Stereographic Map Research Articles

Dating the Sir Francis Drake Silver Maps

The Drake Silver Maps are 68-mm-diameter silver disks with maps of the sixteenth-century known world featuring Drake’s route of circumnavigation. They were probably stamped with dies. The nine existing medallions have weights from 260 to 424 grains (the heaviest one weighs about 28 g). Each of these medallions has a diameter that is about the same as that of a tennis ball. The lightest one is as thin as a thumbnail, and the heaviest one is as thick as a credit card. This article shows that they were most likely created in 1588–89: the strongest evidence for this is that they used the double-hemisphere equatorial stereographic map projection that was first used by Rumold Mercator in 1587.

Stereographic projection to and from the Bloch sphere: Visualizing solutions of the Bloch equations and the Bloch–Riccati equation

Stereographic projection mapping is typically introduced to explain the point at infinity in the complex plane. After this brief exposure in the context of complex analysis, students rarely get an opportunity to fully appreciate stereographic projection mapping as an elegant and powerful technique on its own with many fruitful applications in the physical sciences. Here, using a classical description of nuclear magnetic resonance in the rotating frame, I show how stereographic projection mapping to and from the Bloch sphere can be used for visualizing solutions to Bloch's equation and the Bloch–Riccati equation, respectively. After developing the fundamentals of stereographic projection mapping using examples drawn from nuclear spin precession in the rotating frame, the method is then applied to visualizations of composite pulse excitation of a spin-1/2 system and to radiation damping in a system of isolated spins-1/2. In the case of the radiation-damped system, these visualizations provide particularly vivid illustrations of loxodromic Möbius transformation dynamics.

Big bang maps and color charge force

The Planck and other natural numbers are used for units of forces. They arise also as weights of Gleason operators, defined by 3-dimensional spin-like base triples GF and their weigths. The spin lengths are the spin GF weights for instance. The measuring GF operator triples arise by projective duality from 1-dimensional force vectors in projective to R5 extended Hilbert space H4. Color charges are set as a separate force, using a G-compass (figure 2). For the universes evolution after a big bang several maps are introduced, mostly belonging to the gravity field quantum rgb-graviton. It presents the neutral color charge of nucleons. Orthogonal projections of H4, also in spiralic and angular form, central or stereographic projective maps belong to them. They project also the S³ factor of the strong interation geometry S³xS5 down to the SU(2) geometry S³ of the Hopf map. Fiber bundle maps are added also to S5 with the same fiber S1 to the base space CP² for nucleons and atomic kernels. In octonian coordinates, listed by indices, 01234567, there are three projections from the energy space 123456 of SI to complex quaternionic 2x2-matrix presentations of spacetime 1234, of CP² as 3456 and of GR with mass and rgb-gravitons 1256. GR and CP² are projected into 1234 as the universes spacetime, observable as bubbles for atoms and matter 3456 and GR potentials and actions about and for mass carrying systems 1256.

Analytic treatment of nonparaxial full-Poincaré fields: singularity structure and trapping properties

An analytic extension to the nonparaxial regime of the full-Poincaré (FP) beams is presented. Instead of the stereographic mapping used in the paraxial case, these FP fields are defined in terms of a mapping from the polarization Poincaré sphere onto the sphere of plane-wave directions. It is shown that multipolar fields with complex arguments can be used to implement this mapping and provide closed-form expressions. The three-dimensional polarization singularities of the resulting fields are studied with the help of auxiliary fields presenting vortices at points where the polarization is circular or linear. Finally, the Mie scattering and trapping properties of the FP fields are studied, both of which are greatly simplified by the choice of fields.

