Inversive Planes Research Articles

Flocks and partial flocks of quadrics: a survey

Abstract If O is an ovoid of PG(3, q ), then a partition of all but two points of O into q −1 disjoint ovals is called a flock of O . A partition of a nonsingular hyperbolic quadric Q + (3, q ) into q +1 disjoint irreducible conics is called a flock of Q + (3, q ). Further, if O is either an oval or a hyperoval of PG(2, q ) and if K is the cone with vertex a point x of PG(3, q )⧹PG(2, q ) and base O , then a partition of K ⧹{ x } into q disjoint ovals or hyperovals in the respective cases is called a flock of K . The theory of flocks has applications to projective planes, generalized quadrangles, hyperovals, inversive planes; using flocks new translation planes, hyperovals and generalized quadrangles were discovered. Let Q be an elliptic quadric, a hyperbolic quadric or a quadratic cone of PG(3, q ). A partial flock of Q is a set P consisting of β disjoint irreducible conics of Q . Partial flocks which are no flocks, have applications to k -arcs of PG(2, q ), to translation planes and to partial line spreads of PG(3, q ). Recently, the definition and many properties of flocks of quadratic cones in PG(3, q ) were generalized to partial flocks of quadratic cones with vertex a point in PG( n , q ), for n ⩾3 odd.

Inversive Planes, Minkowski Planes and Regular Sets of Points

New examples of regular sets of points for the Miquelian inversive planes of order q,q a prime power, q≥ 7, are found and connections between such planes and certain Minkowski planes of order q2are presented.

Binary relations as single primitive notions for hyperbolic three-space and the inversive plane

By interpreting J.A. Lester's [9] result on inversive-distance-preserving mappings as an axiomatizability statement, and by using the Liebmann isomorphism between the inversive plane and hyperbolic three-space, we point out that hyperbolic three-spaces (and inversive geometry) coordinatized by Euclidean fields can be axiomatized with planes (or circles) as variables, by using only the plane-orthogonality (or circle-orthogonality) predicate ⊥ p (or ⊥ c ), or by using only the predicate δ′ (or δ), where δ′( p, p′) (or δ( A, B)) is interpreted as ‘the distance between the planes p and p′ is equal to the length of the segment s whose angle of parallelism is π 4 (i. e. II( s) = π 4 )’ (or as ‘the numerical distance between the disjoint circles A and B has the value ϱ, which corresponds to s via Liebmann's isomorphism’).

Largek-arcs in inversive planes of odd order

An infinite family of largek-arcs in the inversive plane\(\mathcal{M}(q)\) over a finite field GF(q), withq ≡ 1 (mod 3),q≥71 orq e {17,23, 27,29,41,47,49,53,59} is constructed.

A classification of finite {0,1,2}-inversive planes

In the present paper we shall classify the finite circular spaces C with the property that for all points P of C the residual space C P is a {0,1,2}-semi-affine plane.

Subregular Spreads of[formula omitted](2n+1,q)

In this paper, we develop some of the theory of spreads of projective spaces with an eye towards generalizing the results of R. H. Bruck (1969,in“Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514, Univ. of North Carolina Press, Chapel Hill). In particular, we wish to generalize the notion of asubregularspread to the higher dimensional case. Most of the theory here was anticipated by Bruck in later papers; however, he never provided a detailed formulation. We fill this gap here by developing the connections between a regular spread of (2n+1)-dimensional projective space and ann-dimensional circle geometry, which is the appropriate generalization of the Miquelian inversive plane. After developing this theory, we provide a fairly general method for constructing subregular spreads ofPG(5,q). Finally, we explore a special case of this construction, which yields several examples of three-dimensional subregular translation planes which are not André planes.

Symplectic spreads in PG(3, q), inversive planes and projective planes

A (line) spread in PG(3, q) is any set of q 2 + 1 disjoint lines in PG(3, q). The spread S is called symplectic if all lines of S are totally isotropic for some symplectic polarity ς of PG(3, q). An ovoid of PG(3, q), q > 2, is a set of q 2 + 1 points, no three of which are collinear; an ovoid of PG(3,2) is the same as an elliptic quadric. An ovoid of the nonsingular quadric Q(4, q) of PG(4, q) is any set O of points of Q(4, q) which has exactly one point in common with each line of Q(4, q). Symplectic spreads of PG(3, q) and ovoids of Q(4, q) are equivalent objects, and, for q even, also ovoids of Q(4, q) and ovoids of PG(3, q) are equivalent objects. With each symplectic spread of PG(3, q) there corresponds a plane of order q 2. A 3 − ( q 2 + 1, q + 1, 1) design is called an inversive plane of order q. With each ovoid of PG(3, q) there corresponds an inversive plane of order q. Here we survey some important characterizations of finite inversive planes, the inversive planes of small order, the planes of small order arising from symplectic spreads of PG(3, q), and all known classes of ovoids of PG(3, q) and Q(4, q). In Appendix A we also survey the ovoids of the nonsingular quadric Q(2 n, q), n ⩾ 3, of PG(2 n, q). In second Appendix B we discuss an interesting relationship between flocks of the quadratic cone K of PG(3, q) and ovoids of Q(4, q).

