These are called chief-tangent curves; on a quadric surface they are the above straight lines.
Every plane section of a quadric surface is a conic or a line-pair.
This plane is called the polar plane of the point P, with regard to the quadric surface.
Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.
The diameter of a quadric surface is a line at the extremities of which the tangent planes are parallel.
The surface itself is therefore called a quadric surface, or a surface of the second order.
On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic.
Every line which has three points in common with a quadric surface lies on the surface.
Every conic which has five points in common with a quadric surface lies on the surface.
Every plane which cuts a quadric surface in a line-pair is a tangent plane.