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This paper recasts four geodetic curves—the great ellipse, the normal section, the loxodrome, and the curve of alignment—into a parametric form of vector-algebra formula. These formulas allow these curves to be drawn using simple, efficient, and robust algorithms. The curve of alignment, which seems to be quite obscure, ought not to be. Like the great ellipse and the loxodrome, and unlike the normal section, the curve of alignment from point A to point B (both on the same ellipsoid) is the same as the curve of alignment from point B to point A. The algorithm used to draw the curve of alignment is much simpler than any of the others and its shape is quite similar to that of the geodesic, which suggests it would be a practical surrogate when drawing these curves. [ABSTRACT FROM AUTHOR]

This paper recasts four geodetic curves—the great ellipse, the normal section, the loxodrome, and the curve of alignment—into a parametric form of vector-algebra formula. These formulas allow these curves to be drawn using simple, efficient, and robust algorithms. The curve of alignment, which seems to be quite obscure, ought not to be. Like the great ellipse and the loxodrome, and unlike the normal section, the curve of alignment from point A to point B (both on the same ellipsoid) is the same as the curve of alignment from point B to point A. The algorithm used to draw the curve of alignment is much simpler than any of the others and its shape is quite similar to that of the geodesic, which suggests it would be a practical surrogate when drawing these curves.

The short communication discusses the interrelationships of loxodromes, isometric latitudes and the normal aspect of Mercator projection. It is shown that by applying the isometric latitude, a very simple equation of the loxodrome on the sphere is reached. The consequence of this is that the isometric latitude can be defined using the generalized longitude, and not only using the latitude, as was common until now. Generalized longitude is the longitude defined for every real number; modulo 2π of generalized and usual longitude are congruent. Since the image of the loxodrome in the plane of the Mercator projection is a straight line, the isometric latitude can also be defined using this projection. Finally, a new definition of the Mercator projection is given, according to which it is a normal aspect cylindrical projection in which the images of the loxodromes on the sphere are straight lines in the plane of the projection that, together with the images of the meridians in the projection, form equal angles, as do the loxodromes with the meridians on the sphere. The short communication provides additional knowledge to all those who are interested in the theory of maps in navigation and have a piece of requisite mathematical knowledge, as well as an interest in map projections. It will be useful for teachers and students studying cartography and GIS, navigation or applied mathematics. Graphic Abstract [ABSTRACT FROM AUTHOR]

The paper analyzes the approximation of the ellipsoid by the sphere. Earth is a space body with a mathematically irregular shape, so idealized smooth surfaces are used for calculations. The first is the geoid, a smooth, equipotential surface that best approximates mean sea level. However, the geoid does not have an analytical form and is unsuitable for many applications, so an ellipsoid is used for approximation. In applications where high accuracy is not required (e.g., with small scale maps), the ellipsoid is approximated by a sphere. The radius of the sphere can be calculated in three ways: according to the equivalent volume criterion, according to the equivalent surface criterion, or as the mean value of the three semi-axes of the ellipsoid. All three methods of approximation were tested by calculating the length of the geodetic line on the ellipsoid, the orthodrome on the spheres and then the error. Also, the influence of latitude on the error value was tested. For three different values of geographic latitude, the lengths of geodetic lines up to one hundred points were calculated (using the Bessel method for solving the second main geodetic task on the ellipsoid), as well as the lengths of the orthodromes on the spheres, with the radii of the spheres determined in the three mentioned ways. The obtained results were then analyzed and discussed.

In this paper, first, we give the general formulas according to first fundamental form of a surface for different types of loxodromes, meridians and surfaces in E 1 3 . After that, we obtain the differential equations of loxodromes on Type-I, Type-II and Type-III twisted surfaces in E 1 3 and also, we state a theorem which generalizes the differential equations of different types of loxodromes on the twisted surfaces for a special case. Finally, we provide several examples for visualizing our obtained results and draw our loxodromes and meridians on twisted surfaces with the aid of Mathematica. [ABSTRACT FROM AUTHOR]