Unitals and inversive planes

We show that if U is a Buekenhout-Metz unital (with respect to a point P) in any translation plane of order q2with kernel containing GF(q), then U has an associated 2-(q2,q+1,q) design which is the point-residual of an inversive plane, generalizing results of Wilbrink, Baker and Ebert. Further, our proof gives a natural, geometric isomorphism between the resulting inversive plane and the (egglike) inversive plane arising from the ovoid involved in the construction of the Buekenhout-Metz unital. We apply our results to investigate some parallel classes and partitions of the set of blocks of any Buekenhout-Metz unital.

Some design-theoretic properties of buekenhout unitals

The purpose of this article is to discuss some questions about parabolic Buekenhout unitals, considered as designs. In this article, we define a parabolic Buekenhout unital to be a unital in any two-dimensional translation plane obtained from the cone over any ovoid. In particular, we discuss resolutions of these designs, inversive plane residuals obtainable from these designs, and also some issues about disjoint Steiner systems. © 1996 John Wiley & Sons, Inc.

Key distribution patterns using Minkowski planes

A key distribution pattern is a combinatorial structure which provides a secure method of distributing secret keys among a number of participants in a cryptographic scheme. Inversive and Laguerre planes have been used to construct key distribution patterns with storage requirements lower than the trivial distribution system. In this note we construct key distribution patterns from Minkowski planes, the third of the so-calledcircle geometries.

The affine planeAG(2,q),q odd, has a unique one point extension

LetJ be a finite inversive plane of odd orderq. If for at least one pointp ofJ the internal affine planeJ p is Desarguesian, thenJ is Miquelian. Other formulation: the finite Desarguesian affine plane of odd orderq has a unique one point extension; this extension is the Miquelian inversive plane of orderq. It follows that there is a unique inversive plane of orderq, withq∈{3, 5, 7}.

Inverse planes with a given collection of common blocks

It is well known that it is possible to construct (q 2−q 2 (miquelian) inversive planes of order q on the same set of points which pairwise intersect in precisely the circles through infinity. Here we give an explicit description of such a family by “projecting” certain Buekenhout-Metz unitals in a well-defined manner. If q⩾8 is an odd power of 2, it is shown that an additional q 2− q (Suzuki-Tits) inverse planes may be added to the family without disturbing the above intersection property. The maximality of the resulting families is then discussed.

Two characterizations of even order miquelian inversive planes

Two characterizations of even order miquelian inversive planes

Point-Primitive Inversive Planes of Odd Order

Point-Primitive Inversive Planes of Odd Order

Configurations of weighted circles in finite inversive planes

A weighted configurationW of circles in I(n), a finite inversive plane of ordern, is an incidence structure in I(n) whose blocks are circles, to each of which is assigned a positive integer called itsweight. W must satisfy the condition that the sum of the weights of the circles meeting at any point must ben + 1. Some properties ofW, particularly whenI(n) is the Miquelian plane M(q), are developed. It is shown that any spreadS in PG(3,q) induces weighted configurations {W(S)} in M(q), calledspecial. Thus properties ofS may be derived from properties of

A Beckman-Quarles Type Theorem for Coxeter's Inversive Distance

AbstractWe prove that a bijective transformation on the set of circles in the real inversive plane which preserves pairs of circles a fixed inversive distance ρ > 0 apart must be induced by a Möbius transformation.

Die von den harmonischen involutionen einer euklidischen M�biusebene erzeugte gruppe

The inversive planes over a Euclidean field are characterized by properties of the group generated by the harmonic involutions.

Automorphisms of (B)-geometries

(B)-Geometries are incidence structures arising from permutation sets. The present paper studies the automorphism groups of (B)-Geometries. In certain cases these automorphisms yield examples of inversive planes and of subplanes which are embedded in Minkowski planes (chapter 2). In chapter 3 we describe the automorphism groups of the (B)-Geometries arising from the groups PΓL(2, pn) and AΓL(1, pn) in their natural representations on the points of the projective and affine line.

Ovoids in finite uniform linear spaces

The concept of an ovoid is extended to linear spaces, and it is shown that a finite uniform linear space with an ovoid is either a trivial structure on a finite set, or else either two or three dimensional. In the three dimensional case, the ovoid is directly related to an inversive plane, in much the same way that an ovoid in a finite projective space is.

Intersection pattern of the classical ovoids in symplectic 3-space of even order

For s = 2 e , e > 1 odd, we determine how the copies of the Suzuki group Sz( s) in the symplectic group Sp(4, s) intersect. Using this information we determine how the classical ovoids in symplectic 3-space W( s) meet and obtain a complete set of double coset representatives of Sz( s) in Sp(4, s). We also note that the permutation representation of Sp(4, s) on the cosets of Sz( s) is multiplicity free, and its irreducible constituents are explicitly determined. Indeed, we show that the complex Hecke algebra of this permutation representation is isomorphic to the center of the complex group algebra of Sz( s). A combinatorial offshoot of this study is the construction of several new series of Buekenhout diagram geometries of type ▪ which are embedded as subgeometries of miquelian and Suzuki-Tits inversive planes.