Axes of an affine conic section




In this chapter we assume an orthonormal coordinate system.

Main direction of a conic section

A direction is a main direction of a conic section if and only if that direction and the orthogonal direction are conjugated relative to the conic section.

Main direction - formula

 
        direction  (r,s,0) is a main direction
<=>
        (r,s,0) and (-s,r,0) are conjugated directions
<=>
        a r(-s) + b"(r.r - s.s) + a' s r = 0
<=>
        b" r2 + (a' - a) r s - b" s2 = 0
If r is not 0, then s/r is the slope of the direction. Then we have the formula
 
        b" + (a' - a) m - b" m2  = 0

Number of main directions

  1. Circle
     
            a = a' and b" = 0
    
    The formula holds for any direction. Each direction is a main direction.
  2. Conic section different from a circle
Remark :
For a parabola, the main directions are the direction of the ideal point of the parabola and the direction orthogonal to that one.

Connection between main directions and characteristic vectors

Suppose the conic section is not a circle.
 
        (r,s,0) is a main direction
<=>
        (r,s,0) and (-s,r,0) are conjugated directions
<=>
        -s. Fx' (r,s,0) + r. Fy' (r,s,0) = 0
<=>
        There is a real number h such that
                / Fx' (r,s,0) = h.r
                \ Fy' (r,s,0) = h.s
<=>
        There is a real number h such that
                / a r + b" s = h r
                \ b"r + a' s = h s
<=>
        There is a real number h such that
                [a   b"][r]     [r]
                [b"  a'][s] = h [s]
<=>
        The main directions are the directions defined by the characteristic
        vectors of the matrix
                [a   b"]
                [b"  a']

Axis of a conic section

An axis of a conic section is a regular center-line with a main direction and who is polar line of the orthogonal main direction.

Vertex of a conic section

A vertex of a conic section is a regular intersection point of the conic section with an axis of the conic section.


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