Center-point of a conic section




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

Theorem 1

If C(xo,yo,zo) is a center-point of a conic section F(x,y,z) = 0, then xo,yo,zo is a solution of Fx' (x,y,z) = 0 and Fy' (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 Fx' (xo,yo,zo) = 0 and Fy' (xo,yo,zo) = 0

Theorem 2

If xo,yo,zo is a solution of Fx' (x,y,z) = 0 and Fy' (x,y,z) = 0, then C(xo,yo,zo) 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 => Fz' (xo,yo,zo) 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



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