The tangent plane to a quadric surface either cuts the surface in two lines, or it has only a single line, or else only a single point in common with the surface.
The surface itself is therefore called a quadric surface, or a surface of the second order.
The theory of poles and polars with regard to a conic is easily extended to quadric surfaces.
No line which does not lie altogether on the surface can have more than two points in common with the surface, which is therefore said to be of the second order or is called a ruled quadric surface.
Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.
These are called chief-tangent curves; on a quadric surface they are the above straight lines.
If through one point on a quadric surface, two, and only two, lines can be drawn on the surface, then through every point two lines may be drawn, and the surface is ruled quadric surface.
The general theory of skews in two linear complexes is identical with that of curves on a quadric in three dimensions and is known.
Hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in SS 89-93, but with one important exception.
If the above four conditions are satisfied by a group throughout the whole of projective space, the congruence group is the continuous group for which some imaginary quadric (with a real equation) is latent.
The first quadric having imaginary generators, no such self-conjugate subgroups can exist for the real group which transforms it into itself; and this real group is in fact simple.
In fact, the second quadric has two real sets of generators; and therefore the real group which transforms it into itself has two self-conjugate subgroups, either of which leaves unchanged each of one set of generators.
The groups simply isomorphic with the totality of the projective transformations which change a quadric of non-vanishing discriminant into itself.
The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius.
Since they are essentially positive thequadric is an ellipsoid; it is called the momental ellipsoid at O.
O to a tangent plane of a certain quadric surface, the tangent plane in question being parallel to (22).
Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above.
Consider first a numerical quadric equation with imaginary coefficients.
Evidently the method gives for L^3 a quadric equation, which is the "resolvent" equation in this particular case.
A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve.
But it is an improper quadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equation which does not break up.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "quadric" in a variety of sentences. We hope that you will now be able to make sentences using this word. Other words: foursquare; quadrilateral; quadruped; quaternary
