Qvist's theorem

Qvist's theorem on finite ovals

The set of Qvist , named after the Finnish mathematician Bertil Qvist makes a statement about ovals in a finite projective plane . Standard examples of ovals are the non-degenerate (projective) conic sections . The theorem indicates how many tangents can go to a given oval through a given point. The answer essentially depends on whether the order (number of points on a straight line -1) of the projective plane is even or odd. In the Pappus case, the theorem offers even order over the term hyperovalan easy way to specify ovals that are not conic sections. (In the Pappus case of odd order, all ovals are already conic sections ( Segre's theorem ).)

Definition of an oval

→ Main article : Oval (projective geometry)

• A set of points in a projective plane is called an oval , if ${\displaystyle {\mathfrak {o}}}$

(1) Any straight line hits a maximum of 2 points. If is, is called passer , if is, is called tangent and if is, is called secant . ${\displaystyle g}$ ${\displaystyle {\mathfrak {o}}}$
${\displaystyle |g\cap {\mathfrak {o}}|=0}$ ${\displaystyle g}$ ${\displaystyle |g\cap {\mathfrak {o}}|=1}$ ${\displaystyle g}$ ${\displaystyle |g\cap {\mathfrak {o}}|=2}$ ${\displaystyle g}$

(2) For every point there is exactly one tangent , i. H. . ${\displaystyle P\in {\mathfrak {o}}}$ ${\displaystyle t}$ ${\displaystyle t\cap {\mathfrak {o}}=\{P\}}$

For finite projective planes (i.e. the set of points and set of lines are finite)

• In a projective plane of order (i.e. every straight line contains points), a set is an oval if and only if is and no three points are collinear (on a straight line). ${\displaystyle n}$ ${\displaystyle n+1}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle |{\mathfrak {o}}|=n+1}$ ${\displaystyle {\mathfrak {o}}}$

Statement and proof of Qvist's theorem

Qvist's theorem

${\displaystyle {\mathfrak {o}}}$be an oval in a finite projective plane of order . ${\displaystyle n}$

(a) If is odd , then: ${\displaystyle n}$

Each point is incised with or tangents. ${\displaystyle P\notin {\mathfrak {o}}}$ ${\displaystyle 0}$ ${\displaystyle 2}$

(b) If is even , then: ${\displaystyle n}$

There is a point , the nucleus or knot , such that the set of tangents to is equal to the tuft of lines from . ${\displaystyle N}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle N}$

Qvist's theorem: For the proof in case n is odd

Qvist's theorem: for the proof in case n even

proof

(a) Let and be the tangent in and . The straight lines are broken down into subsets of thickness 2 or 1 or 0. Since is straight, there is another tangent through each point . The number of tangents is . So go through exactly two tangents, namely and . ${\displaystyle P\in {\mathfrak {o}}}$ ${\displaystyle t_{0}}$ ${\displaystyle P_{0}}$ ${\displaystyle t_{0}=\{P_{0},P_{1},...,P_{n}\}}$ ${\displaystyle P_{i},i\neq 0}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle |{\mathfrak {o}}|=n+1}$ ${\displaystyle P_{i},i\neq 0}$ ${\displaystyle t_{i}}$ ${\displaystyle n+1}$ ${\displaystyle P_{i},i\neq 0,\;}$ ${\displaystyle t_{0}}$ ${\displaystyle t_{i}}$

(b) Let it be a secant, and . Since is odd, it must go through for at least one tangent give. The number of tangents is . So goes through each point for exactly one tangent. If the point of intersection of two tangents, no secant can be used . Because every straight line through the point is a tangent. ${\displaystyle s}$ ${\displaystyle s\cap {\mathfrak {o}}=\{P_{0},P_{1}\}}$ ${\displaystyle s=\{P_{0},P_{1},...,P_{n}\}}$ ${\displaystyle |{\mathfrak {o}}|=n+1}$ ${\displaystyle P_{i}}$ ${\displaystyle i=2,...,n}$ ${\displaystyle t_{i}}$ ${\displaystyle n+1}$ ${\displaystyle P_{i}}$ ${\displaystyle i=2,...,n}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle |{\mathfrak {o}}|=n+1}$ ${\displaystyle N}$

Example Pappus plane of even order

In inhomogeneous coordinates over a body , is even ${\displaystyle K,|K|=n}$

${\displaystyle {\mathfrak {o}}_{1}=\{(x,y)\;|\;y=x^{2}\}\;\cup \{(\infty )\}}$

(projective conclusion of the norm parabola) an oval with the far point as the nucleus (see picture below), d. H. every straight line is a tangent. (Squaring is a bijection in the straight case!) ${\displaystyle N=(0)}$ ${\displaystyle y=c,\;c\in K\;,}$

Definition and characteristics of a hyper-oval

• If an oval is of even order in a finite projective plane , then it has a node . ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle n}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle N}$

The point set is called a hyperoval or (n + 2) arc . (A finite oval is an (n + 1) arc ). ${\displaystyle {\mathfrak {\bar {o}}}:={\mathfrak {o}}\cup \{N\}}$

An essential property of a hyper oval is

• If a hyper- oval is and , then it is an oval. ${\displaystyle {\mathfrak {\bar {o}}}}$ ${\displaystyle R\in {\mathfrak {\bar {o}}}}$ ${\displaystyle {\mathfrak {\bar {o}}}\setminus \{R\}}$

projective conic section ${\displaystyle {\mathfrak {o}}_{1}}$

This property offers a simple way of specifying additional ovals in addition to an oval.

example

In the projective plane above the body straight and , is ${\displaystyle K,|K|=n}$ ${\displaystyle n>4}$

${\displaystyle {\mathfrak {o}}_{1}=\{(x,y)\;|\;y=x^{2}\}\;\cup \{(\infty )\}}$ an oval (conic section) (see picture),

${\displaystyle {\mathfrak {\bar {o}}}_{1}=\{(x,y)\;|\;y=x^{2}\}\;\cup \{(0),(\infty )\}}$ a hyperoval and

${\displaystyle {\mathfrak {o}}_{2}=\{(x,y)\;|\;y=x^{2}\}\;\cup \{(0)\}}$another oval that is not a conic section. (A conic section is clearly defined by 5 points!)

Web links

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

literature

• Bertil Qvist: Some remarks concerning curves of the second degree in a finite plane. In: Ann. Acad. Sci Fenn. No. 134, Helsinki (1952), pp. 1-27.
• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , p. 206.
• Peter Dembowski : Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8 , p. 148.