Azimuthal Projection Research Articles

Lambert Azimuthal Equal-Area Projection

In the present study, we give the proofs about important properties of Lambert azimuthal projection, like conformality, preserve area e.g. which characterizes it. While there are some kind of Lambert projection for instance standard, cylindrical in the literature, we utilize from the south polar aspect and in our proofs, we use this approach. Finally, we give some visualizations of the inverse of projection as an example.

COMPARATIVE ANALYSIS OF DIFFERENT VARIANTS OF AZIMUTHAL CONFORMAL PROJECTION FOR THE TERRITORY OF THE REPUBLIC OF SERBIA

<p>The paper analyzes four variants of conformal azimuthal projections for the territory of the Republic of Serbia. In the procedure, a double conformal mapping was applied. As it is common practice to perform mapping in azimuthal projections from a sphere to a plane, it was initially necessary to perform a conformal mapping from the ellipsoid to the sphere, after which a conformal mapping from the sphere to the plane was executed. The central point of the projection is a point with a value of 44° north latitude and 21° east longitude. The mapping territory is trapezoidal in shape, defined by the values of the geographic latitudes and longitudes of the southernmost (41°53'), northernmost (46°11'), westernmost (18°49'), and easternmost (23°) points in the territory of the Republic of Serbia. The mapping variants are defined based on different values of linear scale at the central point of the projection. Data processing was carried out within the Jupyter Notebook component, part of the Anaconda software package, using the Python programming language. The results are presented numerically, graphically, and visualized using AutoCAD software.</p>

Program VUE: analysing distributions of cryo-EM projections using uniform spherical grids

Three-dimensional cryo electron microscopy reconstructions are obtained by extracting information from a large number of projections of the object. These projections correspond to different `views' or `orientations', i.e. directions in which these projections show the reconstructed object. Uneven distribution of these views and the presence of dominating preferred orientations may distort the reconstructed spatial images. This work describes the program VUE (views on uniform grids for cryo electron microscopy), designed to study such distributions. Its algorithms, based on uniform virtual grids on a sphere, allow an easy calculation and accurate quantitative analysis of the frequency distribution of the views. The key computational element is the Lambert azimuthal equal-area projection of a spherical uniform grid onto a disc. This projection keeps the surface area constant and represents the frequency distribution with no visual bias. Since it has multiple tunable parameters, the program is easily adaptable to individual needs, and to the features of a particular project or of the figure to be produced. It can help identify problems related to an uneven distribution of views. Optionally, it can modify the list of projections, distributing the views more uniformly. The program can also be used as a teaching tool.

Why Mature Galaxies Seem to Have Filled the Universe Shortly After the Big Bang — A New Cosmological Model, that Predicted the JWST Observations

