- Center-line of a conic section

- Corollaries

- Definitions

- Conjugated directions

- Formula for conjugated directions

- Conjugated center-lines of a ellipse or hyperbola

- Corollaries:

In this chapter we consider only affine conic sections.

We say that the center-line is conjugated to the direction defined by the ideal point.

- A center-line of a non-degenerated conic section is the tangent chord of an ideal point.
- If an asymptote is not the ideal line, it is a center-line conjugated to its own direction.
- A center-line conjugated to a non-asymptotic direction is the set of all the midpoints of the chords with that direction.
- All center-lines of a non-degenerated parabola are parallel (With the ideal line as only exception). They all contain the ideal point of the parabola.

- A center-line conjugated to the direction of a chord, is conjugated to the chord itself.
- A center-line conjugated to the direction of a tangent line, is conjugated to the tangent line itself.

(r_{1},s_{1},0) and (r_{2},s_{2},0) are conjugated directions <=> r_{1}.F_{x}' (r_{2},s_{2},0) + s_{1}. F_{y}' (r_{2},s_{2},0) = 0 <=> r_{1}.(a r_{2}+ b" s_{2}) + s_{1}.(b" r_{2}+ a' s_{2}) = 0 <=> a r_{1}r_{2}+ b"(r_{1}s_{2}+ s_{1}r_{2}) + a' s_{1}s_{2}= 0

- Two conjugated center-lines are harmonic conjugated lines with respect to the asymptotes.
- If two lines are harmonic conjugated lines relative to the asymptotes, then these lines are two conjugated center-lines.
- Two conjugated center-lines are coinciding if and only if they are coinciding with an asymptote.
- Two conjugated center-lines of a circle are orthogonal.

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