Quadric (projective Geometry) - Polarity, Tangent Hyperplane, and Singular Points

Polarity, Tangent Hyperplane, and Singular Points

In general, a projective quadric defines a projective polarity: a mapping that takes any point of to a hyperplane of, and vice-versa, while preserving the incidence relation between points and hyperplanes. The coefficient vector of the polar hyperplane, relative to the chosen basis of, is .

If is not on the quadric, the hyperplane is well-defined (that is, not identically zero) and does not contain .

If is on the quadric and the hyperplane is well-defined, and contains (which is said to be a regular point). In fact, it is the hyperplane that is tangent to the quadric at .

If is on the quadric, it may happen that all coefficients are zero. In that case the polar is not defined, and is said to be a singular point or singularity of the quadric.

The tangent hyperplane turns out to be the union of all lines that are either entirely contained in, or intersect at only one point.

The condition for a point to be in the hyperplane that is tangent to at is, which is equivalent to

The condition for a point to be singular is . The quadric has singular points if and only the matrix, in diagonal form, has one or more zeros in its diagonal. It follows that the set of all singular points on the quadric is a projective subspace.