Benz plane

circle plane

A common concept for the geometries of Möbius, Laguerre and Lie, and Minkowski, as given by W. Benz in [a1].

A Benz plane $ {\mathcal B} = ( P, \mathfrak G, \mathfrak C ) $ consist of a set $ P $ of points and two sets of subsets of $ P $, namely the set $ \mathfrak G $ of generators and the set $ \mathfrak C $ of circles, satisfying:

1) each generator intersects each circle in exactly one point;

2) through any three points no two of which are on a common generator there passes exactly one circle;

3) if $ C $ is a circle, $ p $ is a point on $ C $ and $ q $ is a point outside $ C $ and not on a common generator with $ p $, then there is exactly one circle through $ q $ intersecting $ C $ at $ p $ only (the tangency axiom);

4) there is a circle $ C \neq P $, and $ C $ contains at least three points;

5a) $ {\mathcal B} $ is a Möbius plane if $ \mathfrak G = \emptyset $;

5b) $ {\mathcal B} $ is a Laguerre plane if $ \mathfrak G $ is a partition of $ P $;

5c) $ {\mathcal B} $ is a Minkowski plane if $ \mathfrak G = \mathfrak G _ {1} \cup \mathfrak G _ {2} $, with $ \mathfrak G _ {1} $ and $ \mathfrak G _ {2} $ partitions of $ P $, and any generator of $ \mathfrak G _ {1} $ intersects any generator of $ \mathfrak G _ {2} $ in exactly one point.

For a point $ p \in P $ one defines its derived affine plane, $ {\mathcal B} _ {p} $, as follows. The points of $ {\mathcal B} _ {p} $ are the points different from $ p $ and outside all generators through $ p $. The lines of $ {\mathcal B} _ {p} $ are the generators not passing through $ p $ and the circles through $ p $ minus the point $ p $ itself.

So, a Benz plane $ {\mathcal B} $ is a special kind of chain space: Two (different) points of $ {\mathcal B} $ are distant if and only if they do not belong to a common generator.

An affine model of $ {\mathcal B} $ is obtained by adding to $ {\mathcal B} _ {p} $ the traces of all circles not through $ p $.

An imbeddable (or egg-like) Benz plane is isomorphic to a Benz plane of the form $ {\mathcal B} = ( P, \mathfrak G, \mathfrak C ) $ with $ P $ a subset of a (projective or affine) three-dimensional space and $ \mathfrak C $ consist of planar sections of $ P $:

a) $ P $ is an ovoid;

b) $ P $ is a cylindrical surface (cylinder) over an oval as directrix, and hence $ P $ is covered by its set of generators;

c) $ P $ is a ruled quadric in projective $ 3 $- space (a hyperbolic quadric), and hence has two sets of (mutually skew) straight lines covering it. In any case, $ \mathfrak G $ consists of the generators on $ P $ and $ \mathfrak C $ consists of the intersections of $ P $ with all planes containing more than one point of $ P $ but no line of $ P $. Stereographic projection once more gives an affine model of $ {\mathcal B} $.

A classical Benz plane is a chain geometry $ \Sigma ( K,A ) $ where the $ K $- algebra $ A $ is a $ 2 $- dimensional $ K $- vector space. There are three such algebras.

i) $ A $ is a quadratic field extension (cf. Extension of a field). Then $ \Sigma ( K,A ) $ is a Möbius plane.

ii) $ A = K [ \epsilon ] $ with $ \epsilon ^ {2} = 0 $. Then $ \Sigma ( K,A ) $ is a Laguerre plane.

iii) $ A = K \times K $( addition and multiplication componentwise). Then $ \Sigma ( K,A ) $ is a Minkowski plane. The classical Benz planes are imbeddable. Here, $ P $ becomes an elliptic quadric, a quadric cone without its vertex or a hyperbolic quadric, respectively.

The classical affine Benz planes (affine models) are as follows: There is an affine plane $ {\mathcal A} $ over $ K $ and in $ {\mathcal A} $ one takes:

I) all straight lines and circles;

II) all straight lines not belonging to a given parallel class and all parabolas having axis in this parallel class;

III) all straight lines not belonging to two given parallel classes and all hyperbolas having asymptotes in these parallel classes. Many theorems from Euclidean geometry valid in I) (e.g., the theorem on inscribed angles) also hold in II) and III), see [a4], §3.4.

A Benz plane is called Miquelian if the so-called Miquels condition is valid in it: Let $ C _ {1} \dots C _ {4} $ be circles no three of which have a common point but such that $ C _ {i} \cap C _ {i + 1 } = \{ p _ {i} ,q _ {i} \} $ for $ 1 \leq i \leq 4 $( subscripts modulo $ 4 $), where $ p _ {i} $ and $ q _ {i} $ are not necessarily different points. If $ p _ {1} \dots p _ {4} $ lie on a common circle, then so do $ q _ {1} \dots q _ {4} $.

The following theorem was proved by B.L. van der Waerden and L.J. Smid in 1935 for I), II) and by G. Kaerlain in 1970 for III): A Benz plane is classical if and only if it is Miquelian.

T.J. Kahn (1980) has proved that a Benz plane is imbeddable if and only if it satisfies the so-called bundle condition: Let $ p _ {i} $, $ q _ {i} $, $ i = 1 \dots 4 $, be eight points no two of which are on a common generator. If $ p _ {i} ,q _ {i} ,p _ {j} ,q _ {j} $, $ 1 \leq i,j \leq 4 $, belong to a common circle in five cases, then this is also true for the sixth case.

A finite Benz plane is a Benz plane with a finite point set. A finite Benz plane $ {\mathcal B} = ( P, \mathfrak G, \mathfrak C ) $ has order $ | C | -1 $ for any $ C \in \mathfrak C $. This number is independent of the choice of $ C $. The following theorem was proved by P. Dembowski (1964) for I) and by W. Heise (1974) for III): A finite Möbius or Minkowski plane of even order is imbeddable.

A direct consequence of the Segre–Barlotti theorem (that an ovoid or oval in a finite Desarguesian projective space is a quadric or conic) is that a finite imbeddable Benz plane of odd order is Miquelian.

For further information see the surveys [a2], [a5] (e.g., classification of Benz planes by automorphism groups); for topological Benz planes see [a5]. A more general concept, using unitals instead of quadrics (i.e., self-conjugate points of a Hermitian form instead of a quadratic form), arose in [a1]; for subsequent developments, see [a2]. For projectivities in Benz planes see [a3].

References

[a1]	W. Benz, "Vorlesungen über Geometrie der Algebren" , Springer (1973)
[a2]	A. Delandsheer, "Dimensional linear spaces" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995)
[a3]	P. Plaumann, K. Strambach, "Geometry: von Staudt's point of view" , Reidel (1981)
[a4]	E.M. Schröder, "Metric geometry" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995)
[a5]	G.F. Steinke, "Topological circle geometries" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995)