Quadric (projective geometry)
In projective geometry, a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. We shall restrict ourself to the case of finite-dimensional projective spaces.
- (Q1) for any and .
- (Q2) is a bilinear form.
is called quadratic form. The bilinear form is symmetric.
In case of we have , i.e. and are mutually determined in a unique way.
In case of we have always , i.e. is symplectic.
For and ( is a base of ) has the form
Definition and properties of a quadric
Below let be a field, , and the n-dimensional projective space over , i.e.
the set of lines.
Additionally let be a quadratic form on vector space . A point is called singular if . The set
of singular points of is called quadric (with respect to the quadratic form ). For point the set
For the considerations below we assume: .
Example: For we get a conic in .
For the intersection of a line with a quadric we get:
Lemma: For a line (of ) the following cases occur:
- a) and is called exterior line or
- b) and is called tangent line or
- b′) and is called tangent line or
- c) and is called secant line.
Lemma: A line through point is a tangent line if and only if .
- a) is a flat (projective subspace). is called f-radical of quadric .
- b) is a flat. is called singular radical or -radical of .
- c) In case of we have .
A quadric is called non-degenerate if .
Remark: An oval conic is a non-degenerate quadric. In case of its knot is the f-radical, i.e. .
A quadric is a rather homogeneous object:
Proof: Due to the polar space is a hyperplane.
The linear mapping
induces an involutorial central collineation with axis and centre which leaves invariant.
In case of mapping gets the familiar shape with and for any .
- a) The image of an exterior, tangent and secant line, respectively, by the involution of the Lemma above is an exterior, tangent and secant line, respectively.
- b) is pointwise fixed by .
Let be the group of projective collineations of which leaves invariant. We get
Lemma: operates transitively on .
A subspace of is called -subspace if (for example: points on a sphere or lines on a hyperboloid (s. below)).
Lemma: Any two maximal -subspaces have the same dimension .
Let be the dimension of the maximal -subspaces of . The integer is called index of .
Theorem: (BUEKENHOUT) For the index of a non-degenerate quadric in the following is true: .
Let be a non-degenerate quadric in , and its index.
- In case of quadric is called sphere (or oval conic if ).
- In case of quadric is called hyperboloid (of one sheet).
- a) Quadric in with form is non-degenerate with index 1.
- b) If polynomial is irreducible over the quadratic form gives rise to a non-degenerate quadric in .
- c) In the quadratic form gives rise to a hyperboloid.
Remark: It is not reasonable to define formally quadrics for "vector spaces" (strictly speaking, modules) over genuine skew fields (division rings). Because one would get secants bearing more than 2 points of the quadric which is totally different from usual quadrics. The reason is the following statement.
There are generalizations of quadrics: quadratic sets. A quadratic set is a set of points of a projective plane/space, which bears the same geometric properties as a quadric: any line intersects a quadratic set in no or 1 or two lines or is containt in the set.