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

[tex]aX^2+bY^2=0[/tex]

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

[tex]X^2+2XY+Y^2=0[/tex]

is the same as

[tex](X+Y)(X+Y)=0[/tex]

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...

 

[blue_raver22]

It is important to note that in projective geometry, the concept of a conic is defined differently than in Euclidean geometry. In projective geometry, a conic is a subset of the projective plane P2, which is a space where points, lines, and planes are all treated equally. This is why we use homogeneous coordinates, where each point is represented by a triplet (X, Y, Z) rather than just (x, y) as in Euclidean geometry.

The reason for the homogeneity of the conic equation is that it allows us to represent all points on the conic curve, including points at infinity. This is because when we use homogeneous coordinates, the coordinates of a point are only determined up to a scalar multiple. Therefore, the equation aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ = 0 represents all points on the conic, regardless of their coordinates.

Regarding the part of dXY + eXZ + fYZ, this is known as the cross term and can be eliminated by choosing suitable coordinates. This is because in projective geometry, the cross term does not change the shape of the conic, only its orientation. So, by choosing appropriate coordinates, we can simplify the equation and still represent the same conic curve.

The difference between a degenerate and non-degenerate conic lies in the number of points that satisfy the conic equation. A non-degenerate conic has exactly five points that satisfy the equation, while a degenerate conic has fewer than five points. In Euclidean geometry, a degenerate conic would be a pair of intersecting lines or a single point, whereas in projective geometry, it can also be a single line or a point at infinity.

In conclusion, the concept of a conic in projective geometry may seem different at first, but it allows us to study conic curves in a more general and inclusive way. Homogeneous coordinates and the elimination of the cross term are essential tools in understanding and working with conics in projective geometry. The distinction between degenerate and non-degenerate conics is also important in analyzing the properties of these curves.