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
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),In the English literature this term is usually rendered in French (or German) rather than translating it as a passing line. 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 theoremBertil Qvist: Some remarks concerning curves of the second degree in a finite plane, Helsinki (1952), Ann. Acad. Sci Fenn Nr. 134, 1–27 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.)
Notes
References
External links E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 40.