Center-point of a conic section

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

In this chapter we consider only affine conic sections.

Center-point of a conic section

Each real pole of the ideal line is a 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

If C(x_o,y_o,z_o) is a center-point of a conic section F(x,y,z) = 0, then x_o,y_o,z_o is a solution of F_x' (x,y,z) = 0 and F_y' (x,y,z) = 0.

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

If x_o,y_o,z_o is a solution of F_x' (x,y,z) = 0 and F_y' (x,y,z) = 0, then C(x_o,y_o,z_o) is a center-point of the conic section F(x,y,z) = 0.

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

From previous theorems we see that

 
 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