On circular sections of a quadric surface

A brief overview of the history of conic sections is given. Circular sections of ellipsoids and hyperboloids with planes passing through the center of the surface are considered. In general, there are two such secant planes. Generalizing the concept that arose in rigid-body mechanics, a straight line passing through the center of an ellipsoid is called the Galois axis if the orthogonal plane intersects this ellipsoid along a circle. Let us consider the pencil of planes passing through the intermediate principal axis of a triaxial ellipsoid. Each section of an ellipsoid with such a plane is an ellipse, one of the axes of which coincides with the intermediate principal axis of the ellipsoid. When the secant plane rotates around the intermediate principal axis of the ellipsoid, the length of the other axis of the ellipse continuously changes, taking values between the lengths of the minor and major axes of the ellipsoid. Therefore, some such section is a circle whose diameter is the intermediate principal axis of the ellipsoid. A triaxial ellipsoid has two such sections. They transform into each other when mirrored relative to the plane passing through the intermediate and other principal axes of the ellipsoid. Both Galois axes are orthogonal to the intermediate principal axis of the triaxial ellipsoid, and for a non-sphere ellipsoid of rotation, both Galois axes coincide with one axis and are orthogonal to the other principal axes of the ellipsoid. A method for constructing Galois axes from the known principal axes of an ellipsoid is proposed. This construction serves as one of the natural examples of geometric problems. In addition, the Galois axis can be correctly defined not only for the ellipsoid (for which it was originally introduced), but also for some other classes of centrally symmetric surfaces, including hyperboloids.