Qvist's theorem explained

In projective geometry, Qvist's theorem, named after the Finnish mathematician, is a statement on ovals in finite projective planes. Standard examples of ovals are non-degenerate (projective) conic sections. The theorem gives an answer to the question How many tangents to an oval can pass through a point in a finite projective plane? The answer depends essentially upon the order (number of points on a line −1) of the plane.

Definition of an oval

See main article: article and Oval (projective plane).

In a projective plane a set of points is called an oval, if:

Any line meets in at most two points, and
For any point there exists exactly one tangent line through, i.e., .

When the line is an exterior line (or passant),^[1] if a tangent line and if the line is a secant line.

For finite planes (i.e. the set of points is finite) we have a more convenient characterization:

For a finite projective plane of order (i.e. any line contains points) a set of points is an oval if and only if and no three points are collinear (on a common line).

Statement and proof of Qvist's theorem

Qvist's theorem^[2] Let be an oval in a finite projective plane of order .

(a) If is odd,

every point is incident with 0 or 2 tangents.

(b) If is even,

there exists a point, the nucleus or knot, such that, the set of tangents to oval is the pencil of all lines through .

Proof:(a) Let be the tangent to at point and let be the remaining points of this line. For each, the lines through partition into sets of cardinality 2 or 1 or 0. Since the number is even, for any point, there must exist at least one more tangent through that point. The total number of tangents is, hence, there are exactly two tangents through each, and one other. Thus, for any point not in oval, if is on any tangent to it is on exactly two tangents.

(b) Let be a secant, and . Because is odd, through any, there passes at least one tangent . The total number of tangents is . Hence, through any point for there is exactly one tangent. If is the point of intersection of two tangents, no secant can pass through . Because, the number of tangents, is also the number of lines through any point, any line through is a tangent.

Example in a pappian plane of even order:Using inhomogeneous coordinates over a field even, the set

,the projective closure of the parabola, is an oval with the point as nucleus (see image), i.e., any line, with, is a tangent.

Definition and property of hyperovals

Any oval in a finite projective plane of even order has a nucleus .

The point set is called a hyperoval or -arc. (A finite oval is an -arc.)

One easily checks the following essential property of a hyperoval:

For a hyperoval and a point the pointset is an oval.

This property provides a simple means of constructing additional ovals from a given oval.

Example:For a projective plane over a finite field even and, the set

is an oval (conic section) (see image),

is a hyperoval and

is another oval that is not a conic section. (Recall that a conic section is determined uniquely by 5 points.)

External links

E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 40.

Notes and References

In the English literature this term is usually rendered in French (or German) rather than translating it as a passing line.
Bertil Qvist: Some remarks concerning curves of the second degree in a finite plane, Helsinki (1952), Ann. Acad. Sci Fenn Nr. 134, 1–27