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.
Read more about this topic: Quadric (projective Geometry)
Famous quotes containing the words singular and/or points:
“I don’t have any problem with a reporter or a news person who says the President is uninformed on this issue or that issue. I don’t think any of us would challenge that. I do have a problem with the singular focus on this, as if that’s the only standard by which we ought to judge a president. What we learned in the last administration was how little having an encyclopedic grasp of all the facts has to do with governing.”
—David R. Gergen (b. 1942)
“A bath and a tenderloin steak. Those are the high points of a man’s life.”
—Curtis Siodmak (1902–1988)