METHOD TO FIND AXIS OF SYMMETRY OF QUADRICS
Is there a general way to know how many axis of symmetry has a given quadric $Γ:x^tAx+2b^tx+a_{4,4}=0$ with associated $4x4$ matrix $$\overline{A’}= \begin{pmatrix} A & b\\\ b & a_{4,4} \end{pmatrix}$$ where $A$ is the $3x3$ matrix of the quadratic part and $b$ is the $3x1$ vector of the linear terms and $a_{4,4}$ is the constant term (maybe something that depends on the number of same eigenvalues, or on the number of non-null eigenvalues)? If there is, I would like to know the precise method, and how to obtain an equation for them (probably the direction-vector is determined by the vector that generates the eigenspace related to the axis of symmetry).
- If there isn’t a precise method but it is different for each quadric, is there a useful recap that summarizes how to find the axis of symmetry for each quadric?
EXAMPLE OF FINDING THE AXIS OF SYMMETRY, FOR PARABOLOIDS
For paraboloids, we know that they don’t have a center of symmetry, but they do have one axis of symmetry, which is the axis with, as a direction, the vector that generates the eigenspace ($V_0$) related to the null eigenvalue ($λ=0$). Then, we impose that the axis must contain the saddle point (in case of a hyperbolic paraboloid) or the “vertex” (is the name correct? I mean the minimum/maximum point of the paraboloid if it is not rotated) point (in case of an elliptic paraboloid).
But how do I know that there isn’t another axis of symmetry?