Conic Projection Research Articles

Conic projections of the triaxial ellipsoid: the projections for regional mapping of celestial bodies

Abstract. Different distortion classes of the azimuthal and cylindrical projections of the triaxial ellipsoid have been considered in our previous works. These projections make it possible to construct maps of the celestial bodies in planetary scale. However, for regions in the middle latitudes it is advisable to use a conic projection which was not developed until now. In this investigation we describe the development of three conic projections of a triaxial ellipsoid: a conic projection with true scale along meridians, an equal-area conic projection and a quasi-conformal conic projection. In derivation of the projections we use an elliptical cone tangent to a triaxial ellipsoid. The projections are calculated and maps in these projections are created for the first time.

Fibonacci lattices for the evaluation and optimization of map projections

Latitude-longitude grids are frequently used in geosciences for global numerical modelling although they are remarkably inhomogeneous due to meridian convergence. In contrast, Fibonacci lattices are highly isotropic and homogeneous so that the area represented by each lattice point is virtually the same. In the present paper we show the higher performance of Fibonacci versus latitude-longitude lattices for evaluating distortion coefficients of map projections. In particular, we obtain first a typical distortion for the Lambert Conformal Conic projection with their currently defined parameters and geographic boundaries for Europe that has been adopted as standard by the INSPIRE directive. Further, we optimize the defining parameters of this projection, lower and upper standard parallel latitudes, so that the typical distortion for Europe is reduced a 10% when they are set to 36° and 61.5°, respectively. We also apply the optimization procedure to the determination of the best standard parallels for using this projection in Spain, whose values remained unspecified by the National decree that commanded its official adoption, and obtain optimum values of 37° and 42° and a resulting typical distortion of 828 ppm.

Military aeronautical charts in the past and today

Abstract The history of the development of military aeronautical charts began immediately before the First World War. The first charts created at that time did not differ much from topographic maps. Air planes were fairly slow back then and had a small range of action, which meant that the charts were developed at the scale of 1:200,000. When speed of aircraft increased, it soon turned out that this scale was too large. Therefore, many countries began to create charts with smaller scales: 1:300,000 and 1:500,000. The International Map of the World 1:1,000,000 (IMW) was frequently used for continental flights prior to the outbreak of the Second World War, while 1:3,500,000 and 1:5,000,000 maps were commonly used for intercontinental flights. The Second World War brought a breakthrough in the field of aeronautical chart development, especially after 7 December 1941, when the USA entered into the war. The Americans created more than 6000 map sheets and published more than 100 million copies, which covered all continents. In their cartographic endeavours, they were aided foremost by the Brits. On the other hand, the Third Reich had more than 1,500 officers and about 15,000 soldiers and civil servants involved in the development of maps and other geographic publications during the Second World War. What is more, the Reich employed local cartographers and made use of local source materials in all the countries it occupied. The Germans introduced one new element to the aeronautical charts – the printed reference grid which made it easier to command its air force. The experience gained during the Second World War and local conflicts was for the United States an impulse to undertake work on the standardization of the development of aeronautical charts. Initially, standardization work concerned only aeronautical charts issued by the US, but after the establishment of NATO, standardization began to be applied to all countries entering the Alliance. The currently binding NATO STANAGs (Standardization Agreements) distinguish between operational charts and special low-flight charts. The charts are developed in the WGS-84 coordinate system, where the WGS-84 ellipsoid of rotation is the reference surface. The cylindrical transverse Mercator projection was used for the scale of 1:250,000, while the conformal conic projection was used for other scales. The first aeronautical charts issued at the beginning of the 20th century contained only a dozen or so special symbols concerning charts’ navigational content, whereas currently the number of symbols and abbreviations found on such charts exceeds one hundred. The updating documents are published every 28 days in order to ensure that aeronautical charts remain up-to-date between releases of their subsequent editions. It concerns foremost aerial obstacles and air traffic zones. The aeronautical charts published by NATO have scales between 1:50,000 and 1:500,000 and the printed Military Grid Reference System (MGRS), while the aeronautical charts at scales between 1:250,000 and 1:2,000,000 contain the World Geographic Reference System (GEOREF). Nowadays, modern military air planes are characterised by their exceptional combat capabilities in terms of speed, range and manoeuvrability. Aside from aircraft, contemporary armed forces make increasingly frequent use of aerial robots, drones and unmanned cruise missiles. This is why, there has been a noticeable increase, especially in NATO, in the amount of work devoted to the standardization and development of aeronautical charts, as well as deepening of knowledge of navigation and aeronautical information.