Bevel gears are used to transmit torque between intersecting axes. They demonstrate high reliability and durability of work, as well as a constant gear ratio. The disadvantage of such a transmission is the mutual sliding of the surfaces of the teeth of the gears, which leads to the emergence of friction forces and wear of their working surfaces. In this regard, there is a task to design such bevel gears that would have no slip. Non-circular wheels are understood as a pair of closed curves that rotate around fixed centers and at the same time roll over each other without sliding. They can serve as centroids for the design of cylindrical gears between parallel axes. If the axes of rotation of the wheels intersect, then the gears are called conical. An analog of gears between parallel axes, in which centroids are flat closed curves, for gears with intersecting axes are spherical closed curves. For a bevel gear with a constant gear ratio, such spherical curves are circles on the surface of the sphere, and with a variable gear ratio, spatial spherical curves. This paper considers the construction of closed spherical curves that roll around each other without sliding when they rotate around the axes intersecting in the center of the spheres. These curves are formed from symmetrical arcs of the loxodrome, a curve that crosses all the meridians of the ball at a constant angle. This angle should be 45°, which ensures the intersection of the loxodrome at right angles. Analytical dependences have been derived underlying the calculations of profiles of spherical non-circular wheels and their visualization by means of computer graphics. The results could be used to design non-circular wheels for textile machines, hydraulic machine pumps, pump dispensers, etc. [ABSTRACT FROM AUTHOR]

In the present paper, loxodromes, which cut all meridians and parallels of twisted surfaces (that can be considered as a generalization of rotational surfaces) at a constant angle, have been studied in Euclidean 3-space and also some examples have been constructed to visualize and support our theory. [ABSTRACT FROM AUTHOR]

In this article, we first study a class of helicoidal surfaces in Minkowski space–time E 1 4 . Next, we find the parametrizations of loxodromes on the non-degenerate helicoidal surfaces in E 1 4 . In addition, we give some results and examples by using Mathematica. [ABSTRACT FROM AUTHOR]

We investigate the parametrizations of loxodromes on the time-like rotational surfaces by using suitable Lorentzian angles in Minkowski space-time. Also, some examples and corresponding graphs are given by using Mathematica. [ABSTRACT FROM AUTHOR]

Despite the wealth of studies on seasonal movements of birds between southern nonbreeding locations and High Arctic breeding locations, the key mechanisms of navigation during these migrations remain elusive. A flight along the shortest possible route between pairs of points on a sphere ('orthodrome') requires a bird to be able to assess its current location in relation to its migration goal and to make continuous adjustment of heading to reach that goal. Alternatively, birds may navigate along a vector with a fixed orientation ('loxodrome') based on magnetic and/or celestial compass mechanisms. Compass navigation is considered especially challenging for summer migrations in Polar regions, as continuous daylight and complexity in the geomagnetic field may complicate the use of both celestial and magnetic compasses here. We examine the possible use of orientation mechanisms during migratory flights across the Greenland Icecap. Using a novel 2 g solar‐powered satellite transmitter, we documented the flight paths travelled by a female red knot Calidris canutus islandica during two northward and two southward migrations. The geometry of the paths suggests that red knots can migrate across the Greenland Icecap along the shortest‐, orthodrome‐like, path instead of the previously suggested loxodrome path. This particular bird's ability to return to locations visited in a previous year, together with its sudden course changes (which would be appropriate responses to ambient wind fields), suggest a map sense that enables red knots to determine location, so that they can tailor their route depending on local conditions. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the differential equations of the space-like loxodromes on the helicoidal surfaces having space-like meridians and time-like meridians, respectively in Minkowski 3-space. Also we illustrate our main results by using Mathematica. [ABSTRACT FROM AUTHOR]

We introduce a coarse-grained model of DNA with bases modeled as rigid-body ellipsoids to capture their anisotropic stereochemistry. Interaction potentials are all physicochemical and generated from all-atom simulation/parameterization with minimal phenomenology. Persistence length, degree of stacking, and twist are studied by molecular dynamics simulation as functions of temperature, salt concentration, sequence, interaction potential strength, and local position along the chain for both single- and double-stranded DNA where appropriate. The model of DNA shows several phase transitions and crossover regimes in addition to dehybridization, including unstacking, untwisting, and collapse, which affect mechanical properties such as rigidity and persistence length. The model also exhibits chirality with a stable right-handed and metastable left-handed helix. [ABSTRACT FROM AUTHOR]

Voltage-gated potassium (Kv) channels control the electrical excitability of neurons and muscles. Despite this key role, how these channels open and close or gate is not fully understood. Gating is usually attributed to the bending and straightening of pore-lining helices at glycine and proline residues. In this work we focused on the role of proline in the Pro-Val-Pro (PVP) motif of the inner S6 helix in the Kv1.2 channel. We started by developing a simple hinged-rod model to fully explore the configurational space of bent helices and we related these configurations to the degree of pore opening. We then carried out fully atomistic simulations of the S6 helices and compared these simulations to the hinged-rod model. Both methods suggest that Kv1 channels are not tightly closed when the inner helices are straight, unlike what is seen in the non-PVP containing channels KcsA and KirBac. These results invite the possibility that the S6 helices may be kinked when Kv1 channels are closed. Our simulations indicate that the wild-type helix adopts multiple spatially distinct configurations, which is consistent with its role in adopting a closed state and an open state. The two most dominant configurational basins correspond to a 6 Å movement of the helix tail accompanied by the PVP region undergoing a local α-helix to 310-helix transition. We explored how single point mutations affect the propensity of the S6 helix to adopt particular configurations. Interestingly, mutating the first proline, P405 (P473 in Shaker), to alanine completely removed the bistable nature of the S6 helix possibly explaining why this mutation compromises the channel. Next, we considered four other mutations in the area known to affect channel gating and we saw similarly dramatic changes to the helix’s dynamics and range of motion. Our results suggest a possible mechanism of helix pore closure and they suggest differences in the closed state of glycine-only channels, like KcsA, and PVP containing channels. [ABSTRACT FROM AUTHOR]

