General equation of a conic section - properties

Functions of x,y and z

In this chapter we consider real functions from R x R x R to R.
The image of a triplet (x,y,z) is a real number.
Example: f(x,y,z) = 3 x y - x z + 6 y -4

Homogeneous polynomial function F(x,y,z) and equation

If F(x,y,z) is a homogeneous polynomial in x, y, and z then the corresponding function F(x,y,z) is a homogeneous polynomial function and F(x,y,z) = 0 is a homogeneous polynomial equation.
Example: 
 

        F(x,y,z) = x z + y z - x2

Graph of a homogeneous polynomial equation

If (xo,yo,zo) is a solution (different from (0,0,0)) of a homogeneous polynomial equation F(x,y,z) = 0, then (r.xo,r.yo,r.zo) is a solution too.
With all this solutions corresponds exactly one point. The set of all points P(xo,yo,zo) such that F(xo,yo,zo) = 0, is called the graph of the equation F(x,y,z) = 0.

Such graph is an algebraic curve.
F(x,y,z) = 0 is the equation of the algebraic curve.
F(x,y,1) = 0 is the cartesian equation of the algebraic curve.

Degenerated algebraic curve

We say a curve c is degenerated if and only if F(x,y,z) = f(x,y,z) . g(x,y,z) with f(x,y,z) and g(x,y,z) homogeneous polynomial functions in x,y and z.
If c1 is the graph of f(x,y,z)=0 and c2 is the graph of g(x,y,z)=0, then c is the union of c1 and c2. c1 and c2 are called the components of the degenerated curve c.

Example: 

 
F(x,y,z) = (x2  + y2  - 9) (x + y - 1)
The curve of F = 0 is a degenerated curve. The components are 
 
a circle with equation  (x2  + y2  - 9) = 0  and
a line with equation   (x + y - 1) = 0

Conic section

An algebraic curve with an equation of the form 
 
a x2 + 2 b" x y + a' y2 + 2 b' x z + 2 b y z + a" z2= 0
is called a conic section.

In all following pages, we write this equation as F(x,y,z) = 0. The coefficients are real numbers. The cartesian equation of the conic section is : 

 
a x2  + 2 b" x y + a' y2  + 2 b' x + 2 b y + a" = 0

Affine conic section

A conic section is affine if and only if the ideal line is not a component of the conic section.

Partial derivatives and definitions

The three partial derivatives of the equation of a conic section are: 
 
Fx'(x,y,z)  = 2 a x + 2 b" y + 2 b' z = 2 ( a x + b" y + b' z )

Fy'(x,y,z)  = 2 b" x + 2 a' y + 2 b z = 2 ( b" x + a' y + b z )

Fz'(x,y,z)  = 2 b' x + 2 b y + 2 a" z = 2 ( b' x + b y + a" z )
The matrix formed by half the coefficients of x, y and z is 
 
        [ a     b"      b']
  C =   [ b"    a'      b ]
        [ b'    b       a"]
It is the symmetric cubic matrix of the conic section. The determinant of this matrix is usually written as the Greek capital delta. Here we denote this determinant as 'DELTA'. 
 
        | a     b"      b'|
DELTA = | b"    a'      b |
        | b'    b       a"|
The matrix 
 
        [ a     b"]
        [ b"    a']
is the quadratic matrix of the conic section. The determinant of this matrix is usually written as the Greek delta. Here we denote this determinant as 'delta'. 
 
        | a     b"|
delta = | b"    a'| = a a' - b"2
The cofactors of the elements of the cubic matrix are denoted as 
 
        A, A', A", B, B', B"
In the following sections we denote : 
 
        [x]        [x1]        [x2]
    P = [y]   P1 = [y1]   P2 = [y2]
        [z]        [z1]        [z2]

Matrix notation of the equation of a conic section

 

         T                [ a     b"      b'] [x]
        P C P = [x  y  z].[ b"    a'      b ].[y]
                          [ b'    b       a"] [z]


                          [ a x + b" y + b' z ]
              = [x  y  z].[ b" x + a' y + b z ]
                          [ b' x + b y + a" z ]


              = x(a x + b" y + b' z)+y(b" x + a' y + b z)+z(b' x + b y + a" z)

              = a x2  + 2 b" x y + a' y2  + 2 b' x z + 2 b y z + a" z2

So, the equation of a conic section is

        PT C P = 0

Corollaries

It is easy to prove that 
 
F(x1,y1,z1) = P1T C P1

F(kx1,ky1,kz1) =  (kP1)T C (kP1) = k (P1T C P1)

