Degenerated conic sections and classification




Components

We know that a conic section 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.
Since F(x,y,z) is homogeneous with a degree = 2 , f(x,y,z) and g(x,y,z) are homogeneous with a degree = 1. From this we see that a degenerated conic section has 2 lines as components.

Lines through (0,0,1)

Two lines trough (0,0,1) have an equation of the form ux + vy = 0 and u'x + v'y = 0. The conic section who is degenerated in these lines has the equation (ux + vy)(u'x + v'y) = 0.
This equation has the form
 
        a x2  + 2 b" x y + a' y2  = 0
Conversely, every equation of the last form can be written as (ux + vy)(u'x + v'y) = 0 and is a degenerated conic section through the origin (0,0,1). The lines can be imaginary lines.

Example :

 
  9 x2 + 7 y2 - 24 x y = 0
<=>
  (3 x - y) (3 x - 7 y) = 0

Lines through (0,1,0)

In the same way as above, you'll find:
 
The conic section is degenerated in two lines through (0,1,0)

                if and only if

The conic section has equation  a x2  + 2 b' x z + a"z2 = 0

Lines through (1,0,0)

In the same way as above, you'll find:
 
The conic section is degenerated in two lines through (1,0,0)

                if and only if

The conic section has equation  a'y2  + 2 b y z + a"z2 = 0

Direction of two lines through (0,0,1)

The lines have an equation of the form
 
        a x2  + 2 b" x y + a' y2  = 0


         m is the slope of a component

<=>
        (1,m,0) is on the conic section
<=>
        a  + 2 b" m + a' m2  = 0
With this formula, you can calculate the slopes of the lines.
Remark:
If a' = 0, the equation of the conic section is
 
        a x2  + 2 b" x y = 0

Then the two components are

        x = 0 and a x + 2 b" y = 0

If a' = b" = 0, the equation of the conic section is
        a x2   = 0

Then the two components are  x = 0 and x = 0.

Nature of the components of two lines through (0,0,1)

Double points and simple points of a degenerated conic section

Each common point of the two components of a conic section is a double point. All the other points of the conic section are simple points.

Properties:

Theorem

If D(xo,yo,zo) is a double point of a conic section, then
 
        Fx'(xo,yo,zo) = 0 and Fy'(xo,yo,zo) = 0 and Fz'(xo,yo,zo) = 0
Proof:
If D(xo,yo,zo) is a double point, it is on both components of
 
        F(x,y,z) = (ux + vy + wz)(u'x + v'y + w'z) = 0
So, u xo + v yo + w zo = 0 and u'xo + v'yo + w'zo = 0 and then
 
        Fx'(xo,yo,zo) = u(u'xo + v'yo + w'zo ) + u'( u xo + v yo + w zo )
                   = u.0 + u'.0 = 0
Similarly for Fy'(xo,yo,zo) = 0 and Fz'(xo,yo,zo) = 0

Inverse theorem

 
If Fx'(xo,yo,zo) = 0 and Fy'(xo,yo,zo) = 0 and Fz'(xo,yo,zo) = 0
then D(xo,yo,zo) is a double point of a conic section.

Proof:
From Euler's formula we have

 
          2 F(xo,yo,zo)

        = xo.Fx'(xo,yo,zo) + yo.Fy'(xo,yo,zo) + zo.Fz'(xo,yo,zo)

        = xo.0 + yo.0 + zo.0 = 0
Thus, D(xo,yo,zo) is on the conic section.
Now, choose another point P(x1,y1,z1) of the conic section.
A variable point of the line DP is ( kxo + lx1, kyo + ly1, kzo + lz1).
This variable point is permanently on the conic section because
 
F( kxo + lx1, kyo + ly1, kzo + lz1)

        = k2 F(xo,yo,zo)

          + kl(xo.Fx'(x1,y1,z1) + yo.Fy'(x1,y1,z1) + zo.Fz'(x1,y1,z1))

          + l2 F(x1,y1,z1)

        = k2 F(xo,yo,zo)

          + kl(x1.Fx'(xo,yo,zo) + y1.Fy'(xo,yo,zo) + z1.Fz'(xo,yo,zo))

          + l2 F(x1,y1,z1)

        = k2 .0 + kl.0 + l2 .0 = 0
The line DP is a component of the conic section and thus the conic section is degenerated.
If the conic section contains another point P' not on DP then DP' is a component of the conic section and then D is double point.
If P' does not exist, the conic section is degenerated in two coinciding lines and D is double point.

Formula to calculate the double points of a conic section

From previous theorems we conclude:
 
        D(xo,yo,zo) is double point of a conic section

                if and only if

        xo,yo,zo  is a non-trivial solution of the system
                /  Fx'(x,y,z)  = 0
                |
                |  Fy'(x,y,z)  = 0
                |
                \  Fz'(x,y,z)  = 0

Criterion for degenerated conic section

A conic section is degenerated if and only if DELTA = 0

Proof:

 
        The conic section F(x,y,z) = 0 is degenerated

<=>

        The conic section has a double point

<=>
        The system
                /  Fx'(x,y,z)  = 0
                |
                |  Fy'(x,y,z)  = 0
                |
                \  Fz'(x,y,z)  = 0
        has a solution different from (0,0,0)

<=>
        The system
        /  2 ( a x + b" y + b' z ) = 0
        |
        |  2 ( b" x + a' y + b z ) = 0
        |
        \  2 ( b' x + b y + a" z ) = 0
        has a solution different from (0,0,0)

<=>
        DELTA = 0

Classification of the degenerated affine conic sections

 
        The ideal points are the solutions of


        /
        |a x2  + 2 b" x y + a' y2  + 2 b' x z + 2 b y z + a" z2 = 0
        |
        \ z = 0

<=>
        /
        | a x2  + 2 b" x y + a' y2  = 0
        |
        \  z = 0
From above we know that


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