View source for Benz plane

''circle plane''

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

A Benz plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102901.png" /> consist of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102902.png" /> of points and two sets of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102903.png" />, namely the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102904.png" /> of generators and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102905.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102906.png" /> is a circle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102907.png" /> is a point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b1102909.png" /> is a point outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029010.png" /> and not on a common generator with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029011.png" />, then there is exactly one circle through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029012.png" /> intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029013.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029014.png" /> only (the tangency axiom);

4) there is a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029016.png" /> contains at least three points;

5a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029017.png" /> is a Möbius plane if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029018.png" />;

5b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029019.png" /> is a Laguerre plane if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029020.png" /> is a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029021.png" />;

5c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029022.png" /> is a Minkowski plane if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029023.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029025.png" /> partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029026.png" />, and any generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029027.png" /> intersects any generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029028.png" /> in exactly one point.

For a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029029.png" /> one defines its derived affine plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029030.png" />, as follows. The points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029031.png" /> are the points different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029032.png" /> and outside all generators through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029033.png" />. The lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029034.png" /> are the generators not passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029035.png" /> and the circles through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029036.png" /> minus the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029037.png" /> itself.

So, a Benz plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029038.png" /> is a special kind of [[Chain space|chain space]]: Two (different) points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029039.png" /> are distant if and only if they do not belong to a common generator.

An affine model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029040.png" /> is obtained by adding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029041.png" /> the traces of all circles not through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029042.png" />.

An imbeddable (or egg-like) Benz plane is isomorphic to a Benz plane of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029043.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029044.png" /> a subset of a (projective or affine) three-dimensional space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029045.png" /> consist of planar sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029046.png" />:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029047.png" /> is an [[Ovoid(2)|ovoid]];

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029048.png" /> is a [[Cylindrical surface (cylinder)|cylindrical surface (cylinder)]] over an [[Oval|oval]] as directrix, and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029049.png" /> is covered by its set of generators;

c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029050.png" /> is a ruled [[Quadric|quadric]] in projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029051.png" />-space (a hyperbolic quadric), and hence has two sets of (mutually skew) straight lines covering it. In any case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029052.png" /> consists of the generators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029054.png" /> consists of the intersections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029055.png" /> with all planes containing more than one point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029056.png" /> but no line of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029057.png" />. [[Stereographic projection|Stereographic projection]] once more gives an affine model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029058.png" />.

A classical Benz plane is a [[Chain geometry|chain geometry]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029059.png" /> where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029060.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029061.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029062.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029063.png" />-vector space. There are three such algebras.

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029064.png" /> is a quadratic field extension (cf. [[Extension of a field|Extension of a field]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029065.png" /> is a Möbius plane.

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029066.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029067.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029068.png" /> is a Laguerre plane.

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029069.png" /> (addition and multiplication componentwise). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029070.png" /> is a Minkowski plane. The classical Benz planes are imbeddable. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029071.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029072.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029073.png" /> and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029074.png" /> 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|Euclidean geometry]] valid in I) (e.g., the theorem on inscribed angles) also hold in II) and III), see [[#References|[a4]]], §3.4.

A Benz plane is called Miquelian if the so-called Miquels condition is valid in it: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029075.png" /> be circles no three of which have a common point but such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029076.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029077.png" /> (subscripts modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029078.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029080.png" /> are not necessarily different points. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029081.png" /> lie on a common circle, then so do <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029082.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029085.png" />, be eight points no two of which are on a common generator. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029087.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029088.png" /> has order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029089.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029090.png" />. This number is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110290/b11029091.png" />. 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 [[#References|[a2]]], [[#References|[a5]]] (e.g., classification of Benz planes by automorphism groups); for topological Benz planes see [[#References|[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 [[#References|[a1]]]; for subsequent developments, see [[#References|[a2]]]. For projectivities in Benz planes see [[#References|[a3]]].

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