Center-line of a conic section




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

Definitions

Conjugated directions

Two directions are conjugated if and only if the corresponding ideal points are conjugated points relative 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:




Topics and Problems

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