Non-Uniqueness of Einstein’s Special Relativity, and the Inconclusiveness of High Energy (Relativistic) Physics

In this paper, we present a new form of “special relativity” (BSR), which is isomorphic to Einstein’s “special relativity” (ESR). This in turn proves the non-uniqueness of Einstein’s “special relativity” and implies the inconclusiveness of so-called “relativistic physics”. This work presents new results of principal significance for the foundations of physics and practical results for high energy physics, deep space astrophysics, and cosmology as well. The entire exposition is done within the formalism of the Lorentz SL(2C) group acting via isometries on real 3-dimensional Lobachevskian (hyperbolic) spaces L3 regarded as quotients SL(2C)/SU(2). We show via direct calculations that both ESR and BSR are parametric maps from Lobachevskian into Euclidean space, namely a gnomonic (central) map in the case of ESR, and a stereographic map in the case of BSR. Such an identification allows us to link these maps to relevant models of Lobachevskian geometry. Thus, we identify ESR as the physical realization of the Beltrami-Klein (non-conformal) model, and BSR as the physical realization of the Poincare (conformal) model of Lobachevskian geometry. Although we focus our discussion on ball models of Lobachevskian geometry, our method is quite general, and for instance, may be applied to the half-space model of Lobachevskian geometry with appropriate “Lorentz group” acting via isometries on (positive) half space, resulting yet in another “special relativity” isomorphic with ESR and BSR. By using the notion of a homotopy of maps, the identification of “special relativities” as maps from Lobachevskian into Euclidean space allows us to justify the existence of an uncountable infinity of hybrid “special relativities” and consequently an uncountable infinity of “relativistic physics” built upon them. This is another new result in physics and it states that so called “relativistic physics” is unique only up to a homotopy. Finally, we show that “paradoxes” of “special relativities” in either ESR or BSR are simply common distortions of maps between non-isometric spaces. The entire exposition is kept at elementary level accessible to majority of students in physics and/or engineering.

Seasonal Polar Temperatures on the Moon

AbstractThe Diviner Lunar Radiometer Experiment on the Lunar Reconnaissance Orbiter has been acquiring visible and infrared radiance measurements of the Moon for nearly 10 years. These data have been compiled into polar stereographic maps of temperatures poleward of 80° latitude at fixed local times and fixed subsolar longitudes to provide an overview of diurnal temperatures of the polar regions. The data have been divided into winter and summer seasons, defined by the times of year when the subsolar latitude is above or below the equator, to characterize the variations in seasonal temperatures that result from the 1.54° angle between the Moon's spin pole and the ecliptic plane. Since the illumination in the polar regions is perpetually at grazing angles, topography plays a dominate role in surface temperatures. Consequently, the surface and near‐surface thermal environment can vary in complex ways with time of day and season, which produces areas that are seasonally shadowed for prolonged periods and that are much more extensive than the permanently shadowed regions (PSRs). We find that surfaces below 110 K capable of cold trapping water over 1 Gyr increases by factors of 2.8 and 4.3 in the winter for the south and north polar regions, respectively, with seasonal residence times of adsorbed water molecules occurring at higher temperatures and thus larger areas.

Application of Double GPS Multi-rotor UAV in the Investigation of High Slope Perilous Rock-Mass in An Open Pit Iron Mine

In the survey of high slope perilous rock-mass in open-pit iron mine, the traditional unmanned aerial vehicle (UAV) can overcome the shortcomings of high difficulty and high risk factor in artificial geological exploration. However, it is very easy to be disturbed by the magnetic field of iron ore due to its orientation with magnetic compass, which makes the work unable to carry out normally. In order to solve the above problems, the UAV with dual-GPS equipped with laser radar can resist the magnetic field interference to a large extent and scan the high slope. By means of field experiment and point cloud data post-progressing, a set of application method of dual GPS multi-rotor UAV laser measurement technology was proposed for geological survey of high slope perilous rock under the interference of magnetic field. The preliminary conclusions are also obtained: the dual GPS combined orientation method makes the multi-rotor UAV not affected by the iron ore magnetic field and can fly normally according to the flight route. At the same time, the airborne LiDAR technology can realize the long-distance and fast acquisition of the valid terrain data of high-slope dangerous rock mass. Through the fuzzy clustering method (FCM), the joint system of outcrop rock-mass is automatically recognized and categorized based on point cloud data captured. The plane normal vector of fitting structural surface and structural surface occurrence can then be acquired. The normal vector of structural surface and the occurrence of structural surface can be obtained by substituting calculation. The occurrence of the slope and the shape of the structural surface was further drew into the stereographic projection map in order to determine the dominant joint group which affects rock slope stability.

