Qvist's theorem
{{Short description|Theorem in projective geometry}}
In projective geometry, Qvist's theorem, named after the Finnish mathematician {{ill|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
{{main article|Oval (projective plane)}}
- In a projective plane a set {{math|Ω}} of points is called an oval, if:
- Any line {{mvar|l}} meets {{math|Ω}} in at most two points, and
- 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),In the English literature this term is usually rendered in French (or German) rather than translating it as a passing line. 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:{{harvnb|Dembowski|1968|page=147}}
- 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 theoremBertil Qvist: Some remarks concerning curves of the second degree in a finite plane, Helsinki (1952), Ann. Acad. Sci Fenn Nr. 134, 1–27{{harvnb|Dembowski|1968|pages=147–8}}
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|tR}} be the tangent to {{math|Ω}} at point {{mvar|R}} and let {{math|P1, ... , Pn}} be the remaining points of this line. For each {{math|i}}, the lines through {{math|Pi}} 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|Pi}}, 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|Pi}}, {{math|tR}} 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,...,Pn}}}. Because {{math|1={{abs|Ω}} = n + 1}} is odd, through any {{math|1=Pi, i = 2,...,n}}, there passes at least one tangent {{math|ti}}. The total number of tangents is {{math|n + 1}}. Hence, through any point {{math|Pi}} 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
{{reflist}}
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 | url-access=registration | url=https://archive.org/details/finitegeometries0000demb }}
External links
- E. Hartmann: [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.] Skript, TH Darmstadt (PDF; 891 kB), p. 40.
Category:Theorems in projective geometry