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

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 with respect 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

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

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.