Pole and polar line

• Polar line of a point A with respect to a conic section
  
  • Corollaries
    
• Polar line and harmonic conjugate points
  
• If a point is on its polar line then it is on the conic section
  
• Polar line and a degenerated conic section
  
• If A is on a polar line of B, then B is on a polar line of A.
  
• Pole of a line with respect to a conic section
  
• Each line has exactly one pole with respect to a non-degenerated conic section
  
• Calculation of a pole
  
• Conjugated points
  
• Criterion for conjugated points
  
• Conjugated lines
  
• Polar transformation with respect to a non-degenerated conic section
  
• A polar transformation preserves the cross ratio
  
• Polar triangle
  
• Equation of a conic section conjugated to a triangle
  
  • Theorem 1
    
  • Theorem 2
    
  • Theorem 3
    
  • Theorem 4
    

Polar line of a point A with respect to a conic section

1. Point A is a simple point.
   The polar line of A(x₁,y₁,z₁) with respect to a conic section with equation F(x,y,z) = 0 is the line

    
           x1.Fx' (x,y,z) + y1.Fy' (x,y,z) + z1.Fz' (x,y,z) = 0
   <=>
           x.Fx' (x1,y1,z1) + y.Fy' (x1,y1,z1) + z.Fz' (x1,y1,z1) = 0
   
   with matrix notation:
                           [Fx' (x1,y1,z1)]
   <=>        [x   y   z ] [Fy' (x1,y1,z1)] = 0
                           [Fz' (x1,y1,z1)]
   
   <=>
                   PT C P1 = 0
   
   <=>
                   P1T C P = 0
   
   

2. Point A is a double point.
   Each line is a polar line of A.

Corollaries

• If point A is on the conic section, the polar line is the tangent line in A.
• If point A is not on a non-degenerated conic section the polar line is the tangent chord of point A.

Polar line and harmonic conjugate points

Proof:
A(x₁,y₁,z₁) is not on conic section F(x,y,z) = 0.
Say B(x,y,z) is an arbitrary point.
A variable point P of the line AB has coordinates

 
        (x + h x1, y + h y1, z + h z1)

        Point P is on the conic section
<=>
        F(x + h x1, y + h y1, z + h z1) = 0
<=>
        F(x,y,z)

        + h (x.Fx' (x1,y1,z1) + y.Fy' (x1,y1,z1) + z.Fz' (x1,y1,z1))

        + h2  F(x1,y1,z1)  = 0

Points A and B are harmonic conjugate points with respect to the intersection points P₁ and P₂ of the conic section and the variable line AB

 
<=>
        (P1,P2,A,B) = -1
<=>
        (A,B,P1,P2) = -1
<=>
        h1 = - h2
<=>
        h1 + h2 = 0
<=>
        x.Fx' (x1,y1,z1) + y.Fy' (x1,y1,z1) + z.Fz' (x1,y1,z1) = 0
<=>
        B is element of the stated set

If a point is on its polar line then it is on the conic section

If that point A(x₁,y₁,z₁) is a simple point then:

 
        A(x1,y1,z1) is on its polar line
<=>
        x1.Fx' (x1,y1,z1) + y1.Fy' (x1,y1,z1) + z1.Fz' (x1,y1,z1) = 0
<=>
        2.F(x,y,z) = 0
<=>
        point A(x1,y1,z1) is on the conic section

Polar line and a degenerated conic section

Proof:
Take a point A(x₁,y₁,z₁), different from a double point.
The polar line of A is x₁.F_x' (x,y,z) + y₁.F_y' (x,y,z) + z₁.F_z' (x,y,z) = 0 Say D(xo,yo,zo) is a double point.
We investigate if D is on the polar line.

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

      = x1 . 0 + y1 . 0 + z1 . 0

      = 0

If A is on a polar line of B, then B is on a polar line of A.

