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.
Quadratic forms
Let be a field and a vector space over . A mapping from to such that
- (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
- and
- .
For example:
Definition and properties of a quadric
Below let be a field, , and the n-dimensional projective space over , i.e.
the set of points. ( is a (n + 1)-dimensional vector space over the field and is the 1-dimensional subspace generated by ),
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
is called polar space of (with respect to ). Obviously is either a hyperplane or .
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 .
Lemma:
- 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:
Lemma: For any point there exists an involutorial central collineation with center and .
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 .
Remark:
- 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).
Example:
- 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.
Theorem: A division ring is commutative if and only if any equation has at most two solutions.
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.
External links
- Lecture Note Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes, p. 117