non-Euclidean Plane Research Articles

Generalised rigid body motions in non-Euclidean planes with applications to global rigidity

A bar-joint framework (G,p) in a (non-Euclidean) real normed plane X is the combination of a finite, simple graph G and a placement p of the vertices in X. A framework (G,p) is globally rigid in X if every other framework (G,q) in X with the same edge lengths as (G,p) arises from an isometry of X. The weaker property of local rigidity in normed planes (where only (G,q) within a neighbourhood of (G,p) are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for ℓp-norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in X, we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in X.

Commentary to Book I of the Elements. Hartshorne and beyond

(Hartshorne, 2000) interprets Euclid’s Elements provides an interpretation of Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1-I.27, covering the so-called absolute geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Postulate 5, we present a model of nonEuclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the ordered field of hyperreal numbers R*.

Superintegrable systems in non-Euclidean plane: Hidden symmetries leading to linearity

Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [Ballesteros et al., Classical Quantum Gravity 25, 165005 (2008)], the Taub–NUT system [Ballesteros et al., SIGMA 7, 048 (2011)], and all the 17 superintegrable systems for the four types of Darboux spaces as determined by Kalnins et al. [J. Math. Phys. 44, 5811–5848 (2003)].

Double-Distance Frameworks and Mixed Sparsity Graphs

A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint frameworks in a variety of such contexts. The main results are combinatorial characterisations for (i) frameworks restricted to surfaces with both Euclidean and geodesic distance constraints, (ii) frameworks in the plane with Euclidean and non-Euclidean distance constraints, and (iii) direction-length frameworks in the non-Euclidean plane.

Circle inversion and tessellations of the non-Euclidean plane

We introduce the notion of circle inversion in order to construct tessellations in the Poincare’s model of non-Euclidean geometry.

Circle Geometries Modeled in Projective Lines over $$ {{\mathbb{R}}^2} $$ -rings

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of \( {{\mathbb{R}}^2} \)-ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. We believe that even the Euclidean cases of circle geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle geometries might also be of interest in themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Mobius geometry by suitable projections of the Blaschke cylinder.

LOWER BOUND OF LENGTH OF TRIANGLE INSCRIBED IN A CIRCLE ON NON-EUCLIDEAN SPACES

Abstract. Wetzel[5] proved if is a closed curve of length L inE n , then lies in some ball of radius [ L=4]. In this paper, we gener-alize Wetzel’s result to the non-Euclidean plane with much strongerversion. That is to develop a lower bound of length of a triangleinscribed in a circle in non-Euclidean plane in terms of a chord ofthe circle. 1. IntroductionThe problem concerning the smallest circle containing copies of allclosed curves of given length, has been studied from time to time. Amongthem Wetzel[5] proved the following results:If is an arc [closed curve] of length Lin E n , then lies in someball of radius L=2[L=4]. If no smaller ball covers , then is a segmentof length L[L=2 traversed twice].The circumscribed circle of a plane curve has to meet the curve in aset of points of which convex hull contains the center of the circumcir-cle. So there must be a triangle with vertices on the intersection of thecurve with the circle and having the center of the circle in its interior.

On elementary circle-geometry in Cayley–Klein planes

This contribution is sort of an addendum to a recently published paper on circle-geometries in Cayley–Klein planes, see Martini and Spirova (Publicationes Mathematicae Debrecen 72:371–383, 2008b), as it deals with further generalisations and extensions of the author’s results to circle-geometries in all Cayley–Klein planes. The main methods in this paper are the interpretation of planar figures in space and the dualizing according to the duality principle of projective spaces. There are, in principle, only three types of \({\mathbb{R}^2}\)-ring structures and, thus, only three types of corresponding circle-geometries, see Benz (Geometrie der Algebren, 1973a). Therefore, each generalisation to non-Euclidean planes must turn out to be just another representation of the classical Euclidean cases. This point of view gives more insight into why some elementary geometric theorems remain valid when changing the place of action from the Euclidean plane to non-Euclidean circle planes and makes explicit proofs of such elementary geometric theorems in non-Euclidean circle planes superfluous. We believe that even the Euclidean cases of circle-geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle-geometries might also be of interest for their own. For example, among the planar Cayley–Klein geometries the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated, similar to the isotropic Mobius geometry, by suitable projections of the Blaschke cylinder.

Why Euclidean Area Measure Fails in the Noneuclidean Plane

Acknowledgment. We thank Tamas Lengyel (of Occidental College) and the referees for many helpful suggestions. Naimi expresses gratitude to Caltech for its hospitality while he worked there on this paper during a sabbatical leave. Pelayo wishes to thank the California Alliance for Minority Participation and the Undergraduate Research Center at Occidental College for supporting this work. Theorem 4 was obtained by Radoslav Kirov and Naimi as part of a summer undergraduate research program at Occidental College.

