Two-transitive Minkowski planes

In this paper we determine all finite Minkowski planes with an automorphism group which satisfies the following transitivity property: any ordered pair of nonparallel points can be mapped onto any other ordered pair of nonparallel points.     

Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C~2-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C~2-smooth curves on surfaces. We get Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane.     

Fractal analysis of spiral trajectories of some planar vector fields

We study box dimension, Minkowski content and Minkowski measurability of nonrectifiable, smooth spiral trajectories of some dynamical systems in the plane. From this point of view we consider a standard model of Hopf-Takens bifurcation and study the behaviour of trajectories near singular points and limit cycles.     

Projective Minkowski set

The Minkowski set or the central symmetry set (CSS) of a smooth curve Γ on the affine plane is the envelope of chords connecting pairs of points such that the tangents to Γ at them are parallel. Singularities of CSS are of interest, in particular, for applications (for example, in computer graphics). A generalization of the Minkowski set is considered in the paper, namely, the projective Minkowski set with respect to a line on the plane; in the case of general position, we describe its singularities and the bifurcation set of lines corresponding to lines defining the projective Minkowski set having singularities being more degenerate than those of the Minkowski set for a generic line.     

A symmetry theorem for Laguerre planes

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Berührsatz). The same special Pascal theorem now leads to an incidence theorem, , for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.     

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.     

The size of a Minkowski ellipse that contains the unit ball

In this paper we study the minimum radius of Minkowski ellipses (with antipodal foci on the unit sphere) necessary to contain the unit ball of a (normed or) Minkowski plane. We obtain a general upper bound depending on the modulus of convexity, and in the special case of a so-called symmetric Minkowski plane (a notion that we will recall in the paper) we prove a lower bound, and also we obtain that 3 is the exact upper bound.     

关于Finsler流形中的距离函数与测地球

利用 Finsler流形中的切曲率和旗曲率 ,研究了距离函数与测地球的凸性 ;指出了在单连通完备 Minkowski空间中测地球正好是平面的一部分     

Unit-distance graphs in Minkowski metric spaces

The Unit-Distance Graph problem in Euclidean plane asks for the minimum number of colors, so that each point on the Euclidean plane can be assigned a single color with the condition that the points at unit distance apart are assigned different colors. It is well known that this number is between 4 and 7, but the exact value is not known. Here this problem is generalized to Minkowski metric spaces and once again the answer is shown to be between 4 and 7. In extreme special cases where the unit circle is a parallelogram or a hexagon the answer is shown to be exactly 4.     

Finite Minkowski planes and embedded inversive planes

We show that each known finite Minkowski plane of order even, contains embedded Miquelian inversive planes, (cf. Proposition 1). Received 2 July 1999.     

A New Construction for Minkowski Planes

For any pseudo-ordered field F and some mappings f and g of F into itself we can construct a Minkowski plane such that one derived affine plane is a variation on W. A. Pierce's construction. Moreover, such a Minkowski plane induces nearaffine planes described by H. A. Wilbrink.     

A characterization of classical Minkowski planes over a perfect field of characteristic two

We give a new set of axioms defining the concept of (B^*)-plane (i.e. Minkowski plane without the tangency property) and we show that every (B^*)-plane in which a condition similar to the “Fano condition” of Heise and Karzel (see [5, § 3]) holds, is a Minkowski plane over a perfect field of characteristic two. In particular, every finite (B^*)-plane of even order is a Minkowski plane over a field. Consequences for strictly 3-transitive groups are derived from the preceding results; in particular, every strictly 3-transitive set of permutations of odd degree containing the identity is a protective group PGL₂(GF(2 ⁿ )) over a finite field GF(2 ⁿ , for some positive integer n.     

Minkowski planes admitting automorphism groups of small type

We prove that a Minkowski plane with an automorphism group of type 5¹ is of order 5 and, if it is of type 4 or 7 it is of order 3 or 5. Received 5 January 1999.     

A Measure of Non-convexity in the Plane and the Minkowski Sum

In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for “not far from convex” regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no “holes”.     

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G¹ Hermite data; however, one could also obtain higher order algorithms.     

A characterization of special laguerre planes and extended dual affine planes

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure has the residue of each point a dual affine plane, then is either an extended dual affine plane or a special Laguerre plane. This research was partially supported by NSF Grant MCS-8102361.     

Cauchy-Riemann equations in Minkowski plane

The properties of the symmetry and ordering of Minkowski plane are discussed by using hyperbolic imaginary unit and elliptic imaginary unit of Clifford algebra, and the representations of Cauchy-Riemann equations are given in Minkowski plane.     

On delta spaces satisfying Pasch's axiom

We consider partial linear spaces all of whose lines contain at least three points and in which every pair of intersecting lines generates a subspace isomorphic to a projective or dual affine plane. In particular, we classify in this paper those spaces that contain a projective plane.