It is a standard exercise in mechanical engineering to infer the external forces and torques on a body from a given static shape and known elastic properties. Here we apply this kind of analysis to distorted double-helical DNA in complexes with proteins: We extract the local mean forces and torques acting on each base pair of bound DNA from high-resolution complex structures. Our analysis relies on known elastic potentials and a careful choice of coordinates for the well-established rigid base-pair model of DNA. The results are robust with respect to parameter and conformation uncertainty. They reveal the complex nanomechanical patterns of interaction between proteins and DNA. Being nontrivially and nonlocally related to observed DNA conformations, base-pair forces and torques provide a new view on DNA-protein binding that complements structural analysis. [ABSTRACT FROM AUTHOR]

This work tackles the problem of misconception when using sophisticated mathematical tools, nonlinear optimization in this particular case, to solve a navigational problem. Namely, to reach the Great Circle vertex with two rhumb line legs ensuing the optimized distance, an initial rhumb line course equal to the orthodromic course at middle latitude may be used. The initial course is thereupon optimized by the incremental value steps. The optimized distance is achieved if the rhumb line course is altered towards the vertex at the orthodrome-loxodrome intersection point. As determination of this point cannot be formulated in a closed form, an iterative solution is to be applied. The derived transcendental equation forms a basis for an iterative solution of intersection using the Newton-Raphson method. To the contrary, finding solutions to a system of nonlinear equations can mislead a researcher unable to comprehend and grasp the mathematical meanings of the algorithm. The gist of this essay is a novel concept showing an intrinsic property i.e. orthodrome-loxodrome correlation using a well-known formula.

In this paper, we generalize the equations of loxodromes on helicoidal surfaces and canal surfaces in E³ to the case of 4-dimension (E4). Also we give some examples via Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the equations of loxodromes on helicoidal surfaces and canal surfaces in E³ to the case of 4-dimension (E4). Also we give some examples via Mathematica. [ABSTRACT FROM AUTHOR]

Artin–Tits groups act on a certain delta-hyperbolic complex, called the “additional length complex”. For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that “most” elements of Artin–Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin–Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin–Tits groups of spherical type, their pure subgroups and some of their commutator subgroups. [ABSTRACT FROM AUTHOR]

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary. [ABSTRACT FROM AUTHOR]

We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. “to be orthogonal”, “to be tangent”, etc.), as new objects in an extended Möbius–Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterize many other conformally-invariant families of objects, e.g. loxodromes or continued fractions. The paper describes a method, which reduces a collection of conformally invariant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. To show its usefulness, the method is implemented as a C++ library. It operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive Python wrapper of the library is provided as well. [ABSTRACT FROM AUTHOR]

We show that if a finitely generated group $G$ has a nonelementary WPD action on a hyperbolic metric space $X$ , then the number of $G$ -conjugacy classes of $X$ -loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$. As an application we prove that for $N\geq 3$ the number of distinct $\text{Out}(F_{N})$ -conjugacy classes of fully irreducible elements $\unicode[STIX]{x1D719}$ from an $R$ -ball in the Cayley graph of $\text{Out}(F_{N})$ with $\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$ of the order of $R$ grows exponentially in $R$. [ABSTRACT FROM AUTHOR]

Let G ↷ X be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X-loxodromics, approaches 1 as n→∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN) and right-angled Artin groups. [ABSTRACT FROM AUTHOR]

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Г) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Г). In addition, we identify a milder condition for a finitely generated subgroup of A(Г) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Г. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups. [ABSTRACT FROM AUTHOR]

