General equation of a conic section - properties

• Functions of x,y and z
• Homogeneous polynomial function F(x,y,z) and equation
• Graph of a homogeneous polynomial equation
• Degenerated algebraic curve
• Conic section
• Affine conic section
• Partial derivatives and definitions
• Matrix notation of the equation of a conic section
• Corollaries
• Switching property
• Euler's formula
• Taylor's formula
• New equation of a conic section after a projective coordinate transformation
• New DELTA of a conic section after a projective coordinate transformation
• Corollary
• New delta of a conic section after an affine coordinate transformation
• Corollary:
• Property of a metric coordinate transformation

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 (x_o,y_o,z_o) is a solution (different from (0,0,0)) of a homogeneous polynomial equation F(x,y,z) = 0, then (r.x_o,r.y_o,r.z_o) is a solution too.
With all this solutions corresponds exactly one point. The set of all points P(x_o,y_o,z_o) such that F(x_o,y_o,z_o) = 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.