Qvist's theorem

In projective geometry, Qvist's theorem, named after the Finnish mathematician Bertil Qvist [de], 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 $Ω$ of points is called an oval, if:

Any line $l$ meets $Ω$ in at most two points, and
For any point $P \in Ω$ there exists exactly one tangent line $t$ through $P$ , i.e., $t \cap Ω = {P$ }.

For finite planes (i.e. the set of points is finite) we have a more convenient characterization:^[2]

For a finite projective plane of order $n$ (i.e. any line contains $n + 1$ points) a set $Ω$ of points is an oval if and only if $| Ω | = n + 1$ and no three points are collinear (on a common line).

Statement and proof of Qvist's theorem

Qvist's theorem^[3]^[4]

Let $Ω$ be an oval in a finite projective plane of order $n$ .

(a) If

n

is odd,

every point

P \notin Ω

is incident with 0 or 2 tangents.

(b) If

n

is even,

there exists a point

N

, the nucleus or knot, such that, the set of tangents to oval

Ω

is the pencil of all lines through

N

.

Qvist's theorem: to the proof in case of n odd

Qvist's theorem: to the proof in case of n even

Proof

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

(b) Let $s$ be a secant, $s \cap Ω = {P 0, P 1$ } and $s = {P 0, P 1,..., P n$ }. Because $| Ω | = n + 1$ is odd, through any $P i, i = 2,...,n$ , there passes at least one tangent $t i$ . The total number of tangents is $n + 1$ . Hence, through any point $P i$ for $i = 2,..., n$ there is exactly one tangent. If $N$ is the point of intersection of two tangents, no secant can pass through $N$ . Because $n + 1$ , the number of tangents, is also the number of lines through any point, any line through $N$ is a tangent.

Example in a pappian plane of even order

Using inhomogeneous coordinates over a field $K, | K | = n$ even, the set

Ω 1 = {(x, y) | y = x 2} \cup {(\infty)

},

the projective closure of the parabola $y = x 2$ , is an oval with the point $N = (0)$ as nucleus (see image), i.e., any line $y = c$ , with $c \in K$ , is a tangent.

Definition and property of hyperovals

Any oval $Ω$ in a finite projective plane of even order $n$ has a nucleus $N$ .

The point set

Ω := Ω \cup {N

} is called a hyperoval or (

n + 2

)-arc. (A finite oval is an (

n + 1

)-arc.)

One easily checks the following essential property of a hyperoval:

For a hyperoval $Ω$ and a point $R \in Ω$ the pointset $Ω \ {R$ } 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 $K, | K | = n$ even and $n > 4$ , the set

Ω 1 = {(x, y) | y = x 2} \cup {(\infty)

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

Ω 1 = {(x, y) | y = x 2} \cup {(0), (\infty)

} is a hyperoval and

Ω 2 = {(x, y) | y = x 2} \cup {(0)

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

Notes

^ In the English literature this term is usually rendered in French (or German) rather than translating it as a passing line.
^ Dembowski 1968, p. 147
^ Bertil Qvist: Some remarks concerning curves of the second degree in a finite plane, Helsinki (1952), Ann. Acad. Sci Fenn Nr. 134, 1–27
^ Dembowski 1968, pp. 147–8

References

Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry / from foundations to applications, Cambridge University Press, ISBN 978-0-521-48364-3
Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275

External links

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

[1] In the English literature this term is usually rendered in French (or German) rather than translating it as a passing line.

[2] Dembowski 1968, p. 147

[3] Bertil Qvist: Some remarks concerning curves of the second degree in a finite plane, Helsinki (1952), Ann. Acad. Sci Fenn Nr. 134, 1–27

[4] Dembowski 1968, pp. 147–8

[1]

[2]

[3]

[4]