H. Havlicek,B. Odehnal, and M. Saniga. On invariant notions of Segre varieties in binary projective spaces. Des. Codes Cryptogr. 62 (2012), 343-356.

Invariant notions of a class of Segre varieties S_(m) of PG(2^m-1,2) that are direct products of m copies of PG(1,2), m being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains S_(m) and is invariant under its projective stabiliser group Stab_m(2). By embedding PG(2^m-1,2) into PG(2^m-1,4), a basis of the latter space is constructed that is invariant under Stab_m(2) as well. Such a basis can be split into two subsets of an odd and even parity whose spans are either real or complex-conjugate subspaces according as m is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a Stab_m(2)-invariant geometric spread of lines of PG(2^m-1,2). This spread is also related with a Stab_m(2)-invariant non-singular Hermitian variety. The case m= is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under Stab₃(2), while the points of PG(7,2) form five orbits.

B. Odehnal. Die Linienelemente des P³. Österreich. Akad. Wiss. Math.-Naturw. Kl. S.-B. II 215 (2006) 155-171.

Die vorliegende Arbeit befaßt sich mit der Abbildung der Linienelemente des dreidimensionalen projektiven Raumes P³ auf die Punkte einer fünfdimensionalen rationalen Fläche M⁵, die in den neundimensionalen projektiven Raum P⁹ eingebettet ist. Ferner wird der Zusammenhang zwischen der Segre-Varietät S_3,5 und der Mannigfaltigkeit M⁵ untersucht. Auf der Modellfläche erscheinen einfache Linienelementmannigfaltigkeiten als projektive Unterräume wieder. Das Dualitätsprinzip der projektiven Geometrie gestattet die Übertragung der Ergebnisse über Linienelemente auf ihre dualen Gegenstücke.
[The present paper deals with a mapping of line elements (point and incident line) of a projective three-space P³ to points on a five-dimensional rational surface M⁵ embedded in a projective nine-space P⁹. The relation between the Segre variety S_3,5 and M⁵ is studied in detail. On the model manifold linear submanifolds of line elements can be found as linear subspaces entirely contained in the model surface. The principal of duality allows us to treat the dual objects (plane with inicdent line) simultaneously.]

B. Odehnal, H. Pottmann, and J. Wallner. Equiform kinematics and the geometry of line elements. Beitr. Algebra Geom. 47/2 (2006), 567-582.

The geometry of lines is a classical topic which is of interest not only for its own sake. Given the nature of its object of study, the lines of Euclidean or projective three-space, it is natural that frequently problems on the borderlines between mathematics, computer science, and engineering are solved with line-geometric methods. The number of applications where lines and points on them (i.e., line elements) appear together raises interest in the geometry of line elements. This paper generalizes the concept of Plücker coordinates to the case of line elements and establishes some basic facts. We emphasize the relation with equiform kinematics, thus generalizing the well known relations between Euclidean kinematics and classical line geometry.

B. Odehnal, H. Pottmann, and J. Wallner. Equiform kinematics and the geometry of line elements. Beitr. Algebra Geom. 47/2 (2006), 567-582.

This paper presents a new method for the recognition and reconstruction of surfaces from 3D data. Line element geometry, which generalizes both line geometry and the Laguerre geometry of oriented planes, enables us to recognize a wide class of surfaces (spiral surfaces, cones, helical surfaces, rotational surfaces, cylinders, etc.) by fitting linear subspaces in an appropriate seven-dimensional image space. In combination with standard techniques such as PCA and RANSAC, line element geometry is employed to effectively perform the segmentation of complex objects according to surface type. Examples show applications in reverse engineering of CAD models and testing mathematical hypotheses concerning the exponential growth of sea shells.

B. Odehnal. Flags in Euclidean three-space. Math. Pannon. 17/1 (2006), 29-48.

This paper is devoted to the study of flags in Euclidean three-space. Coordinates of flags are defined and a point model in a projective space is discussed. It turns out that the six-dimensional manifold of flags in Euclidean three-space can be embedded into a nine-dimensional projective space. There this manifold is a delPezzo surface of degree five and thus it allows a rational parametrization. The group of Euclidean motions acting transitively on the flags in Euclidean three-space induces a subgroup of automorphic collineations of the model surface. We study special subsets of the manifold of flags and show the close relations to Euclidean kinematics and Non-Euclidean geometries.