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Bird migration has fascinated humans for centuries and routes crossing the globe are now starting to be revealed by advanced tracking technology. A central question is what compass mechanism, celestial or geomagnetic, is activated during these long flights. Different approaches based on the geometry of flight routes across the globe and route simulations based on predictions from compass mechanisms with or without including the effect of winds have been used to try to answer this question with varying results. A major focus has been use of orthodromic (great circle) and loxodromic (rhumbline) routes using celestial information, while geomagnetic information has been proposed for both a magnetic loxodromic route and a magnetoclinic route. Here, we review previous results and evaluate if one or several alternative compass mechanisms can explain migration routes in birds. We found that most cases could be explained by magnetoclinic routes (up to 73% of the cases), while the sun compas s could explain only 50%. Both magnetic and geographic loxodromes could explain <25% of the routes. The magnetoclinic route functioned across latitudes (1°S-74°N), while the sun compass only worked in the high Arctic (61-69°N). We discuss the results with respect to orientation challenges and availability of orientation cues. [ABSTRACT FROM AUTHOR]

A loxodrome is a curve on a parametrized surface that intersects one family of parametric lines at a constant angle. In this paper, we investigate loxodromes on rotational surfaces in the 3-dimensional simply isotropic space $${I^3}$$ and give an example of loxodromes. Also, we completely classify geodesics on rotational surfaces in $${I^3}$$ . [ABSTRACT FROM AUTHOR]

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references. [ABSTRACT FROM AUTHOR]

The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(픽n) for n ≥ 2. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Osin asks whether every finitely generated group has an acylindrical action on a hyperbolic space for which all generalized loxodromic elements are loxodromic. We answer this question in the negative, using Dunwoody's example of an inaccessible group as a counterexample. [ABSTRACT FROM AUTHOR]

In this paper, we examine the pseudo-spherical curves, which are equivalent to each other under the conformal maps preserving a fixed point in the de Sitter 2-space, by using the Clifford algebra Cl . Also, we find the parametric equations of de Sitter loxodromes. [ABSTRACT FROM AUTHOR]

Birds migrating between distant locations regularly perform long continuous flights lasting several days. What compass mechanism they use is still a mystery. Here, we use a novel approach, applying an individual-based model, taking compass mechanisms based on celestial and geomagnetic information and wind into account simultaneously, to investigate what compass mechanism likely is used during long continuous flights and how wind drift or compensation affects the resulting tracks. We found that for the 6 cases of long continuous migration flights, the magnetoclinic route could best explain the route selection in all except one case compared with the alternative compass mechanisms. A flight strategy correcting for wind drift resulted most often in routes ending up closest to the predicted destinations. In only half of the cases could a time-compensated sun compass explain the migration routes observed with sufficient precision. Migration from Europe to the Siberian tundra was especially challenging to explain by one compass mechanism alone, suggesting a more complex navigation strategy. Our results speak in favor of a magnetic compass based on the angle of inclination used by birds during continuous long-distance migration flights, but also a capacity to detect and correct for drift caused by winds along the route. [ABSTRACT FROM AUTHOR]

The class of p-loxodromic functions (meromorphic functions, satisfying the condition f (qz) = pf (z) for some q Є C\{0} and all z Є C\{0}) is studied. Each p-loxodromic function is Julia exceptional. The representation of these functions as well as their zero and pole distribution are investigated. [ABSTRACT FROM AUTHOR]

The recent advances in the field of computer-vision have opened the doors of many opportunities for taking advantage of these techniques and technologies in many fields and applications. Having a high demand for these systems in today and future vehicles implies a high production volume of video cameras. The above criterions imply that it is critical to design test systems which deliver fast and accurate calibration and optical-testing capabilities. In this paper we introduce new generation of test-stands delivering high calibration quality in single-shot calibration of fisheye surround-view cameras. This incorporates important geometric features from bundle-block calibration, delivers very high (sub-pixel) calibration accuracy, makes possible a very fast calibration procedure (few seconds), and realizes autonomous calibration via machines. We have used the geometrical shape of a Spherical Helix (Type: 3D Spherical Spiral) with special geometrical characteristics, having a uniform radius which corresponds to the uniform motion. This geometrical feature was mechanically realized using three dimensional truncated icosahedrons which practically allow the implementation of a spherical helix on multiple surfaces. Furthermore the test-stand enables us to perform many other important optical tests such as stray-light testing, enabling us to evaluate the certain qualities of the camera optical module. [ABSTRACT FROM AUTHOR]

In nature, science and engineering, we often come across helicoidal surfaces. A curve on a helicoidal surface in Euclidean 3-space is called a loxodrome if the curve intersects all meridians at a constant azimuth angle. Thus loxodromes are important in navigation. In this paper, we find the differential equation of the loxodrome on a helicoidal surface in Euclidean 3-space. Also we give some examples and draw the corresponding pictures via the Mathematica computer program to aid understanding of the mathematics of navigation. [ABSTRACT FROM AUTHOR]