The Azimulthal Hyper-Projection (AHP) onto spacetime Model of Cosmology is just an extension of the familiar Atlas map of a globe in geography. The Atlas is an azimuthal projection of 3 spacial dimensions projected onto a 2 dimensional plane. The important feature of an Atlas is that the geometry expands asymptotic toward the horizon, and is therefore POSITIONALLY DEPENDENT, such that if North America is centrally positioned, then Siberia and Alaska are viewed as extremely remote. Thus if the Bering Strait were centrally positioned, then longitudinal meridians of North America would likewise be viewed as being remotely separated. AHP extends this concept to a hyper-projection of 5 dimensions onto spacetime. However as viewed in spacetime, geodesics appear to be expanding outward in 3 dimensional space away from the observer. Thus by extension, AHP is also POSITIONALLY DEPENDENT, such that the separations between remote galaxies appear (measures) to be much greater than they would appear from their mutual local group, and vice versa our local group of galaxies would appear to be mutually separated with extreme distances from a remote perspective. A proposed experiment (figure 14) is offered to prove that two separated objects, as measure from a remote distance \((\sim 40 AU)\) are greater than their respective distance, as measured locally. Red-shifting is alternatively proposed as azimuthal angular projections of wavelengths \(\lambda\). Magnitude is alternatively proposed as a function of Lambert's cosine law of illumination, over expanding geodesics. From established \(z\) values, the Universal hypersphere is calculated (figure 9). A function of this projected hypersphere is shown to match the Universal expansion rate, as established from supernova cosmology survey points, and also satisfies the Cosmological Constant \(\Lambda\), without dark energy. Flattened galaxy rotation curves are simply and accurately explained as measured expanding geodesics and apparent increasing density, along the observer's line of sight (figure 11, equation 20). As well, this model provides a reasonable explanation of the James Web Space Telescope findings: of maturely developed galaxies at the time of the early universe. The justification of such a radically novel model is from recent observations from the first dataset, provided by NASA's James Webb Space Telescope (JWST) of six massive galaxies, at a time in the early universe, seem to defy conventional cosmological models, as they appear to be as mature and developed as our own local group. Such unexpected discoveries JUSTIFY A RADICALLY NOVEL MODEL OF COSMOLOGY. TO QUOTE JOEL LEJA, ASSISTANT PROFESSOR OF ASTRONOMY AND ASTROPHYSICS AT PENN STATE “IT TURNS OUT WE FOUND SOMETHING SO UNEXPECTED IT ACTUALLY CREATES PROBLEMS FOR SCIENCE. IT CALLS THE WHOLE PICTURE OF EARLY GALAXY FORMATION INTO QUESTION”.

Why Mature Galaxies Seem to Have Filled the Universe Shortly After the Big Bang — A New Cosmological Model, that Predicted the JWST Observations

The Azimulthal Hyper-Projection onto spacetime Model (AHOSM) of Cosmology is just an extension of the familiar Atlas map of a globe in geography class. The Atlas is an azimuthal projection of 3 spacial dimensions projected onto a 2 dimensional plane. The important feature of an Atlas is that the geometry expands asymptotic toward the horizon, and is therefore POSITIONALLY DEPENDENT, such that if North America is centrally positioned, then Siberia and Alaska are viewed as extremely remote. Thus if the Bering Strait were centrally positioned, then longitudinal meridians of North America would likewise be viewed as being extremely remote. AHOSM extends this concept to a hyper-projection of hyper-spacetime (4 spacial dimensions) onto spacetime. However as viewed in spacetime, geodesics appear to be expanding outward in 3 dimensional space away from the observer. Thus by extension, AHOSM is also POSITIONALLY DEPENDENT, such that the most remote galaxies only appear to be mutually separated (from each other) with extreme distance, as viewed from our central position, and vice versa our local group of Galaxies would appear to be mutually separated with extreme distances from a remote perspective. A proposed experiment is offered to prove that the distance from Earth to Mars is measured greater from the perspective of the outer solar system and vice versa. Red-shifting is alternatively proposed as azimuthal angular projections of wavelengths \(\lambda\). Magnitude is alternatively proposed as a function of Lambert's cosine law of illumination, over expanding geodesics. This simple parsimonious model requires only a few assumptions, excluding dark energy to satisfy the Cosmological Constant \(\Lambda\), and is shown to match the Universal expansion rate, as established from supernova cosmology survey points. Galaxy rotation curves are simply and accurately explained as measured expanding geodesics and density, along the observer's line of site The justification of such a radically novel model is from recent observations from the first dataset, provided by NASA's James Webb Space Telescope (JWST) of six massive galaxies, at a time in the early universe, seem to defy conventional cosmological models, as they appear to be as mature and developed as our own local group. Such unexpected discoveries JUSTIFY A RADICALLY NOVEL MODEL OF COSMOLOGY. TO QUOTE JOEL LEJA, ASSISTANT PROFESSOR OF ASTRONOMY AND ASTROPHYSICS AT PENN STATE “IT TURNS OUT WE FOUND SOMETHING SO UNEXPECTED IT ACTUALLY CREATES PROBLEMS FOR SCIENCE. IT CALLS THE WHOLE PICTURE OF EARLY GALAXY FORMATION INTO QUESTION”.

