Axes of an affine conic section

• Main direction of a conic section
  
• Main direction - formula
  
• Number of main directions
  
• Connection between main directions and characteristic vectors
  
• Axis of a conic section
  
• Vertex of a conic section
  

Main direction of a 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

 
        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
   • b" is not 0
     

      
             b" + (a' - a) m - b" m2  = 0
     
             The discriminant  = (a' - a)2 + 4 b"2 > 0
     

     So, there are two different main directions
   • b" = 0
     The formula gives now r s = 0 <=> r = 0 or s = 0
     So, there are two different main directions

Connection between main directions and characteristic vectors

 
        (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

Vertex of a conic section