Since it is very hard to find good and organized scripts online for quadrics I came here to maybe find some answers to questions that've been bugging me lately:
- If a quadric has two axes of symmetry, are they always perpendicular?
- Is the intersection of two axis of symmetry always a centre of the quadric?
- If 1. and 2. are false, can we make 2. correct if we assume the two axes to be perpendicular?
If there were two axis not perpendicular, by reflection there would be more than two axis.
If an axis was not by the center, there would be two centers.
Note that by a similarity transformation (which preserves axis and center), a centered conic can be reduced to
$$\lambda x^2+\mu y^2=1.$$
In this representation the symmetry axis are obviously $x=0$ and $y=0$, which are orthogonal and meet at the origin. This generalizes to quadrics.