my-server
← Wiki

Qvist's theorem

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

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

When the line is an exterior line (or passant), 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

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