Basic thing of Conic in Projective Geometry

[wawar05]

I am new about conic in projective geometry since it seems to be really different in euclidean plane.

A conic is a subset of P2 given by a homogenous quadratic equation:

aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ = 0

why is it homogeneous?

meanwhile, it suitable coordinates we have aX^2 + bY^2 + cZ^2 = 0, with a, b, c element {0, 1, -1}.

why the part of dXY + eXZ + fYZ can be erased?

what is the difference of degenerate conic and non-degenerate conic?

 

[micromass]

I am new about conic in projective geometry since it seems to be really different in euclidean plane.

A conic is a subset of P2 given by a homogenous quadratic equation:

aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ = 0

why is it homogeneous?

An equation is homogeneous if all the terms have the same degree. A term of the form X² or XY all have degree 2.

meanwhile, it suitable coordinates we have aX^2 + bY^2 + cZ^2 = 0, with a, b, c element {0, 1, -1}.

why the part of dXY + eXZ + fYZ can be erased?

The point is that you can change coordinates in such a way such that the d, e and f can be dropped. See http://home.scarlet.be/~ping1339/reduc.htm for a reduction of a conic section to it's reduced form.

what is the difference of degenerate conic and non-degenerate conic?

A degenerate conic consists of lines and points, whilme a non-degenerate conic is a nice curve.

For example, the conic

aX^2+bY^2=0

has only (0,0) as a solution, thus the conic is just a point. The conic

X^2+2XY+Y^2=0

is the same as

(X+Y)(X+Y)=0

Thus the conic is two times the line X=-Y. Such a conics are degenerate because they are not the nice smooth curves we expect.

 

[phinds]

I always find it helpful to go back to the basics when thinking about the conics. the geometry of it all is much simpler than equations and is SO easy to visualize. I've done up a drawing for you:

 

[wawar05]

^^, thank you for the helps... I am now having good understanding of conic related to projective plane...