Ternary Operations as Primitive Notions for Constructive Plane Geometry V

In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes (generalizations of Euclidean planes). The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry. Mathematics Subject Classification: 51M05, 51M15, 03F65.     

A new type of orthogonality for normed planes

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d ⩾ 3.     

Polyhedral approximation and practical convex hull algorithm for certain classes of voxel sets

In this paper we introduce an algorithm for the creation of polyhedral approximations for certain kinds of digital objects in a three-dimensional space. The objects are sets of voxels represented as strongly connected subsets of an abstract cell complex. The proposed algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some practical improvements to the discussed convex hull algorithm to reduce computation time.     

Consistent Digital Rays

Given a fixed origin o in the d-dimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment \(\overline {op}\) between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log?n) bound in the n×n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital star-shaped regions centered at o, which we use to design efficient algorithms for image processing problems.     

The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes

In their paper ``An Introduction to Finsler Geometry,' J. C. Alvarez and C. Duran asked if there are other Minkowski planes besides the Euclidean for which the ratio of the Minkowski length of the unit ``circle' to the Holmes-Thompson area of the unit disc equals 2. In this paper we show that this ratio is greater than 2, and that the ratio 2 is achieved only for Minkowski planes that are affine equivalent to the Euclidean plane. In other words, the ratio is 2 only when the unit ``circle' is an ellipse.

     

Equi-isoclinic planes of Euclidean spaces

In a Euclidean space, a p-set of equi-isoclinic planes is a set of p isoclinic planes of which each pair has the same non-zero angle.The Euclidean 4-space E⁴ contains a unique congruence class of quadruples of equi-isoclinic planes, whereas quintuples of equi-isoclinic planes do not exist in E⁴.In the following a method is given to derive sets of equi-isoclinic planes in Euclidean spaces. We find again the well-known sets of equi-isoclinic planes of E⁴. The quadruples of equi-isoclinic planes in E⁵ are derived. It turns out that E⁵ contains one congruence class of such quadruples which are not flat quadruples and one congruence class of quintuples of equi-isoclinic planes, whereas sextuples of equi-isoclinic planes do not exist in E⁵.It appears that the symmetry group of that quintuple is isomorphic to the symmetric group S₅.     

Geometric algebra of strictly convex Minkowski planes

Imposing geometric or group-theoretical conditions on left reflections or the group \({\mathfrak{G}}\) generated by them, we obtain many characterizations of the Euclidean plane and of Radon planes within the framework of strictly convex Minkowski planes. In particular, Bachmann’s view of geometry provides a rich source of pertinent conditions on \({\mathfrak{G}}\) . A special role in characterizing the Euclidean plane and Radon planes is played by the shape of the locus of images of a point x under the set of left reflections in lines having a point distinct from x in common.     

Parallel-P-Äquidistante Regelflächen

In this paper we study pairs of ruled surfaces in the Euclidean space E³, which have the following property: The corresponding generators are parallel and the distance of the corresponding polar planes is constant.     

On the Napoleon-Torricelli configuration in Affine Cayley-Klein planes

There are three affine Cayley-Klein planes (see [5]), namely, the Euclidean plane, the isotropic (Galilean) plane, and the pseudo-Euclidean (Minkow-skian or Lorentzian) plane. We extend the generalization of the well-known Napoleon theorem related to similar triangles erected on the sides of an arbitrary triangle in the Euclidean plane to all affine Cayley-Klein planes. Using the Ω_k-and anti-Ω_k-equilateral triangles introduced in [28], we construct the Napoleon and the Torricelli triangle of an arbitrary triangle in any affine Cayley-Klein plane. Some interesting geometric properties of these triangles are derived. The author is partially supported by grant VU-MI-204/2006.     

Planar sections of the quadric of Lie cycles and their Euclidean interpretations

All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.     

On Reduced Triangles in Normed Planes

A convex body is reduced if it does not properly contain a convex body of the same minimal width. In this paper we present new results on reduced triangles in normed (or Minkowski) planes, clearly showing how basic seemingly elementary notions from Euclidean geometry (like that of the regular triangle) spread when we extend them to arbitrary normed planes. Via the concept of anti-norms, we study the rich geometry of reduced triangles for arbitrary norms giving bounds on their side-lengths and on their vertex norms. We derive results on the existence and uniqueness of reduced triangles, and also we obtain characterizations of the Euclidean norm by means of reduced triangles. In the introductory part we discuss different topics from Banach Space Theory, Discrete Geometry, and Location Science which, unexpectedly, benefit from results on reduced triangles.     

Ebenen mit Kongruenz

Euclidean planes are characterized as affine planes with a congruence relation on the set of the pairs of points. Furthermore we study the consequences of the congruence axioms for other incidence structures e.g. finite or half ordered ones, and discuss the question when the 3-reflection theorem is valid.

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Herrn Professor Adriano Barlotti zum 60. Geburtstag     

Verallgemeinerung des Wendekreises der Ebenen Euklidischen Kinematik auf Cayley/Klein-Zwangläufe 1. Art

In [12] the local kinematics of all seven Cayley/Klein-planes (CK-planes) was developed in a largely uniform way. The motions of Euclidean, pseudo-Euclidean, elliptic and hyperbolic planes were called CK-motions of 1^st kind. They are fundamentally different from quasielliptic, quasihyperbolic and isotropic motions (CK motions of 2^nd kind). In this paper we consider a uniform generalization of the inflection circle of plane Euclidean kinematics to CK-motions of 1^st kind, which turns out to be a curve of 2^nd order. It can be shown that many important properties of the Euclidean inflection circle are retained. We also generalize the Euclidean cuspidal circle, inflection point and cuspidal point.

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Herrn Prof. Dr.Oswald Giering zum 60. Geburtstag gewidmet     

Two close sets of bounded variation

If two subsets of bounded variation in Euclidean space are close in the deviation metric, then on almost all k-dimensional planes, except perhaps on a set of planes of small measure, their intersections with k-dimensional planes are also close.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 653–656, April, 1976.The author acknowledges the comments of L. D. Ivanov.     

Über drei zwangläufig gegeneinander bewegte Cayley/Klein-Ebenen

If three Euclidean planes move relatively to each other, the three poles of rotation are either identical or pairwise distinct and collinear. In the second case the distances of the poles are in the ratio of the motions' angular velocities. These known facts of Euclidean kinematics can be generalized in a largely uniform way to plane Cayley/Klein motions with finite poles. For the angular velocities we give a representation which is valid for all considered Cayley/Klein motions. Application of the duality principle of the projective plane yields a proposition about concurrent fixed lines. We also generalize the generation of a pair of envelope curves of a Euclidean motion as paths of a point of a third moved plane.

     

Die von den harmonischen involutionen einer euklidischen Möbiusebene erzeugte gruppe

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

Pseudo-Euklidische Ebenen und euklidische Räume

Euclidean planes and spaces as well as pseudo-euclidean planes are characterized solely by their incidence structure and a congruence relation on the set of the pairs of points.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

An axiomatic description of the Strambach planes

In this paper we generalize the construction of Strambach planes to an arbitrary ordered Euclidean field, we describe collineation groups of the obtained structures, and we give an axiomatic characterization of them.