This article follows a "rhumb" along four nonrhetorical observations on the concepts of date, moment, time, and semelfactive, and nine rhetorical theorems concerning "date" in relation to eidos, eugeneia, credibility, kairos, anagnôrisis, Innerzeitigkeit, evidence, différend, and the sublime, so as to explode our "idiocy" about the topic of date and to offer a rhetorical and phenomenological critique of "date." [ABSTRACT FROM AUTHOR]

Aminomethylenehelicene oligomers up to the ( M)-heptamer were synthesized by reductive amination from a formylhelicene building block. The oligomers containing more than three helicenes formed a double helix in 1,3-dilfuorobenzene. The ( M)-tetramer and ( M)-pentamer unfolded into a random coil by heating to 60 °C, whereas the ( M)-hexamer only slightly unfolded at the same temperature. A two-sided thermal hysteresis was detected in the structural change of the ( M)-tetramer and ( M)-pentamer during cooling and heating. The ( M)-pentamer and ( M)-hexamer unfolded with the addition of trifluoroacetic acid and regenerated a double helix with the addition of triethylamine. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the tangent indicatrix, the principal normal indicatrix and the binormal indicatrix of a timelike curve in Minkowski 3-space E13 and we construct their Prenet equations and curvature functions. Moreover, we obtain some differential equations which characterize a timelike curve to be a slant helix by using the Prenet apparatus of a spherical indicatrix of the curve. Also, related examples and their illustrations are given. [ABSTRACT FROM AUTHOR]

In this paper we show that, for historical reasons, the most likely way for Mercator to have drawn his map was by using a table of rhumbs. This is argued by examining in some detail the development of the concept of a rhumb line from its initial articulation by Pedro Nunes in 1537 to its advanced treatment in 1566. We also describe the intellectual debate that took place around this topic in Europe during that period. A hitherto neglected Portuguese source of about 1540, where tables of rhumbs are discussed for the first time, has been found to be an important link in the chain connecting the initial idea of a rhumb line to Mercator’s achievement of 1569. [ABSTRACT FROM PUBLISHER]

We introduce the R-package ‘birdring’, which provides a collection of R-functions to help with the analysis of ring re-encounter data. At present, it contains functions to read EURING data into R, to draw maps for visualising recovery data, to re-code EURING code into interpretable names, and to calculate the loxodromic and orthodromic distances between two encounter locations. The package also allows spatially different re-encounter probabilities to be estimated using the division-coefficient method. A function to obtain the proportional overlap between prior and posterior distributions facilitates parameter estimability, which is often an issue in mark–recapture models. The package is a work-in-progress and we welcome additional contributions of functions for the analysis of ring re-encounter data. [ABSTRACT FROM PUBLISHER]

The analyzed projections from this paper are cartographical projections of great importance in practice, solving cartographical problems regarding map drawing based on practical needs. These are the ones referring to terrain surfaces that will be represented, to the accuracy that must be obtained, as well as the ways to accomplish favorable and precise links with other kinds of cartographical projections. Cylindrical Projection (Mercator): is based on a cylinder tangent to the equator. Good for equatorial regions but greatly distorted at high latitudes. This one of the oldest and most common projections The maps in Mercator projection have a great importance in maritime and air navigation, due to the fact that it is a conform projection and the cartographic network formed in perpendicular lines, the loxodrome will be a straight line. The same line makes with each projection of the meridians the same azimuth. The UTM system that uses the Mercator projection can be used all over the world having the advantage that it reduces the errors of representation in plan due to introducing a scale factor that makes that the linear distortions from the margin of the spindle projected in plan to reduce to half. In this projection it is impossible to represent whole surface of the earth on the same plan, the projection being made on different plans, each of them along one meridian called center meridian. The UTM projection is used especially in military activities. UTM is a commonly used projection for USGS maps ranging in scale from 1:24,000 to 1:250,000. The UTM projections are based on 60 UTM Zones each defined by a central meridian and covering 3 degrees of Longitude to the East and West. Maps based on the UTM Projection have a Cartesian Coordinate grid system which is used to define any point on the map. Positions are defined by the UTM Zone, an 'X' coordinate called the Easting (in meters) and a 'Y' coordinate called the Northing (in meters). [ABSTRACT FROM AUTHOR]

Letbe a non-elementary subgroup of. In this paper, we show that if each loxodromic element inis hyperbolic, thenis Fuchsian. We also construct two examples, which imply that the condition “each loxodromic element inis hyperbolic” is necessary whenand this condition coincides with “for all” when. [ABSTRACT FROM PUBLISHER]