- 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.

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" rIf r is not 0, then s/r is the slope of the direction. Then we have the formula^{2}+ (a' - a) r s - b" s^{2}= 0

b" + (a' - a) m - b" m^{2}= 0

- Circle
a = a' and b" = 0

The formula holds for any direction. Each direction is a main direction. - Conic section different from a circle
- b" is not 0

b" + (a' - a) m - b" m

So, there are two different main directions^{2}= 0 The discriminant = (a' - a)^{2}+ 4 b"^{2}> 0 - b" = 0

The formula gives now r s = 0 <=> r = 0 or s = 0

So, there are two different main directions

- b" is not 0

For a parabola, the main directions are the direction of the ideal point of the parabola and the direction orthogonal to that one.

(r,s,0) is a main direction <=> (r,s,0) and (-s,r,0) are conjugated directions <=> -s. F_{x}' (r,s,0) + r. F_{y}' (r,s,0) = 0 <=> There is a real number h such that / F_{x}' (r,s,0) = h.r \ F_{y}' (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']

The tutorial address is http://home.scarlet.be/math/

Copying Conditions

Send all suggestions, remarks and reports on errors to Johan.Claeys@ping.be The subject of the mail must contain the flemish word 'wiskunde' because other mails are filtered to Trash