Why Mature Galaxies Seem to Have Filled the Universe Shortly After the Big Bang — A New Cosmological Model, that Predicted the JWST Observations

Recent observations from the first dataset, provided by NASA's James Webb Space Telescope (JWST) of six massive galaxies, at a time in the early universe, seem to defy conventional cosmological models, as they appear to be as mature and developed as our own local group. Such unexpected discoveries JUSTIFY A RADICALLY NOVEL MODEL OF COSMOLOGY. TO QUOTE JOEL LEJA, ASSISTANT PROFESSOR OF ASTRONOMY AND ASTROPHYSICS AT PENN STATE ''IT TURNS OUT WE FOUND SOMETHING SO UNEXPECTED IT ACTUALLY CREATES PROBLEMS FOR SCIENCE. IT CALLS THE WHOLE PICTURE OF EARLY GALAXY FORMATION INTO QUESTION''. This article provides an alternative mathematical model of cosmological redshifting (\(z\)), which actually predicted such mature galaxies in a 2022 preprint, prior to these recent observations. As well, this model also predicts discrepancies between theoretical and observed galaxy rotation curves with apparent increased energy density. The Azimuthal Projection Model of the Universe is conceptualized as an \(\mathbb{R}^5\) spacetime, with a four spatial dimensional hypersphere azimuthally projected onto a three-dimensional spatial sphere. This simple parsimonious model requires only a few assumptions, excluding dark energy to satisfy the Cosmological Constant \(\Lambda\), and is shown to match the Universal expansion rate, as established from supernova cosmology survey points. This novel model conceives the universe as a higher dimensional dynamic with spacetime as a projection, rather than as an arrow from the absolute beginning of the Big Bang. Redshifting is alternatively proposed as azimuthal angular projections of wavelengths \(\lambda\). Accelerated Universal Expansion is alternatively proposed as azimuthal projections of meridians, asymptotical to a horizon, and Lambert's cosine law of luminous intensity. A radical implication of this model is that azimuthal angular projections are positional dependent, and thus it's conceivable that apparent distances between galaxies vary with the location of the observer (see figure 4). Supportive mathematical evidence is described from the Hubble Tension; Discrepancies between visible spectra redshifting of cepheid variables (the most recent calculation is \(Ho=74.03 \pm 1.42km/sec/Mpc\)), and from temperature fluctuations in the Cosmic Microwave Background (CMB) (which are calculated to be \(Ho=68.7 \pm 1.3km/sec/Mpc\)), which resolves the discrepancy by recalibrating redshift data from supernova Cosmology survey points.

Decoding motor imagery based on dipole feature imaging and a hybrid CNN with embedded squeeze-and-excitation block

Motor imagery (MI) decoding is the core of an intelligent rehabilitation system in brain computer interface, and it has a potential advantage by using source signals, which have higher spatial resolution and the same time resolution compared to scalp electroencephalography (EEG). However, how to delve and utilize the personalized frequency characteristic of dipoles for improving decoding performance has not been paid sufficient attention. In this paper, a novel dipole feature imaging (DFI) and a hybrid convolutional neural network (HCNN) with an embedded squeeze-and-excitation block (SEB), denoted as DFI-HCNN, are proposed for decoding MI tasks. EEG source imaging technique is used for brain source estimation, and each sub-band spectrum powers of all dipoles are calculated through frequency analysis and band division. Then, the 3D space information of dipoles is retrieved, and by using azimuthal equidistant projection algorithm it is transformed to a 2D plane, which is combined with nearest neighbor interpolation to generate multi sub-band dipole feature images. Furthermore, a HCNN is designed and applied to the ensemble of sub-band dipole feature images, from which the importance of sub-bands is acquired to adjust the corresponding attentions adaptively by SEB. Ten-fold cross-validation experiments on two public datasets achieve the comparatively higher decoding accuracies of 84.23% and 92.62%, respectively. The experiment results show that DFI is an effective feature representation, and HCNN with an embedded SEB can enhance the useful frequency information of dipoles for improving MI decoding.