A local projection for integrating geodetic and terrestrial coordinate systems

This paper develops a system for projecting coordinates from a geocentric coordinate system to a topocentric coordinate system defined by terrestrial measurements. The proposed system uses an extension of the Stereographic Double Map Projection and a parametric least squares estimation to calculate the parameters for the map projection. The Extended Stereographic Double Projection and the projection parameter estimation are implemented and tested. Tests are performed to determine rate that discrepancies occur and the maximum extents of the Extended Stereographic Double Projection. This maximum area is determined to be ∼10 km2 with RMS values of residuals of 0.0029 m in northing and 0.0020 m in easting. Tests are also performed to determine the factor that limits the effective area of the Projection.

Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection

In this paper the geometry of two- and three-qubit states under local unitary groups is discussed. We first review the one-qubit geometry and its relation with Riemannian sphere under the action of group SU(2). We show that the quaternionic stereographic projection intertwines between local unitary group $$SU(2)\otimes SU(2)$$SU(2)?SU(2) and quaternionic Mobius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining stereographic projection for much more global group Sp(2), generalizing the familiar Bloch sphere in two-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic Mobius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps are entanglement sensitive.

Two species of vortices in massive gauged non-linear sigma models

Non-linear sigma models with scalar fields taking values on $\mathbb{C}\mathbb{P}^n$ complex manifolds are addressed. In the simplest $n=1$ case, where the target manifold is the $\mathbb{S}^2$ sphere, we describe the scalar fields by means of stereographic maps. In this case when the $\mathbb{U}(1)$ symmetry is gauged and Maxwell and mass terms are allowed, the model accommodates stable self-dual vortices of two kinds with different energies per unit length and where the Higgs field winds at the cores around the two opposite poles of the sphere. Allowing for dielectric functions in the magnetic field, similar and richer self-dual vortices of different species in the south and north charts can be found by slightly modifying the potential. Two different situations are envisaged: either the vacuum orbit lies on a parallel in the sphere, or one pole and the same parallel form the vacuum orbit. Besides the self-dual vortices of two species, there exist BPS domain walls in the second case. Replacing the Maxwell contribution of the gauge field to the action by the second Chern-Simons secondary class, only possible in $(2+1)$-dimensional Minkowski space-time, new BPS topological defects of two species appear. Namely, both BPS vortices and domain ribbons in the south and the north charts exist because the vacuum orbit consits of the two poles and one parallel. Formulation of the gauged $\mathbb{C}\mathbb{P}^2$ model in a Reference chart shows a self-dual structure such that BPS semi-local vortices exist. The transition functions to the second or third charts break the $\mathbb{U}(1)\times\mathbb{S}\mathbb{U}(2)$ semi-local symmetry, but there is still room for standard self-dual vortices of the second species. The same structures encompassing $N$ complex scalar fields are easily generalized to gauged $\mathbb{C}\mathbb{P}^N$ models.

Uniting the Arctic frontiers – International cooperation on circum-Arctic geological and geophysical maps

ABSTRACTFollowing an initiative taken by the Russian Ministry of Natural resources and Ecology and by the Federal Agency of Mineral Resources (Rosnedra) in 2003 international cooperation on compiling a new generation of circum-Arctic geological and geophysical maps (in scale 1: 5 000 000) was undertaken by a consortium of national agencies from Canada, Denmark, Finland, Norway, Russia, Sweden and the USA. The polar stereographic maps include onshore and offshore geological coverage to 60° N. The bedrock map and database was first published in 2008, the geophysical maps were completed in 2010, while a tectonic map is currently in press. The new circum-Arctic maps are formally published under the Comission for the Geological Map of the World (CGMW/CCGM). A metallogenic map and database of the main occurrences of onshore and offshore metal deposits is scheduled to be completed in 2016.

