Oval (projective geometry)

To define an oval:
p: passer-by,
t: tangent,
s: secant

In projective geometry, an oval is a circle-like curve in a projective plane . The standard examples are the non-degenerate conic sections . While a conic section is only defined in a pappus plane , there can be ovals in any projective plane. There are many criteria in the literature for determining when an oval is a conic section (in a pappus plane). A remarkable result is Buekenhout's theorem : If an oval has the Pascal property (comparable to Pappus' theorem), the projective plane is pappusian and the oval is a conic section.

An oval is defined in projective geometry with the help of incidence properties (see below). In contrast to an oval in differential geometry , where differentiability is used to define it.

The higher-dimensional analogue of the oval is the ovoid in projective spaces.

Definition of an oval

• 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). ${\displaystyle n}$ ${\displaystyle n+1}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle |{\mathfrak {o}}|=n+1}$ ${\displaystyle {\mathfrak {o}}}$

The proof of this characterization in the finite case follows from the property of a projective plane of the order that every straight line contains points and straight lines pass through every point . The total number of points is . If the plane is a Pappus plane over a body , then the following applies . ${\displaystyle n}$ ${\displaystyle n+1}$ ${\displaystyle n+1}$ ${\displaystyle n^{2}+n+1}$ ${\displaystyle K}$ ${\displaystyle |K|=n}$

If a set of points is an affine plane with the defining properties (1), (2) of an oval (now with affine straight lines), then one calls an affine oval . ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle {\mathfrak {o}}}$

An affine oval is always a projective oval in the projective closure (addition of a distance line).

An oval can also be defined as a special square set .

Examples

Conic sections

Projective conic section in inhomogeneous coordinates: parabola and far point of the axis

Projective conic section in inhomogeneous coordinates: hyperbola and far points of the asymptotes

In every pappus plane there are non-degenerate conics, and every non-degenerate conic section is an oval. The easiest way to do this is to use one of the two inhomogeneous representations of a projective conic section (see pictures).

Non-hardened conic sections are ovals with special properties:

• The principle of Pascal and its deviations applies .
• There are many symmetries (collineations that leave the conic section invariant).

A non-degenerate conic section can always be represented in inhomogeneous coordinates as a parabola + far point of the axis or hyperbola + far points of the asymptotes. (The representation as a circle (affine oval) in the affine part is only possible if the projective conic section has passers-by, which is not the case, for example, in the complex plane.)

Ovals that are not conic sections

in the real projective plane

1. If you put together a smooth semicircle (continuous tangent) with a semellipse, an oval is created that is not a conic section.
2. If you replace the inhomogeneous representation of a non-degenerate conic section as a parable + farthest point the term by , the result is an oval. ${\displaystyle x^{2}}$ ${\displaystyle x^{4}}$
3. If you replace the inhomogeneous representation of a non-degenerate conic section as hyperbole + remote points the term by , the result is an oval, not a conic is. ${\displaystyle 1/x}$ ${\displaystyle 1/x^{3}}$
4. The implicit curve is an oval. ${\displaystyle x^{4}+y^{4}=1}$

in a finite plane of even order

1. In a finite Pappusian plane of even order, a conic section has a nucleus (see Qvist's theorem ) that can be exchanged for any point on the conic section. This creates an oval that is not a conic section.
2. If the body is with elements, then it is ${\displaystyle K=GF(2^{m})}$ ${\displaystyle 2^{m}}$

${\displaystyle {\mathfrak {o}}=\{(x,y)\in K^{2}\;|y=x^{(2^{k})}\;\}\;\cup \;\{(\infty )\}}$

for and too coprime, an oval that is not a conic section. ${\displaystyle k\in \{2,...,m-1\}}$ ${\displaystyle k}$ ${\displaystyle m}$

Further finite examples:

When is an oval a conic section?

In order for an oval to be a non-degenerate conic section in a projective plane, the oval and possibly the projective plane must meet further conditions. Here are some results:

1. An oval in any projective plane that fulfills the 6-point or 5-point Pascal condition is a conic section (in a Pappus plane) (see Pascal's theorem ).
2. An oval in a Pappus projective plane is a conic section if the group of projectivities that leave invariant operate transitive on 3-fold, i.e. H. to 2 triples of points there is a projectivity with . In the finite case, double transitive is sufficient. ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle A_{1},A_{2},A_{3},\;B_{1},B_{2},B_{3}}$ ${\displaystyle \pi }$ ${\displaystyle \pi (A_{i})=B_{i},\;i=1,2,3}$
3. An oval in a pappus projective plane of the characteristic is a conic section if there is an involutorial perspective with a center that leaves invariant at every point of a tangent (or secant) . ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle \neq 2}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle {\mathfrak {o}}}$
4. An oval in a finite pappus projective plane of odd order is a conic section ( Segre's theorem ).

For topological ovals:

5. Each completed oval of the complex projective plane is a conic section.

literature

• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , p. 141.
• Peter Dembowski : Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8 . P. 147

Individual evidence

1. ↑ B. Segre : Sui k-Archi nei Piani Finiti di Caracteristica Due , Re. Math. Pure Appl. 2 (1957) p. 289-300.
2. ^ P. Dembowski : Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8 , p. 51
3. ^ E. Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 45.
4. ↑ J. Tits : Ovoides à Translations , Rend. Mat. 21 (1962), pp. 37-59.
5. ↑ H. Mäurer: Ovoide with symmetries at the points of a hyperplane , Abh. Math. Sem. Hamburg 45 (1976), pp. 237-244.
6. ↑ Th. Buchanan: Ovals and conic sections in the complex projective plane , Math.-phys. Semester reports 26, 1979, pp. 244–260.