From Conic to Cylindrical Map Projections

In books and textbooks on map projections, cylindrical, conic and azimuthal projections are usually considered separately. It is sometimes mentioned that cylindrical and azimuthal projections can be interpreted as limiting cases of conic, but this is rarely proven. The goal of this article is to show in a rigorous and systematic way how to generally approach solving the problem of transition from a conic to a corresponding cylindrical projection. This article points to the fact that J. H. Lambert showed as early as 1772 that a conformal cylindrical projection is created from a conformal conic projection. Following his idea, this paper shows that not only conformal, but also equal-area and equidistant cylindrical projections can be derived from corresponding conic map projections. Although it seems that the paper deals with quite well known and intuitive property of conic projections, it will also show that the transition from the conic to the corresponding cylindrical projection is not always possible.

Topology and Morphology Design of Spherically Reconfigurable Homogeneous Modular Soft Robots.

Imagine a swarm of terrestrial robots that can explore an environment, and, on completion of this task, reconfigure into a spherical ball and roll out. This dimensional change alters the dynamics of locomotion and can assist them in maneuvering variable terrains. The sphere-plane reconfiguration is equivalent to projecting a spherical shell onto a plane, an operation that is not possible without distortions. Fortunately, soft materials have the potential to adapt to this disparity of the Gaussian curvatures. Modular Soft Robots (MSoRos) have promise of achieving dimensional change by exploiting their continuum and deformable nature. However, the design of such soft robots has remained unexplored thus far. Here, for the first time, we present the topology and morphology design of MSoRos that are capable of reconfiguring between spherical and planar configurations. Our approach is based in geometry, where a platonic solid determines the number of modules required for plane-to-sphere reconfiguration and the radius of the resulting sphere, for example, four "tetrahedron-based" or six "cube-based" MSoRos are required for spherical reconfiguration. The methodology involves: (1) inverse orthographic projection of a "module-topology curve" onto the circumscribing sphere to generate the spherical topology; (2) azimuthal projection of the spherical topology onto a tangent plane at the center of the module resulting in the planar topology; and (3) adjusting the limb stiffness and curling ability by manipulating the geometry of cavities to realize a physical finite-width, Motor-Tendon Actuated MSoRo that can actuate between the sphere-plane configurations. The topology design is shown to be scale invariant, that is, the scaling of base platonic solid is reflected linearly in spherical and planar topologies. The module-topology curve is optimized for the reconfiguration and locomotion ability using an intramodular distortion metric that quantifies sphere-to-plane distortion. The geometry of the cavity optimizes for the limb stiffness and curling ability without compromising the actuator's structural integrity.

MRI and FUNDUS image fusion for improved ocular biometry in Ocular Proton Therapy

IntroductionOcular biometry in Ocular Proton Therapy (OPT) currently relies on a generic geometrical eye model built by referencing surgically implanted markers. An alternative approach based on image fusion of volumetric Magnetic Resonance Imaging (MRI) and panoramic fundus photography was investigated. Materials and methodsEighteen non-consecutive uveal melanoma (UM) patients, who consented for an MRI and had their tumour base visible on panoramic fundus photography, were included in this comparative analysis. Through generating digitally-reconstructed projections from MRI images using the Lambert azimuthal equal-area projection, 2D-3D image fusion between fundus photography and an eye model delineated on MRI scans was achieved and allowed for a novel definition of the target base (MRI + FCTV). MRI + FCTV was compared with MRI-only delineation (MRIGTV) and the conventional (EyePlan) target definition (EPCTV). ResultsThe combined use of fundus photography and MRI to define tumour volumes reduced the average discrepancies by almost 65% with respect to the MRI only tumour definitions when comparing with the conventionally planned EPCTV. With the proposed method, shallow sub-retinal tumour infiltration, otherwise invisible on MRI, can be included in the target volume definition. Moreover, a novel definition of the fovea location improves the accuracy and personalisation of the 3D eye model. ConclusionMRI and fundus image fusion overcomes some of the limitations of ophthalmological MRI for tumour volume definition in OPT. This novel eye tumour modelling method might improve treatment planning personalisation, allowing to better anticipate which patients could benefit from prophylactic treatment protocols for radiation induced maculopathy.

