Every plane section of a quadric surface is a conic or a line-pair.
Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.
The surface itself is therefore called a quadric surface, or a surface of the second order.
If through one point on a quadric surface no line on the surface can be drawn, then the surface contains no lines.
On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic.
This plane is called the polar plane of the point P, with regard to the quadric surface.
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.
These are called chief-tangent curves; on a quadric surface they are the above straight lines.