12)                                                 u^tMv = 0.
Conjugate directions are defined by the middles of chords drawn in a fixed parallel direction u.
In fact, if x₁ and x₂=x₁+ku are two points of the conic, then:
                                      x₁^tMx₁ = 0,  x₂^tMx₂ = 0.
Later implies by substituting x₂:
                       ( x₁+ku)^tM(x₁+ku) = 0, which is equivalent to:
(13)                                   2 x₁^tMu + k u^tMu = 0.
On the other side the middles of the segments defined by x₁ and x₂ are defined by:
                                      y = (x₁+x₂)/2 =  x₁+(k/2)u.
Thus taking the product and taking (13) into account we find:
                            y^tMu   =   (x₁+(k/2)u)^tMu = 0.
This shows that the middles y of parallel chords in direction u satisfy the linear equation:
(14)                                                        y^tMu   = 0.
Writing the line described by  y in parametric form  y = y₀+tv we get:
                                                 y₀^tMu + t v^tMu = 0.
Since this is identically satisfied for all t it implies both:
(15)                           y₀^tMu =0       and       v^tMu = 0.
The first shows that the tangent at y₀ coincides with the direction of the parallel chords.
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 M₂ are the only orthogonal directions which are also conjugate.
Since the parallel chords to the one direction are bisected by the conjugate diameter it follows that
the directions of the eigen-vectors of M₂ are the symmetry axes of the conics. These are called
axes of the conic.

11. The cases of parabola and rectangular hyperbola

Regarding the conjugacy of diameters the case of parabola deserves some special attention.
First it should be noticed that the relation concerns really the submatrix:

This because of the zero third coordinate of u, v. In the parabola case the determinant
|M₂|=0, hence, the rank of the matrix is one and by elementary linear algebra there is a vector
w such that the matrix is representable as:

This means that for all directions u, all v=M₂u are collinear and consequently
for all possible directions u the equation:
v^tMu = v^tM₂u=0,
defines vectors v pointing in the same direction. This is the axis-direction of the parabola,
and it happens that the conjugates of all directions in the case of parabola coincide with the axis direction.
The case of rectangular hyperbola occurs when the two asymptotic directions determined by equation (5)
are orthogonal. This amounts to the condition:
a + b = 0.

12. Reduction to canonical form

By selecting the axes of the conic as coordinate axes and the center as the origin of coordinates, we obtain
the canonical form of a central conic, which is then represented by a diagonal matrix of the form:

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

Bibliography

[Eisenhart] Luther Pfahler Eisenhart Coordinate Geometry Dover, New York
[Loney] Loney, S. L. Coordinate Geometry. Delhi, AITBS Publishers, 1991

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