Center-line of a
conic section
- 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.
Center-line of a conic section
A center-line of a conic section is a polar line of an ideal point.
We say that the center-line is conjugated to the direction defined by the ideal point.
Corollaries
- 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.
Definitions
- 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.
Conjugated directions
Two directions are conjugated if and only if the corresponding ideal points are conjugated points with respect to the conic section.
Formula for conjugated directions
(r1,s1,0) and (r2,s2,0) are conjugated directions<=>
r1.Fx' (r2,s2,0) + s1. Fy' (r2,s2,0) = 0<=>
r1.(a r2 + b" s2) + s1.(b" r2 + a' s2) = 0<=>
a r1 r2 + b"(r1 s2 + s1 r2) + a' s1 s2 = 0
Conjugated center-lines of a ellipse or hyperbola
Two center-lines are conjugated center-lines of a ellipse or hyperbola if and only if one center-line is conjugated to the direction of the other center-line.
Corollaries:
- Two conjugated center-lines are harmonic conjugated lines with respect to the asymptotes.
- If two lines are harmonic conjugated lines with respect 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.