In this chapter we consider only affine conic sections.

- The ideal line has exactly one pole relative 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

Proof:

- Point C is a simple point
C is a simple center-point => The polar line of c is the ideal line => x.F

_{x}' (x_{o},y_{o},z_{o}) + y.F_{y}' (x_{o},y_{o},z_{o}) + z.F_{z}' (x_{o},y_{o},z_{o}) = 0 is 0.x + 0.y + 1.z = 0 => F_{x}' (x_{o},y_{o},z_{o}) = 0 and F_{y}' (x_{o},y_{o},z_{o}) = 0 - 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

Proof:

- Point C is a double point

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

- Point C is a simple point => F
_{z}' (x_{o},y_{o},z_{o}) is not 0The polar line of C(x

_{o},y_{o},z_{o}) is x.F_{x}' (x_{o},y_{o},z_{o}) + y.F_{y}' (x_{o},y_{o},z_{o}) + z.F_{z}' (x_{o},y_{o},z_{o}) = 0 <=> The polar line of C(x_{o},y_{o},z_{o}) is x.0 + y.0 + z.F_{z}' (x_{o},y_{o},z_{o}) = 0 <=> The polar line of C(x_{o},y_{o},z_{o}) is z = 0 => Point C is center-point

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

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