Definiton and associated matrixes of analytical quadric

Given a real quadratic polynomial $$\begin{gathered} q(x,y,z)=a{x}^2+b{y}^2+c{z}^2+2fxy+2gxz+2hyz+ \\ +2px+2qy+2rz+d \end{gathered}$$ 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 $q(x,y,z)$ $$q_2(x,y,z)=a{x}^2+b{y}^2+c{z}^2+2fxy+2gxz+2hyz$$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 $(x^{\prime },y^{\prime }.z^{\prime })$ is another system of rectangular coordinates and $$q(x^{\prime },y^{\prime },z^{\prime })=a^{\prime }{x}^{\prime 2}+b^{\prime }{y}^{\prime 2}+c^{\prime }{z}^{\prime 2}+2f^{\prime }{x}^{\prime }{y}^{\prime }+2g^{\prime }{x}^{\prime }{z}^{\prime }+$$ $$+2h^{\prime }{y}^{\prime }{z}^{\prime }+2p^{\prime }{x}^{\prime }+2q^{\prime }{y}^{\prime }+2r^{\prime }{z}^{\prime }+d^{\prime }$$ it is a real quadratic polynomial in (x',y',z'), then we will say that $Q$ it coincides with $Q^{\prime }$, or that the equations $q(x,y,z)=0$ and $q^{\prime }(x^{\prime },y^{\prime },z^{\prime })=0$ define the same quadric, if and only if a non zero real number $K$ exists, such that $$q^{\prime }(x^{\prime },y^{\prime },z^{\prime })=Kq(x^{\prime },y^{\prime }{z}^{\prime })$$ where $q^{\prime }(x^{\prime },y^{\prime },z^{\prime })=0$ denotes the polynomial in $(x^{\prime },y^{\prime },z^{\prime })$ 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 $(x^{\prime },y^{\prime },z^{\prime })$.

Associated matrixes

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

Similarly, we define $$\bar{A}=\left[\begin{array}{cc}A& {\omega }^T\\ \omega d\end{array}\right],\omega =(p,q,r)$$ 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 $\bar{A}$. 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 $\bar{A}$ is equivalent to the knowledge of $q(x,y,z)$ since $$q(x,y,z)=(x,y,z,1)\bar{A}(x,y,z,1{)}^T$$

Let's observe, nevertheless, that the quadric $Q$ only determines $\bar{A}$ 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 $\bar{A}$, the polynomial $q(X^{\prime })=q(x^{\prime },y^{\prime },z^{\prime })$ defined by the formula $q(X^{\prime })=q(X^{\prime }{M}^t+P)$ has matrix ${\bar{A}}^{\prime }={\bar{M}}^T\bar{A}\bar{M}$ and main matrix $A^{\prime }=M^{T}AM$.

  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 $X^{\prime }=(x^{\prime },y^{\prime },z^{\prime })$ such that the main part of the polynomial $q(x^{\prime },y^{\prime },z^{\prime })$ has the form ${\lambda }_1{x}^{\prime 2}+{\lambda }_2{y}^{\prime 2}+{\lambda }_3{z}^{\prime 2}$ which we will call a diagonal form. Also,${\lambda }_1$,${\lambda }_2$ and ${\lambda }_3$ are the eigenvalues of the main matrix of $q(x,y,z)$.