Projective classification of the 2-nd order curves

Projective classification of the 2-nd order curves


	Projective classification of the 2-nd order curves

We call two points

$\begin{displaymath} (p_1,\; p_2,\; p_3) \mbox{ and } (q_1,\; q_2,\; q_3) \end{displaymath}$

equivalent

$\begin{displaymath} (p_1,\; p_2,\; p_3) \simeq (q_1,\; q_2,\; q_3) \end{displaymath}$

if there exists a real number

$\lambda \not= 0$

such that

$\begin{displaymath} (p_1,\; p_2,\; p_3) = \lambda \cdot (q_1,\; q_2,\; q_3) \end{displaymath}$

The set of all equivalence classes of points from

${\rm R}^3$

is called the projective plane,

${\rm P}^2.$

From a geometrical point of view, the class of points that are equivalent to

$p= (p_1,\; p_2,\; p_3)$

is a straight line that joins the origin and p (Fig.1.1).

Consider the second order curve

$\begin{displaymath} a_{11} x^2 + a_{12} xy + a_{22} y^2 + b_1 x + b_2 y + c =0, \end{displaymath}$

where

$a_{11},\; a_{12}, \; a_{22}$

and

$b_1,\; b_2, \; c$

are real numbers.

We can consider the equivalence classes for points

$\begin{displaymath} (x,\; y,\; 1) \end{displaymath}$

where

$(x,\; y)$

belongs to the curve. Then we obtain the second order curve on the projective plane

${\rm P}^2,$

defined as

$\begin{displaymath} a_{11} x^2 + a_{12} xy + a_{22} y^2 + b_1 xz + b_2 yz + cz^2 =0, \end{displaymath}$

where

$\begin{displaymath} (x,\; y,\; z) \simeq (\frac{x}{z},\; \frac{y}{z}, 1) \end{displaymath}$

and the point

$(\frac{x}{z},\; \frac{y}{z})$

belongs to our original 2-nd order curve.

Consider the matrix of the 2-nd order curve from

${\rm P}^2,$

$\begin{displaymath} A=\left( \begin{array}{ccc} a_{11} & \frac{a_{12}}{2} & \fr... ... \frac{b_{1}}{2} & \frac{b_{2}}{2} & c \end{array} \right) \end{displaymath}$

One can make A diagonal by the change of coordinates

$\begin{displaymath} \left( \begin{array}{c} x \ y \ z \end{array}\right)=... ...gin{array}{c} \bar x \ \bar y \ \bar z \end{array}\right) \end{displaymath}$

where

$\xi_1,$ $\xi_2$

and

$\xi_3$

are eigenvectors of A.

This change of coordinates corresponds to a rotation of

${\rm R}^3$

around the origin if we choose

$\xi_1,\;\;\xi_2, \;\;\xi_3$

so that

$\begin{displaymath} \vert \xi_1 \vert =1 ,\;\; \vert \xi_2 \vert =1,\;\; \vert \xi_3 \vert =1 \mbox{ and } \det(\xi_1 \;\;\xi_2 \;\; \xi_3)=1, \end{displaymath}$