- 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

- 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

- New delta of a conic section after an affine coordinate transformation

- Property of a metric coordinate transformation

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

Example:

F(x,y,z) = x z + y z - x^{2}

With all this solutions corresponds exactly one point. The set of all points P(x

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.

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 1:

F(x,y,z) = (xThe curve of F = 0 is a degenerated curve. The components are^{2}+ y^{2}- 9 z^{2}) (x + y - z)

a circle with equation (xExample 2:^{2}+ y^{2}- 9 z^{2}) = 0 and a line with equation (x + y - z) = 0

2 x^{2}+ 3 x y - 2 y^{2}- 3 x z - y z + z^{2}= (x + 2 y - z) (2 x - y - z) De curve with equation 2 x^{2}+ 3 x y - 2 y^{2}- 3 x z - y z + z^{2}= 0 is degenerated in two lines with equation x + 2 y - z = 0 and 2 x - y - z = 0

a xis called a conic section.^{2}+ 2 b" x y + a' y^{2}+ 2 b' x z + 2 b y z + a" z^{2}= 0

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 x^{2}+ 2 b" x y + a' y^{2}+ 2 b' x + 2 b y + a" = 0

FThe matrix formed by half the coefficients of x, y and z is_{x}'(x,y,z) = 2 a x + 2 b" y + 2 b' z = 2 ( a x + b" y + b' z ) F_{y}'(x,y,z) = 2 b" x + 2 a' y + 2 b z = 2 ( b" x + a' y + b z ) F_{z}'(x,y,z) = 2 b' x + 2 b y + 2 a" z = 2 ( b' x + b y + a" z )

[ 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"The cofactors of the elements of the cubic matrix are denoted as^{2}

A, A', A", B, B', B"In the following sections we denote :

[x] [x_{1}] [x_{2}] P = [y] P_{1}= [y_{1}] P_{2}= [y_{2}] [z] [z_{1}] [z_{2}]

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 x^{2}+ 2 b" x y + a' y^{2}+ 2 b' x z + 2 b y z + a" z^{2}So, the equation of a conic section is P^{T}C P = 0

F(x_{1},y_{1},z_{1}) = P_{1}^{T}C P_{1}F(kx_{1},ky_{1},kz_{1}) = (kP_{1})^{T}C (kP_{1}) = k^{2}(P_{1}^{T}C P_{1}) F(x_{1}+ x_{2}, y_{1}+ y_{2}, z_{1}+ z_{2}) = (P_{1}+ P_{2})^{T}C (P_{1}+ P_{2}) [ a b" b'] [x_{1}] [F_{x}'(x_{1},y_{1},z_{1})] C.P_{1}= [ b" a' b ]. [y_{1}] = (1/2).[F_{y}'(x_{1},y_{1},z_{1})] [ b' b a"] [z_{1}] [F_{z}'(x_{1},y_{1},z_{1})]

If F(x,y,z) = a x^{2}+ 2 b" x y + a' y^{2}+ 2 b' x z + 2 b y z + a" z^{2}then x_{1}.F_{x}'(x_{2},y_{2},z_{2}) + y_{1}.F_{y}'(x_{2},y_{2},z_{2}) + z_{1}.F_{z}'(x_{2},y_{2},z_{2}) = x_{2}.F_{x}'(x_{1},y_{1},z_{1}) + y_{2}.F_{y}'(x_{1},y_{1},z_{1}) + z_{2}.F_{z}'(x_{1},y_{1},z_{1}) proof: x_{1}.F_{x}'(x_{2},y_{2},z_{2}) + y_{1}.F_{y}'(x_{2},y_{2},z_{2}) + z_{1}.F_{z}'(x_{2},y_{2},z_{2}) [F_{x}'(x_{2},y_{2},z_{2})] = [x_{1}y_{1}z_{1}] [F_{y}'(x_{2},y_{2},z_{2})] = 2 P_{1}^{T}C P_{2}[F_{z}'(x_{2},y_{2},z_{2})] and x_{2}.F_{x}'(x_{1},y_{1},z_{1}) + y_{2}.F_{y}'(x_{1},y_{1},z_{1}) + z_{2}.F_{z}'(x_{1},y_{1},z_{1}) [F_{x}'(x_{1},y_{1},z_{1})] = [x_{2}y_{2}z_{2}] [F_{y}'(x_{1},y_{1},z_{1})] = 2 P_{2}^{T}C P_{1}[F_{z}'(x_{1},y_{1},z_{1})] But P_{1}^{T}C P_{2}is a number; and the transpose of a number is that number itself. So, P_{1}^{T}C P_{2}= (P_{1}^{T}C P_{2})^{T}= P_{2}^{T}C P_{1}

If F(x,y,z) = a x^{2}+ 2 b" x y + a' y^{2}+ 2 b' x z + 2 b y z + a" z^{2}then x_{1}.F_{x}'(x_{1},y_{1},z_{1}) + y_{1}.F_{y}'(x_{1},y_{1},z_{1}) + z_{1}.F_{z}'(x_{1},y_{1},z_{1}) = 2 F(x_{1},y_{1},z_{1}) Proof: x_{1}.F_{x}'(x_{1},y_{1},z_{1}) + y_{1}.F_{y}'(x_{1},y_{1},z_{1}) + z_{1}.F_{z}'(x_{1},y_{1},z_{1}) [F_{x}'(x_{1},y_{1},z_{1})] = [x_{1}y_{1}z_{1}] [F_{y}'(x_{1},y_{1},z_{1})] [F_{z}'(x_{1},y_{1},z_{1})] = [x_{1}y_{1}z_{1}] .2 C P_{1}= 2 P_{1}^{T}C P_{1}= 2 F(x_{1},y_{1},z_{1})

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(kx_{1}+ lx_{2}, ky_{1}+ ly_{2}, kz_{1}+ lz_{2}) = k^{2}F(x_{1},y_{1},z_{1}) + kl(x_{1}.F_{x}'(x_{2},y_{2},z_{2}) + y_{1}.F_{y}'(x_{2},y_{2},z_{2}) + z_{1}.F_{z}'(x_{2},y_{2},z_{2})) + l^{2}F(x_{2},y_{2},z_{2}) Proof: F(kx_{1}+ lx_{2}, ky_{1}+ ly_{2}, kz_{1}+ lz_{2}) = (k P_{1}+ l P_{2})^{T}C (k P_{1}+ l P_{2}) = (k P_{1}^{T}+ l P_{2}).(k C P_{1}+ l C P_{2}) = k^{2}P_{1}^{T}C P_{1}+ kl(P_{1}^{T}C P_{2}+ P_{2}^{T}C P_{1}) + l^{2}P_{2}^{T}C P_{2}= k^{2}F(x_{1},y_{1},z_{1}) + kl(x_{1}.F_{x}'(x_{2},y_{2},z_{2}) + y_{1}.F_{y}'(x_{2},y_{2},z_{2}) + z_{1}.F_{z}'(x_{2},y_{2},z_{2})) + l^{2}F(x_{2},y_{2},z_{2})

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) <=> P^{T}C P = 0 <=> (P = M P') P'^{T}M^{T}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 = M^{T}C M

C1 = M^{T}C M We take the determinant of both sides DELTA1 = determinant(M^{T}) DELTA determinant(M) <=> DELTA1 = DELTA . (determinant(M))^{2}

delta1 = delta . (determinant(M))^{2}

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