 
        A(x1,y1,z1) is on the polar line of B(x2,y2,z2)
<=>
        x1.Fx' (x2,y2,z2) + y1.Fy' (x2,y2,z2) + z1.Fz' (x2,y2,z2) = 0
<=>
        x2.Fx' (x1,y1,z1) + y2.Fy' (x1,y1,z1) + z2.Fz' (x1,y1,z1) = 0
<=>
        B(x2,y2,z2) is on the polar line of A(x1,y1,z1)

Pole of a line with respect to a conic section

 
        Point A is a pole of line a with respect to a conic section
<=>
        Line a is a polar line of A.

Each line has exactly one pole with respect to a non-degenerated conic section

 
The conic section has equation  PT C P = 0

The line a has equation
        u x + v y + w z = 0
<=>
                  [x]
        [u  v  w] [y] = 0
                  [z]

<=>
        U.P = 0         with U = [u  v  w]



         [x1]
Say P1 = [y1]  are the coordinates of  a possible pole of line a.
         [z1]

The polar line  is      P1T C P = 0

This polar line is also the line U.P = 0

From this we have :     k U =   P1T C

<=>                     P1T = k U C-1

From that last result, we see that there is exactly one pole of the line a.

Calculation of a pole

 
Take the conic section  y2  - 2 x = 0 and the line x + y + 1 = 0

• First method : w use the formula from previous theorem.

   
              [ 0,  0, -1 ]
          C = [ 0,  1, 0  ]       U = [1   1   1]
              [ -1, 0, 0  ]
  
           -1  [ 0,  0, -1 ]
          C  = [ 0,  1, 0  ]
               [ -1, 0, 0  ]
  
            T                  [ 0,  0, -1 ]
          P1  = k [1   1   1]. [ 0,  1, 0  ] = k [-1   1   -1]
                               [ -1, 0, 0  ]
  
          The pole is point (-1,1,-1) or (1,-1,1)
  

• Second method If P(x₁,y₁,z₁) is the pole, then the polar line is

   
          x1.Fx' (x,y,z) + y1.Fy' (x,y,z) + z1.Fz' (x,y,z) = 0
  <=>
          x1(-2z) + y1(2y) + z1(-2x) = 0
  <=>
          z1 x - y1 y + x1 z = 0
  

  This line has to be the line x + y + z = 0. So the pole is point (1,-1,1).
• Third method We choose two simple points P₁(0,1,-1) and P₂(1,0,-1) on line x + y + 1 = 0. Say P is the pole of that line.
  The polar line of P₁ x + y = 0 contains point P. The polar line of P₂ z - x = 0 contains point P. Thus, P is the intersection point of these polar lines. So the pole is point (1,-1,1).

Conjugated points

 
        The points A and B are conjugated with respect to a conic section
<=>
        A is on a polar line of point B
<=>
        B is on a polar line of point A.

Criterion for conjugated points

1. Points A and B aren't double points

    
           A(x1,y1,z1) and B(x2,y2,z2) are conjugated points
   <=>
           A is on a polar line of point B
   <=>
           x1.Fx' (x2,y2,z2) + y1.Fy' (x2,y2,z2) + z1.Fz' (x2,y2,z2) = 0
   

2. A is a double point

    
           A(x1,y1,z1) and B(x2,y2,z2) are conjugated points
   <=>
           B is an arbitrary point
   <=>
           x2.0 + y2.0 + z2.0 = 0
   <=>
           x2.Fx' (x1,y1,z1) + y2.Fy' (x1,y1,z1) + z2.Fz' (x1,y1,z1) = 0
   <=>
           x1.Fx' (x2,y2,z2) + y1.Fy' (x2,y2,z2) + z1.Fz' (x2,y2,z2) = 0
   

3. A is a double point
   Analogous as in the previous case.

 
        A(x1,y1,z1) and B(x2,y2,z2) are conjugated points
<=>
        x2.Fx' (x1,y1,z1) + y2.Fy' (x1,y1,z1) + z2.Fz' (x1,y1,z1) = 0
<=>
        x1.Fx' (x2,y2,z2) + y1.Fy' (x2,y2,z2) + z1.Fz' (x2,y2,z2) = 0