F(x1 + x2, y1 + y2, z1 + z2) = (P1 + P2)T  C (P1 + P2)


        [ a     b"      b']  [x1]        [Fx'(x1,y1,z1)]
C.P1 =  [ b"    a'      b ]. [y1] = (1/2).[Fy'(x1,y1,z1)]
        [ b'    b       a"]  [z1]        [Fz'(x1,y1,z1)]

Switching property

Theorem:
 
If F(x,y,z) = a x2  + 2 b" x y + a' y 2 + 2 b' x z + 2 b y z + a" z2

then

     x1.Fx'(x2,y2,z2) + y1.Fy'(x2,y2,z2) + z1.Fz'(x2,y2,z2)

  =  x2.Fx'(x1,y1,z1) + y2.Fy'(x1,y1,z1) + z2.Fz'(x1,y1,z1)

proof:

     x1.Fx'(x2,y2,z2) + y1.Fy'(x2,y2,z2) + z1.Fz'(x2,y2,z2)

                        [Fx'(x2,y2,z2)]
        =  [x1  y1  z1] [Fy'(x2,y2,z2)] = 2 P1T C P2
                        [Fz'(x2,y2,z2)]
and
     x2.Fx'(x1,y1,z1) + y2.Fy'(x1,y1,z1) + z2.Fz'(x1,y1,z1)

                        [Fx'(x1,y1,z1)]
        =  [x2  y2  z2] [Fy'(x1,y1,z1)] = 2 P2T C P1
                        [Fz'(x1,y1,z1)]

But P1T C P2 is a number; and the transpose of a number is that number itself.

So,
        P1T C P2 = (P1T C P2)T  = P2T C P1

Euler's formula

 
If F(x,y,z) = a x2  + 2 b" x y + a' y2  + 2 b' x z + 2 b y z + a" z2

then

x1.Fx'(x1,y1,z1) + y1.Fy'(x1,y1,z1) + z1.Fz'(x1,y1,z1) = 2 F(x1,y1,z1)

Proof:

        x1.Fx'(x1,y1,z1) + y1.Fy'(x1,y1,z1) + z1.Fz'(x1,y1,z1)


                        [Fx'(x1,y1,z1)]
        =  [x1  y1  z1] [Fy'(x1,y1,z1)]
                        [Fz'(x1,y1,z1)]

        =   [x1  y1  z1] .2 C P1 = 2 P1T C P1 = 2 F(x1,y1,z1)

Taylor's formula

 
If F(x,y,z) = a x  + 2 b" x y + a' y  + 2 b' x z + 2 b y z + a" z

then

        F(kx1 + lx2, ky1 + ly2, kz1 + lz2)

        = k2 F(x1,y1,z1)

          + kl(x1.Fx'(x2,y2,z2) + y1.Fy'(x2,y2,z2) + z1.Fz'(x2,y2,z2))

          + l2  F(x2,y2,z2)
Proof:
        F(kx1 + lx2, ky1 + ly2, kz1 + lz2)

        = (k P1 + l P2)T C (k P1 + l P2)

        = (k P1T + l P2 ).(k C P1 + l C P2)

        = k2 P1T C P1 + kl(P1T C P2 + P2T C P1) + l2 P2T C P2

        = k2 F(x1,y1,z1)

          + kl(x1.Fx'(x2,y2,z2) + y1.Fy'(x2,y2,z2) + z1.Fz'(x2,y2,z2))

          + l2 F(x2,y2,z2)

New equation of a conic section after a projective coordinate transformation

 
We know that the transformation formulas are
                                                              [x']
        P = M P' with M = the transformation matrix and P' =  [y']
                                                              [z']
Then

        F(x,y,z) = 0    (condition for old x,y,z)

<=>
        PT C P = 0

<=>                     (P = M P')
        P'T MT C M P' = 0     (condition for new x',y',z')

<=>
        P'T C1 P' = 0  (New equation of a conic section)

So the connection between the old C and the new C1 is
          C1 =  MT C M

New DELTA of a conic section after a projective coordinate transformation

 
          C1 =  MT C M

We take the determinant of both sides
        DELTA1 = determinant(MT ) DELTA determinant(M)
<=>
        DELTA1 =  DELTA . (determinant(M))2

Corollary

The sign of DELTA is invariant by a projective coordinate transformation.

New delta of a conic section after an affine coordinate transformation

It can be proved that
 
        delta1 =  delta . (determinant(M))2

Corollary:

The sign of delta is invariant by a affine coordinate transformation.

Property of a metric coordinate transformation

It can be proved that a + a' is invariant for a metric transformation.

Topics and Problems

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