Map Projections Classification

Many books, textbooks and papers have been published in which the classification of map projections is based on auxiliary (developable) surfaces and projections are divided into conic, cylindrical and azimuthal projections. We argue that such a classification of map projections is unacceptable and give many reasons for that. Many authors wrote in more detail about the classification of map projections, and our intention is to give a new refined and rectified insight into the classification of map projections. Our approach can be included in map projection publications of general and thematic cartography. Doing this, misconceptions and unnecessary insistence on conceptuality instead of reality will be avoided.

Filtering states with total spin on a quantum computer

Starting from a general wave function described on a set of spins/qubits, we propose several quantum algorithms to extract the components of this state on eigenstates of the total spin ${\bf S}^2$ and its azimuthal projection $S_z$. The method plays the role of total spin projection and gives access to the amplitudes of the initial state on a total spin basis. The different algorithms have various degrees of sophistication depending on the requested tasks. They can either solely project onto the subspace with good total spin or completely uplift the degeneracy in this subspace. After each measurement, the state collapses to one of the spin eigenstates that could be used for post-processing. For this reason, we call the method Total Quantum Spin filtering (TQSf). Possible applications ranging from many-body physics to random number generators are discussed.

Spectral Galerkin method for solving Helmholtz boundary integral equations on smooth screens

Abstract We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin–Bubnov methods take as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the unit disk. By exactly depicting edge singular behaviors we show that these spectral or high-order bases yield super-algebraic error convergence in the corresponding energy norms whenever the screen is an analytic deformation of the unit disk. Moreover, we provide a fully discrete analysis of the method, including quadrature rules, based on analytic extensions of the spectral basis to complex neighborhoods. Finally, we include numerical experiments to support our claims as well as appendices with computational details for treating the associated singular integrals.

Historical change in the large-scale population distribution of the United States

Measures of central tendency and dispersion are the mainstay of statistical description. Their spatial analogues can be used to describe geographic distributions, and here they are employed to describe the historical path of the large-scale spatial distribution of the US population. The azimuthal equidistant projection and a three-dimensional coordinate approach are used to describe centrality, and Bachi's standard distance, and the standard deviational ellipse are used to describe dispersion. Other measures are used to describe demographic features such as differences in population density, and the degree to which the population distribution has become bicoastal. Decennial US Census data for the period 1790–2010 is combined with novel use of several measures. The analysis reveals that both the dispersion of the population and the growth of coastal populations has slowed dramatically during the last few decades. In addition to implications for changes in the demand for services such as housing and health care, the results suggest that the geographic distribution of the US population is reaching, at least temporarily, a mature equilibrium characterized by slower changes.

Analiza kartografske projekcije karte Nova Lusitania

The map named Carta Geographica de Projecçaõ Espherica Orthogonal da Nova Lusitania ou America Portugueza e Estado do Brazil from 1798, together with its 1795 (?), 1797 and 1803 versions, is undoubtedly one of the cartographic monuments developed by Portuguese cartography from the late eighteenth century. Its organizer was the geographer, astronomer, and frigate captain Antonio Pires da Silva Pontes Leme, who relied on the work of 34 people, including astronomers, geographers, and engineers, who, although only mentioned in the 1798 version, contributed to the creation of all versions. All of them are similar in appearance, but differ in size, content, details, amount, and distribution of toponyms, which will be the subject of another paper. The greatest similarity, however, concerns the defined map projection. The objective of this paper is to analyse and present the possible hypotheses and conclusions about which map projection was adopted for all versions of Nova Lusitania, through the identification of characteristics that allowed to infer and prove the adopted projection. The applied methodology verified that in the bibliographic search, the information about the map structure is insufficient. An article presented by General Djalma Polli Coelho in October 1950 states that the projection suggested by its title, as orthogonal spherical, appeared to be the Sanson-Flamsteed equal-area projection. However, the expression Carta Geographica de Projecçaõ Espherica Orthogonal allows us to infer also the transverse orthographic projection. Through parameters defined for the two projections, it was possible to establish the comparative elements for a cartographic analysis, which would allow us to conclude and prove the structure adopted for the map, allowing to conclude if the adopted projection for the Nova Lusitania was an azimuthal orthographic equatorial projection, or a Sanson-Flamsteed, sinusoidal projection on the meridian 315°, defined west-east, (counterclockwise), from the El Hierro (Ferro) Island. This meridian is referenced approx. –62°39'46" off the Greenwich meridian.

