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Qvist's theorem

释义

Definition of an oval
Statement and proof of Qvist's theorem
Definition and property of hyperovals
Notes
References
External links

In projective geometry Qvist's theorem, named after the Finnish mathematician Bertil Qvist, 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 {{math|Ω}} of points is called an oval, if:
1. Any line {{mvar|l}} meets {{math|Ω}} in at most two points, and
2. For any point {{math|P ∈ Ω}} there exists exactly one tangent line {{mvar|t}} through {{mvar|P}}, i.e., {{math|1=t ∩ Ω = {P}}}.

When {{math|1={{abs|l ∩ Ω }} = 0}} the line {{mvar|l}} is an exterior line (or passant),^[1] if {{math|1={{abs|l ∩ Ω}} = 1}} a tangent line and if {{math|1={{abs|l ∩ Ω}} = 2}} the line is a secant line.

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 {{mvar|n}} (i.e. any line contains {{math|n + 1}} points) a set {{math|Ω}} of points is an oval if and only if {{math|1={{abs|Ω}} = n + 1}} and no three points are collinear (on a common line).

Statement and proof of Qvist's theorem

Qvist's theorem^[: //#3'>3]^[4]

Let {{math|Ω}} be an oval in a finite projective plane of order {{mvar|n}}.

(a) If {{mvar|n}} is odd,

every point {{math|P ∉ Ω}} is incident with 0 or 2 tangents.

(b) If {{mvar|n}} is even,

there exists a point {{mvar|N}}, the nucleus or knot, such that, the set of tangents to oval {{math|Ω}} is the pencil of all lines through {{mvar|N}}.

Proof

(a) Let {{math|t_R}} be the tangent to {{math|Ω}} at point {{mvar|R}} and let {{math|P₁, ... , P_n}} be the remaining points of this line. For each {{math|i}}, the lines through {{math|P_i}} partition {{math|Ω}} into sets of cardinality 2 or 1 or 0. Since the number {{math|1={{abs|Ω}} = n + 1}} is even, for any point {{math|P_i}}, there must exist at least one more tangent through that point. The total number of tangents is {{math|n + 1}}, hence, there are exactly two tangents through each {{math|P_i}}, {{math|t_R}} and one other. Thus, for any point {{mvar|P}} not in oval {{math|Ω}}, if {{mvar|P}} is on any tangent to {{math|Ω}} it is on exactly two tangents.

(b) Let {{mvar|s}} be a secant, {{math|1=s ∩ Ω = {P₀, P₁}}} and {{math|1=s= {P₀, P₁,...,P_n}}}. Because {{math|1={{abs|Ω}} = n + 1}} is odd, through any {{math|1=P_i, i = 2,...,n}}, there passes at least one tangent {{math|t_i}}. The total number of tangents is {{math|n + 1}}. Hence, through any point {{math|P_i}} for {{math|1=i = 2,...,n}} there is exactly one tangent. If {{mvar|N}} is the point of intersection of two tangents, no secant can pass through {{mvar|N}}. Because {{math|n + 1}}, the number of tangents, is also the number of lines through any point, any line through {{mvar|N}} is a tangent.

Example in a pappian plane of even order

Using inhomogeneous coordinates over a field {{math|1=K, {{abs|K}} = n}} even, the set

{{math|1=Ω₁ = {(x, y) {{!}} y = x²} ∪ {(∞)}}},

the projective closure of the parabola {{math|1=y = x²}}, is an oval with the point {{math|1=N = (0)}} as nucleus (see image), i.e., any line {{math|1=y = c}}, with {{math|c ∈ K}}, is a tangent.

Definition and property of hyperovals

Any oval {{math|Ω}} in a finite projective plane of even order {{mvar|n}} has a nucleus {{mvar|N}}.

The point set {{math|1={{overline|Ω}} := Ω ∪ {N}}} is called a hyperoval or ({{math|n + 2}})-arc. (A finite oval is an ({{math|n + 1}})-arc.)

One easily checks the following essential property of a hyperoval:

For a hyperoval {{math|{{overline|Ω}}}} and a point {{math|R ∈ {{overline|Ω}}}} the pointset {{math|{{overline|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 {{math|1=K, {{abs|K}} = n}} even and {{math|n > 4}}, the set

{{math|1=Ω₁ = {(x, y) {{!}} y = x²} ∪ {(∞)}}} is an oval (conic section) (see image),

{{math|1={{overline|Ω}}₁ = {(x, y) {{!}} y = x²} ∪ {(0), (∞)}}} is a hyperoval and

{{math|1=Ω₂ = {(x, y) {{!}} y = x²} ∪ {(0)}}} is another oval that is not a conic section. (Recall that a conic section is determined uniquely by 5 points.)

Notes

1. ^In the English literature this term is usually rendered in French (or German) rather than translating it as a passing line.
2. ^{{harvnb|Dembowski|1968|page=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. ^{{harvnb|Dembowski|1968|pages=147–8}}

References

{{citation|first1=Albrecht|last1=Beutelspacher|author1-link=Albrecht Beutelspacher|first2=Ute|last2=Rosenbaum|title=Projective Geometry / from foundations to applications|year=1998|publisher=Cambridge University Press|isbn=978-0-521-48364-3}}
{{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}}

External links

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

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