An improved Monte Carlo simulation method for discontinuity orientations based on Fisher distribution and its program implementation

Fisher distribution is the most commonly used probability density function for discontinuity orientations. Based on Fisher distribution, Monte Carlo simulation method for discontinuity orientations was reviewed and improved. Those orientations extending beyond the edge of an upper hemisphere projection (OEBEUHP) often have an important influence on both the mean orientation and Fisher constant K, thus affecting simulation results. The detailed algorithms for identifying and adjusting those OEBEUHP were developed in this paper. Based on the improved method, a program for generating discontinuity orientations and plotting their stereographic projection maps, named as MCSDO, was developed. Due to the aforementioned adjustment, the generated orientations by MCSDO are close to the original discontinuity orientations, which were mapped in field, and satisfactory. Only the original orientations and target number of generated orientations need to be input. By running the program we can directly obtain the follows: orientations of generated discontinuities, mean orientations of both original and generated discontinuities, Fisher constant K of both original and generated discontinuities, and stereographic projection maps of both original and generated discontinuities. MCSDO is a freeware designed for researchers and practicing engineers, and can be easily mastered with a little computer knowledge.

Formation of Cu domains on [formula omitted] revealed by stereographic maps of elastically backscattered electrons

The results of investigations of the Cu/Ru101¯0 adsorption system obtained by using directional elastic peak electron spectroscopy (DEPES), Auger electron spectroscopy and low energy electron diffraction are presented. The measurements were performed for the clean and Cu covered ruthenium surface at a coverage of 7 bilayers at room temperature. At early stages of growth the formation of two flat Cu bilayers was observed. Further adsorption leads to the nucleation of Cu(111) islands of the face centered cubic (fcc) structure. However, the experimental results reveal the formation of two fcc(111) domains which are mutually rotated by 180° arranged in the 21¯1¯111Cu0001101¯0Ru and 2¯11111Cu0001101¯0Ru orientations with respect to the substrate. Moreover, the quantitative analysis of DEPES data reveals nonequivalent (45%/33%) concentrations of these domains.

Recurrent Anholonomy in Curved Space Navigation Solved by the Riemann Zeta Function

Navigation on curved surfaces has to handle additional nontrivial nonlinear geometric signals from (an)holonomy. Things may be even worse since propulsion recurrently couples back to the sensor signal. This effect can be experienced with orbiting gyroscopic sensors or spinning particles subject to spin-orbit coupling, which are just Gsensors of rotational acceleration. A solution to the rather simple orbital situation on spherically curved surfaces can be obtained by applying a Gudermannian mapping from precession patterns on the sphere onto holographic geometric phase patterns on the Poincaré disc by interference. This kind of stereographic mapping between surfaces of different constant curvatures induces the geometric extra rotations recurrently coupling back to the precession signal, where the fixed point solution to the emerging strange attractor is given by the functional equation of the Riemann zeta function.

Map-projection-independent crater size-frequency determination in GIS environments—New software tool for ArcGIS

