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: $$$\begin{bmatrix} x & y & z & 1\end{bmatrix} \begin{bmatrix}1 & 2 & 0 & 1 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 5\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}=\begin{bmatrix} x & y & z & 1\end{bmatrix} \begin{bmatrix}x+ 2y + 1 \\ 2x + 2y \\ z + 1 \\ x +z+ 5\end{bmatrix}=$$$ $$$=x^2+2y^2+z^2+4xy+2x+2z+5$$$