Generalized distance-squared mappings of the plane into the plane (original) (raw)

A Note on “Distance Transformations in Arbitrary Dimensions”

The upper limit of the absolute error between Euclidean and m-neighbor distance has been presented as a generalization of the results given in [l]. The conjecture regarding least upper Iimit has been proved under a restriction. A counterexample has been formulated to show that d18 in 3D may be greater than the Euclidean distance. The uniqueness of the counterexample has been established. 0 1988 Academic PW, hc.

Mappings onto the plane

Transactions of the American Mathematical Society, 1971

In this paper, we show that if X is a connected, locally connected, locally compact topological space and/is a 1-1 mapping of X onto E2, then/is a homeomorphism. Using this result, we obtain theorems concerning the compactness of certain mappings onto E2.

Maps between buildings that preserve a given Weyl distance

Indagationes Mathematicae, 2004

Let A and A t be two buildings of the same type (W, S), viewed as sets of chambers endowed with "distance" functions ~ and ~, respectively, admitting values in the common Weyl group W, which is a Coxeter group with standard generating set S. For a given element w 6 W, we study surjective maps ~o : A-+ A ~ with the property that 6 (C, D) = w if and only if 6'(~o(C), ~o(D)) = w. The result is that the restrictions of ~o to all residues of certain spherical types-determined by w-are isomorphisms. We show with counterexamples that this result is optimal. We also demonstrate that, in many cases, this is enough to conclude that ~o is an isomorphism. In particular, q~ is an isomorphism if A and A ~ are 2-spherical and every reduced expression of w involves all elements of S. 1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS The theorem of Beckman and Quarles (see, for instance, [3]) states that one can identify an isometry of real Euclidean space by checking whether a given surjective mapping preserves a certain fixed distance. Generalizations and analogous statements for other (types of) spaces have been considered in the literature. For discrete geometries, one is led to consider the distance in one of the graphs associated with the geometry. For instance, for the class of generalized n-gons (which are the spherical buildings of rank 2), it is shown in [5] that, up to some well understood exceptions, surjective maps preserving a certain arbitrary, but fixed, distance i, with

The Algebro-geometric Study of Range Maps

Journal of Nonlinear Science, 2016

Localizing a radiant source is a widespread problem to many scientific and technological research areas. E.g. localization based on range measurements stays at the core of technologies like radar, sonar and wireless sensors networks. In this manuscript we study in depth the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummer's and Cayley's surfaces. Our work gives new insights also on the localization based on range differences.

On Best Proximity Point Results for Some Type of Mappings

Journal of Function Spaces

In this paper, we give new conditions for existence and uniqueness of a best proximity point for Geraghty- and Caristi-type mappings. The presented results are most valuable generalizations of the Geraghty and Caristi fixed point theorems.

A CLASSIFICATION OF MAP PROJECTIONS 1

Annals of the Association of American Geographers, 1962

HE desire for a classification of map pro-T jections stems from the fact that an infinite number of distinct projections are possible. Hence, the fundamental problem in classifying map projections is the partitioning of this infinite set into a comprehensible and useful finite number of all-inclusive and preferably non-overlapping classes.

Cyclic Constructions of Distance-Preserving Maps

IEEE Transactions on Information Theory, 2005

Distance-preserving maps (DPMs) from binary vectors to permutations are useful for the construction of permutation arrays. A new construction of DPMs is presented. We study the distance increasing property of the new DPMs that this construction yields. Then we show that new DPMs contain a class of nicely structured DPMs of even length with good distance increasing property.

Distance-preserving maps in generalized polygons, part I : maps on flags

2002

In this paper, we characterize isomorphisms of generalized polygons (in particular automorphisms) by maps on flags which preserve a certain fixed distance. In Part II, we consider maps on point and/or lines. Exceptions give rise to interesting properties, which on their turn have some nice applications.

Unit-sphere preserving mappings

Glasnik Matematicki, 2004

We prove that if a one-to-one mapping f : R n → R n (n ≥ 2) preserves the unit n − 1 spheres (S n−1), then f is a linear isometry up to translation. 2000 Mathematics Subject Classification. 51K05. Key words and phrases. Isometry, unit-circle preserving mapping.