Center-line of a conic section
In this chapter we consider only affine conic sections.
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.
- 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.
Two directions are conjugated if and only if the corresponding ideal
points are conjugated points relative to the conic section.
- 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.
(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
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
- 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.
Topics and Problems
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