Statistical analysis of crater size-frequency distributions (CSFDs) of impact craters on planetary surfaces is a well-established method to derive absolute ages on the basis of remotely-sensed image data. Although modelling approaches and the derivation of absolute ages from a given CSFD have been described and discussed in considerable depth since the late 1960s, there is no standardised methodology or guideline for the measurement of impact-crater diameters and area sizes that are both needed to determine absolute ages correctly. Distortions of distances (i.e., diameters) and areas within different map projections are considerable error sources during crater and area measurements. In order to address this problem and to minimize such errors, a software extension for Environmental Systems Research Institute's (ESRI's) ArcMap (ArcGIS) has been developed measuring CSFDs on planetary surfaces independently of image and data frame map projections, which can also be theoretically transferred to every Geographic Information System (GIS) capable of working with different map projections. Using this new approach each digitized impact crater is internally projected to a stereographic map projection with the crater's central-point set as the projection center. In this projection, the circle is defined without any distortion of its shape (i.e., conformality). Using a sinusoidal map projection with a center longitude set to the crater's central-point, the diameter of the impact crater is measured along this central meridian which is true-scale and does not show any distortion. The crater is re-projected to the map projection of the current data frame and stored as vector geometry with attributes. Output from this workflow comprises correct impact-crater diameters and area sizes in sinusoidal map projections and can be used for further processing, i.e. absolute age determinations (e.g., using the software CraterStats). The ArcMap toolbar CraterTools developed in this context significantly helps to improve and simplify the crater size-frequency (CSF) measurement process. For GIS-based measurements, we strongly recommend our procedure as the standard method for determining CSFDs on planetary surfaces to minimize map distortion effects for further analysis.

Crystalline structure of the Au(1 1 1) surface revealed by stereographic intensity DEPES maps

The intensity of elastically backscattered electrons at the primary electron beam energy 1.9 keV was used to obtain a stereographic map of Au(1 1 1) by means of the directional elastic peak electron spectroscopy (DEPES). An experimental result is compared with the theoretical data obtained by using multiple scattering calculations (MS) performed for both not-reconstructed and model-reconstructed clusters. The lateral lattice misfit of the first layer leads to quantitative changes of theoretical intensities showing a sensitivity of DEPES to the short atomic chain axial order. This comparison proves that a main contribution of the experimental contrast originates from a higher background level. Moreover an anisotropy of the inelastic mean free path is discussed in the paper.

Global faceting behavior of strained Ge islands on Si

The evolution of crystallographic facets of strained heteroepitaxial Ge islands on Si isinvestigated. Islands growing on Si(001), (111), (110) and (113) are bound by anequilibrium set of facets that includes only shared stable surfaces between bulk Si andGe—{105}, {113}, {15 3 23} and {111}. The formation of a stereographic map from these indicesfacilitates the prediction of Ge faceted-island shapes on any Si substrate at different stagesof growth. The analysis presented here can be applied to other heteroepitaxial islandingsystems where a finite set of shared equilibrium facets exists for the bulk starting materials.

QCD-Instantons and conformal space-time inversion symmetry

In this paper, we explore the appealing possibility that the strong suppression of large-size QCD instantons - as evident from lattice data - is due to a surviving conformal space-time inversion symmetry. This symmetry is both suggested from the striking invariance of high-quality lattice data for the instanton size distribution under inversion of the instanton size rho --> ^2 / rho and from the known validity of space-time inversion symmetry in the classical instanton sector. We project the instanton calculus onto the four-dimensional surface of a five-dimensional sphere via conformal stereographic mapping, before investigating conformal inversion. This projection to a compact, curved geometry is both to avoid the occurence of divergences and to introduce the average instanton size from the lattice data as a new length scale. The average instanton size is identified with the radius b of this 5d-sphere and acts as the conformal inversion radius. For b = , our corresponding results are almost perfectly symmetric under space-time inversion and in good qualitative agreement with the lattice data. For rho / b --> 0, we recover the familiar results of instanton perturbation theory in flat 4d-space. Moreover, we illustrate that a (weakly broken) conformal inversion symmetry would have significant consequences for QCD beyond instantons. As a further successful test for inversion symmetry, we present striking implications for another instanton dominated lattice observable, the chirality-flip ratio in the QCD vacuum.

Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque

The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular, the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type, which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.

Conformal Powers of the Laplacian via Stereographic Projection

A new derivation is given of Branson's factorization formula for the confor- mally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the correspon- ding conformally invariant operator on Euclidean space (the k-th power of the Euclidean Laplacian) via conjugation by the stereographic projection mapping.