Why Euclidean Area Measure Fails in the Noneuclidean Plane

Why Euclidean Area Measure Fails in the Noneuclidean Plane

Once More: Proportionality in the Non-Euclidean Plane

"Once More: Proportionality in the Non-Euclidean Plane." Mathematics Magazine, 72(4), pp. 319–320

Once More: Proportionality in the Non-Euclidean Plane

Once More: Proportionality in the Non-Euclidean Plane

Philon lines in non-Euclidean planes

One approach to the ancient problem of “duplicating the cube” is related to a peculiar extremal problem in Euclidean and non-Euclidean planes. The non-Euclidean instances of this problem lead to the study of a family of octavic curves in the real projective plane.

Harmonic analysis on the quantized Riemann sphere

We extend the spectral analysis of differential forms on the disk (viewed as the non-Euclidean plane) in recent work by J. Peetre L. Peng G. Zhang to the dual situation of the Riemann sphereS2. In particular, we determine a concrete orthogonal base in the relevant Hilbert spaceLν,2(S2), where−ν2-is the degree of the form, a section of a certain holomorphic line bundle over the sphereS2. It turns out that the eigenvalue problem of the corresponding invariant Laplacean is equivalent to an infinite system of one dimensional Schrödinger operators. They correspond to the Morse potential in the case of the disk. In the course of the discussion many special functions (hypergeometric functions, orthogonal polynomials etc.) come up. We give also an application to “Ha-plitz” theory.

Beltrami's model and the independence of the parallel postulate

E. Beltrami in 1868 did not intend to prove the consistency of non-euclidean plane geometry nor the independence of the euclidean parallel postulate. His approach would have been unsuccessful if so intended. J. Houel in 1870 described the relevance of Beltrami's work to the issue of the independence of the euclidean parallel postulate. Houel's method is different from the independence proofs using reinterpretation of terms deployed by Peano about 1890, chiefly in using a fixed interpretation for non-logical terms. Comparing the work of Beltrami and Houel with the treatment of non-euclidean geometry after the development of the axiomatic method in the 1890s indicates an important shift in mathematicians’ attitudes towards mathematical theories.

Symmetrology of art: coloured and generalized symmetries

The paper describes general features and analyses a number of examples of various categories of colored and generalized symmetries used in visual arts. After a thorough discussion of the problems of symmetry analysis and construction of dichroic and colored patterns, examples of South American. Mexican, Egyptian, Celtic, New Guinean, Australian Aboriginal and Moorish, as well as 20th century ornamental art are analysed. It is shown that with the exception of the modern creations by Hinterreiter and Escher, colored symmetry has been used much less than dichroic symmetry and has remained largely limited to unidirectional color modulation of uncolored or originally dichroic patterns. From among the categories of generalized symmetry, similarity, affine- and perspective-projective transformations are discussed and illustrated as well as catamorphy and the plane groups in non-Euclidean planes. Finally, the notions of submotif, submesh, supermesh, twinning, pseudosymmetry, noncommensurability of patterns, positional and “occupational” order-disorder phenomena, local high symmetry, dissymetry, enhancement of symmetry, dichotomy, modulation, antiphase boundaries and the hierarchy and superposition of symmetries are applied to the world of visual arts.

The foundations of geometry and the non-Euclidean plane, by George E. Martin. Pp 509. $25·90. 1982. ISBN 3-540-90694-0 (Springer)

The foundations of geometry and the non-Euclidean plane, by George E. Martin. Pp 509. $25·90. 1982. ISBN 3-540-90694-0 (Springer)

The Foundations of Geometry and the Non-Euclidean Plane.

The Foundations of Geometry and the Non-Euclidean Plane.

H. L. Royden. Remarks on primitive notions for elementary Euclidean and non-Euclidean plane geometry. The axiomatic method with special reference to geometry and physics, Proceedings of an International Symposium held at the University of California, Berkeley, December 26,1957-January 4, 1958, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1959, pp. 86–96.

H. L. Royden. Remarks on primitive notions for elementary Euclidean and non-Euclidean plane geometry. The axiomatic method with special reference to geometry and physics, Proceedings of an International Symposium held at the University of California, Berkeley, December 26,1957-January 4, 1958, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1959, pp. 86–96.

GEOMETRIES OVER THE ALGEBRA OF ANTIOCTAVES

We give a rigorous construction for a projective and a noneuclidean geometry over the alternative algebra of antioctaves (split octaves). This construction generalizes Freudenthal's definition of the projective plane over the algebra of octaves (Cayley numbers). It is proved that the groups of automorphisms of the projective and the noneuclidean plane are simple noncompact Lie groups of types E6 and F4, respectively.