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

*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''), 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

*
* {{Citation , last1=Dembowski , first1=Peter , title=Finite geometries , publisher=Springer-Verlag , location=Berlin, New York , series=Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44 , mr=0233275 , year=1968 , isbn=3-540-61786-8 , url-access=registration , url=https://archive.org/details/finitegeometries0000demb

External links

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

Conic sections
Theorems in projective geometry
Articles containing proofs
Projective geometry
Incidence geometry