Definiton and associated matrixes of analytical quadric

Given a real quadratic polynomial $$q(x,y,z)=ax^2+by^2+cz^2+2fxy+2gxz+2hyz+ \\ +2px+2qy+2rz+d$$ in the rectangular coordinates $(x,y,z)$, we will say that the equation $q(x,y,z)=0$ defines a quadric, which we will denote by $Q$.

Let's remember that the definition of quadratic polynomial includes the condition for which the main part of $q(x,y,z)$ $$q_2(x,y,z)=ax^2+by^2+cz^2+2fxy+2gxz+2hyz$$does not equal zero.

A point $(a, b, c)$ belongs to the quadric $Q$ if and only if $Q (a, b, c) = 0$. The point is called real if $a, b, c$ are real, and imaginary if some of its coordinates are complex.

It is obvious that if $(a, b, c)$ is an imaginary point belonging to the quadric, as $q(x,y,z)$ is a real polynomial, $Q$ contains the conjugate of $(a, b, c)$.

If $(x',y'.z')$ is another system of rectangular coordinates and $$q(x',y',z')=a'x'^2+b'y'^2+c' z'^2+2f'x'y'+2g'x'z'+$$ $$+2h'y'z'+2p'x'+2q'y'+2r'z'+d'$$ it is a real quadratic polynomial in (x',y',z'), then we will say that $Q$ it coincides with $Q'$, or that the equations $q(x,y,z)=0$ and $q'(x',y',z')=0$ define the same quadric, if and only if a non zero real number $K$ exists, such that $$q'(x',y',z')=Kq(x',y'z')$$ where $q'(x',y',z')=0$ denotes the polynomial in $(x',y',z')$ which is obtained by replacing the coordinates $(x,y,z)$ of the polynomial $q(x,y,z)$ by the expressions of the change of coordinates $(x',y',z')$.

Associated matrixes

We set $$A= \begin{bmatrix} a & f & g \\ f & b & h \\ g & h & c \end{bmatrix}$$ and we say that it is the main matrix of the polynomial $q(x,y,z)$.

Similarly, we define $$\overline{A}=\begin{bmatrix} A & \omega^T \\ \omega d \end{bmatrix}, \omega=(p,q,r)$$ and we say that it is the matrix of the polynomial $q(x,y,z)$.

Also we say that $A$ is the main matrix of $\overline{A}$. These two matrices are also called infinity matrix and projection matrix of the conical curve.

The knowledge of $A$ is equivalent to the knowledege of the main part of $q(x,y,z)$ (that is to say to $q_2(x,y,z)$), since $$q_2(x,y,z)=(x,y,z)A(x,y,z)^T $$

Similarly, the knowledge of $\overline{A}$ is equivalent to the knowledge of $q(x,y,z)$ since $$q(x,y,z)=(x,y,z,1)\overline{A}(x,y,z,1)^T$$

Let's observe, nevertheless, that the quadric $Q$ only determines $\overline{A}$ except for a non-zero real factor.

Next, we are going to give two results that are going to allow us to reduce the general equation of a quadric:

• Given a polynomial $q(X)=q(x,y,z)$ in the coordinates $X=(x,y,z)$, with matrix $A$ and main matrix $\overline{A}$, the polynomial $q(X')=q(x',y',z')$ defined by the formula $q(X')=q(X'M^t+P)$ has matrix $\overline{A}'=\overline{M}^T\overline{A}\overline{M}$ and main matrix $A' = M^TAM$.

Note that in this result we used the notation $X=(x,y,z)$ to indicate that $X$ is the three-dimensional vector that takes as its coordinates $X$. We use this notation for simplifcation.

• Given a system of rectangular coordinates $X=(x,y,z)$ and a quadratic polynomial $q(x,y,z)$, there exists a system of rectangular coordinates $X'=(x',y',z')$ such that the main part of the polynomial $q(x',y',z')$ has the form $\lambda_1 x'^2+\lambda_2y'^2+\lambda_3z'^2$ which we will call a diagonal form. Also,$\lambda_1$,$\lambda_2$ and $\lambda_3$ are the eigenvalues of the main matrix of $q(x,y,z)$.

Consider the matrix $$\overline{A}=\begin{bmatrix} 1 & 2 & 0 & 1 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 5\end{bmatrix}$$ the equation of the quadric associated with the above mentioned matrix is calculated in the following way: $$\begin{bmatrix} x & y & z & 1\end{bmatrix} \begin{bmatrix}1 & 2 & 0 & 1 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 5\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}=\begin{bmatrix} x & y & z & 1\end{bmatrix} \begin{bmatrix}x+ 2y + 1 \\ 2x + 2y \\ z + 1 \\ x +z+ 5\end{bmatrix}=$$ $$=x^2+2y^2+z^2+4xy+2x+2z+5$$