Conjugated lines

 
        The lines a and b are conjugated with respect to a conic section
<=>
        a contains each pole of line b AND
        b contains each pole of line a

 
        The lines a and b are conjugated with respect to a conic section
<=>
        a contains the  pole of b
<=>
        b contains the  pole of a

Polar transformation with respect to a non-degenerated conic section

Remark :
Each ordered quartet of concurrent lines a,b,c,d is transformed in an ordered quartet of collinear points A,B,C,D.
Each ordered quartet of concurrent points A,B,C,D is transformed in an ordered quartet of collinear lines a,b,c,d .

A polar transformation preserves the cross ratio

 
        A(x1,y1,z1) B(x2,y2,z2)

        C(x1 + h x2, y1 + h y2, z1 + h z2)

        D(x1 + h'x2, y1 + h'y2, z1 + h'z2)

 
        a( Fx' (x1,y1,z1) , Fy' (x1,y1,z1) , Fz' (x1,y1,z1) )
        b( Fx' (x2,y2,z2) , Fy' (x2,y2,z2) , Fz' (x2,y2,z2) )
        c( Fx' (x1,y1,z1) + h Fx' (x2,y2,z2) , ..., ...)
        d( Fx' (x1,y1,z1) + h' Fx' (x2,y2,z2) , ..., ...)

Polar triangle

We say that the polar triangle is conjugated to the conic section or that the conic section is conjugated to the polar triangle.

Equation of a conic section conjugated to a triangle

Theorem 1

 
        k A2  + l B2  + m C2  = 0

Proof:

 
        B = 0 is polar line of point B
=>
        (B,C,S1,S2) = -1
=>
        (AB,AC,AS1,AS2) = -1
=>
        There is a value of h such that
        Line AS1 has equation B + h C = 0  and
        Line AS2 has equation B - h C = 0

• The degenerated conic section with equation (B + h C).(B - h C) = 0
• The degenerated conic section with equation A.A = 0

 
        (B + h C).(B - h C) + k A.A = 0
<=>
        B2  - h2  C2  + k A2  = 0

Theorem 2

 
        k A2  + l B2  + m C2  = 0

Proof:
Since the conic section is non-degenerated, the parameter l is not 0.
Dividing the equation by l, the conic section has equation

 
        B2 + (m/l) C2  + (k/l) A2 = 0
                                    2
                We denote m/l as - h
<=>
        B2 - h2 C2  + (k/l) A2 = 0
<=>
        (B - h C) (B + h C) + (k/l) A2 = 0

• The degenerated conic section with equation (B + h C).(B - h C) = 0
• The degenerated conic section with equation A.A = 0

Theorem 3

 
        k A2  + l B2  + m C2  = 0

Proof:.. Say the conic section is degenerated in the lines d1 and d2.

1. First case: d1 is different from d2
   

   

    
           C = 0 is polar line of poin C.
   
   =>      (B,C,S1,S2) = -1
   
   =>      (AB,AC,AS1,AS2) = -1
   
   =>      There is a value of h such that
           Line AS1 has equation B + h C = 0  and
           Line AS2 has equation B - h C = 0
   
   =>      The degenerated conic section has equation
   
           B2  - h2  C2  = 0
   

2. Second case: d1 coinciding with d2
   The equation of the conic section is

    
           A2 = 0  or  B2 = 0  or C2 = 0
   

Theorem 4

 
        k A2  + l B2  + m C2  = 0

Proof:

1. First case: 2 parameters are zero ; take l = m = 0
   The equation of the conic section is A.A = 0 and the triangle is conjugated with the conic section.
2. Second case : 1 parameter is zero ; take k = 0
   The equation of the conic section can be written as

    
           B2  - h2  C2  = 0  <=> (B - h C) (B + h C) = 0
   

   Then the conic section is conjugated to the triangle.
3. Third case: no parameter is zero. It can be proved that this case cannot occur.