Center-point of a conic section

• Center-point of a conic section
  
• Corollaries
  
• Theorem 1
  
• Theorem 2
  
• Formula
  

Center-point of a conic section

Corollaries

• The ideal line has exactly one pole with respect to a non-degenerated conic section. Thus, each non-degenerated conic section has exactly one center-point.
• It can be proved that

   
          point P is a regular center-point
  <=>
          point P is a symmetric point of the conic section
  

• The center-point of a non-degenerated parabola is the ideal point of the parabola.
• Each double point is a center-point

Theorem 1

Proof:

1. Point C is a simple point

    
           C is a simple center-point
   
   =>      The polar line of c is the ideal line
   
   =>      x.Fx' (xo,yo,zo) + y.Fy' (xo,yo,zo) + z.Fz' (xo,yo,zo) = 0
           is 0.x + 0.y + 1.z = 0
   
   =>      Fx' (xo,yo,zo) = 0 and Fy' (xo,yo,zo) = 0
   

2. Point C is a double point
   In this case it is trivial that F_x' (x_o,y_o,z_o) = 0 and F_y' (x_o,y_o,z_o) = 0

Theorem 2

Proof:

1. Point C is a double point
   

    
   =>      C is a pole of the ideal line
   
   =>      C is a center-point
   

2. Point C is a simple point => F_z' (x_o,y_o,z_o) is not 0

    
           The polar line of C(xo,yo,zo) is
           x.Fx' (xo,yo,zo) + y.Fy' (xo,yo,zo) + z.Fz' (xo,yo,zo) = 0
   <=>
           The polar line of C(xo,yo,zo) is
           x.0 + y.0 + z.Fz' (xo,yo,zo) = 0
   <=>
           The polar line of C(xo,yo,zo) is
            z = 0
   
   =>      Point C is center-point
   

Formula

 
        C is center-point of conic section F(x,y,z) = 0
<=>
        The coordinates of C are solutions of the system
                / Fx' (x,y,z) = 0
                \ Fy' (x,y,z) = 0