Transversal azimutal projeksiyonlarda Aitoff ve Wagner dönüşümleri

An interesting method has been applied by David Alexandrovich Aitoff modifying the equidistant transversal azimuthal projection so that it can be used for planisphere representations. Such transformations were generalized by Karheinz Wagner long after Aitoff. In this study, the transformation method is discussed, and it is applied to streographic projection. This modification has been examined in terms of distortions and compared with Aitoff and Hammer projections. The results show that this new option can be used in planisphere representations.

Проекции меридианного сечения: новый класс проекций для трехосного эллипсоида

Historically the conformal projections have been used for mapping not only the Earth, but other celestial bodies as well. Their application enables preserving the shape of the relief features on the maps, which is extremely important for various analyses of celestial bodies’ surfaces. For many small bodies of the Solar system the International Astronomical Union recommends to apply a triaxial ellipsoid as a reference surface. But if the conformal projections for the reference surfaces of a sphere and an ellipsoid of revolution already exist, obtaining these projections for a triaxial ellipsoid will be significantly complicated, and the task of preserving the shape of relief features still actual. In general, the article deals with cylindrical and azimuthal projections of the meridian section for global mapping the celestial body surface in accordance with the idea formulated by prof. L. M. Bugaevsky. The projections are implemented for mapping of Phobos, moon of Mars.

地図投影法を応用した多軸関節の可動域“joint sinus cone”の表現方法 －肩関節可動域表示例から検証するランベルト正積方位図法の適切性－

地図投影法を応用した多軸関節の可動域“joint sinus cone”の表現方法 －肩関節可動域表示例から検証するランベルト正積方位図法の適切性－

Visualization of flat and curved spacetimes with simple cartography tools

Well-developed tools from cartography are adapted and used to visualize curved and flat spacetime to further understanding of relativistic effects, while interpretation of these tools is aided through their application to the familiar world map. First spacetime is cut into two-dimensional subspaces that are mapped to the Minkowski or Euclidean plane using map projections. Then, exploiting the cartographic technique of map interruptions, the two-dimensional subspaces are further cut into ‘wedges’ to minimize the distortion. The mental exercise of sewing the wedges back together aids in visualizing and interpreting the geometry of the spacetime slice. Map distortion indicators alert the reader to the inevitable effects of distortion. This wedge diagram provides a simple yet powerful tool to visualize both flat and curved spacetime geometry. It is more general than the familiar embedding diagram and avoids the pitfall of confusing the space in question with the embedding space. The wedge diagram is applied first to Minkowski spacetime with the cartographers’ azimuthal equidistant map projection and, when combined with the wedge technique, provides a simple visualization of the twin paradox. The wedge diagram is then applied to the exterior Schwarzschild spacetime to visualize the physical significance of spacetime curvature. In particular it is shown how gravitational time dilation results from intrinsic curvature of a radial-time slice of spacetime. Then this diagram is applied to the Friedmann–Robertson–Walker (FRW) spacetime revealing how expansion of the Universe results from a qualitatively similar curvature of a radial-time slice, but with the roles of space and time reversed. The analogy between the two spacetimes suggests novel interpretations of the two spacetimes. We interpret the Schwarzschild spacetime as time expanding with increasing distance from the horizon. And we interpret the FRW spacetime as the Universe expanding as the influence of the big bang peters out.

The azimuthal equidistant projection for a Finsler manifold by the exponential map

The azimuthal equidistant projection for a Finsler manifold by the exponential map