Conic Projections of the Triaxial Ellipsoid: The Projections for Regional Mapping of Celestial Bodies

In our previous works, we described the projections that make it possible to construct maps of the celestial bodies in planetary scale – the azimuthal and cylindrical projections of different distortion classes. However, for regions in the middle latitudes, it is advisable to use a conic projection, which has not been developed previously. In this investigation, we describe the development of three conic projections of a triaxial ellipsoid: a conic projection with true scale along meridians, an equal-area conic projection, and a quasi-conformal conic projection. The quasi-conformal conic projection is a projection close to the conformal projection in the neighbourhood of each meridian corresponding to a meridian section. We treat conic projections as projections in which the meridians are a bundle of straight lines emanating from a single point, and parallels are curves constructed in accordance with the selected character of distortion. This definition of conic projections of the triaxial ellipsoid allows us to connect various classes of projections in a system. Thus, cylindrical projections can be considered as a limiting case of conic projections, and azimuthal projections as a special case. For the triaxial ellipsoid as a surface that can be projected on a plane without distortions, we use a direct elliptic cone tangent to the ellipsoid. The projections are calculated, and maps in these projections are created for the first time.

Enhancing adaptive composite map projections: Wagner transformation between the Lambert azimuthal and the transverse cylindrical equal-area projections

ABSTRACTThe adaptive composite map projection technique changes the projection to minimize distortion for the geographic area shown on a map. This article improves the transition between the Lambert azimuthal projection and the transverse equal-area cylindrical projection that are used by adaptive composite projections for portrait-format maps. Originally, a transverse Albers conic projection was suggested for transforming between these two projections, resulting in graticules that are not symmetric relative to the central meridian. We propose the alternative transverse Wagner transformation between the two projections and provide equations and parameters for the transition. The suggested technique results in a graticule that is symmetric relative to the central meridian, and a map transformation that is visually continuous with changing map scale.

Mexican biogeographic provinces: Map and shapefiles.

We provide a map of the 14 biogeographic provinces of Mexico based on the ecoregions recognized for the country, which combine climatic, geological and biotic criteria. These provinces belong to the Nearctic region (Californian, Baja Californian, Sonoran, Chihuahuan Desert and Tamaulipas provinces), Neotropical region (Pacific Lowlands, Balsas Basin, Veracruzan and Yucatán Peninsula provinces) and the Mexican transition zone (Sierra Madre Occidental, Sierra Madre Oriental, Transmexican Volcanic Belt, Sierra Madre del Sur and Chiapas Highlands provinces). In order to facilitate future biogeographic analyses, we provide a file of the biogeographical regionalisation of Mexico by converting the map into a polygon shapefile and a raster file with all provinces. We also separately provide each of the provinces in vector and raster format. All the maps are in geographical and Lambert Conformal Conic projections.

Daily estimation of ground-level PM2.5 concentrations at 4km resolution over Beijing-Tianjin-Hebei by fusing MODIS AOD and ground observations.

The satellite-borne Moderate Resolution Imaging Spectroradiometer (MODIS) aerosol optical depth (AOD) is widely used to estimate ground-level fine ambient particulate matter (PM2.5) concentrations to evaluate their health effects. The associated estimation accuracy is often reduced by AOD missing values and by insufficiently accounting for the spatio-temporal PM2.5 variations. In this study, we aim to estimate ground-level PM2.5 concentrations at a fine resolution with improved accuracy by fusing fine-scale satellite and ground observations in the populated and polluted Beijing-Tianjin-Hebei (BTH) area of China in 2014. We employed a Bayesian-based statistical downscaler to model the spatio-temporal linear AOD-PM2.5 relationships. We used a 3km MODIS AOD product, which was resampled to a 4km resolution in a Lambert conic conformal projection, to assist comparison and fusion with predictions by atmospheric chemistry models. A two-step method was used to fill the missing AOD values to obtain a full AOD dataset with complete spatial coverage. The downscaler has a good performance in the fitting procedure (R2=0.75) and in the cross validation procedure (R2=0.58 by random method and R2=0.47 by city-specific method). The number of missing AOD values was serious and related to elevated PM2.5 concentrations. The gap-filled AOD values corresponded well with our understanding of PM2.5 pollution conditions in BTH. The prediction accuracy of PM2.5 concentrations were improved in terms of their annual and seasonal mean. As a result of its fine spatio-temporal resolution and complete spatial coverage, the daily PM2.5 estimation dataset could provide extensive and insightful benefits to related studies in the BTH area. This may include understanding the formation processes of regional PM2.5 pollution episodes, evaluating daily human exposure, and establishing pollution controlling measures.

Iterative conic beam tomography based on Bayesian approach to radiation therapy

The key problem in increasing efficiency of radiotherapy of malignant tumors in brain and other dangerous neoplasms is the problem of increasing the quality of 3D positioning of a patient before radiotherapy. We consider the principles of development of fast parallel iterative algorithms based on graphic accelerators and the OpenGL library. The proposed approaches provide simultaneous residual minimization for the sought solution and total variation of the reconstructed 3D image. In this case, the number of required initial data, i.e., conic X-ray projections, can be reduced several times, and therefore, the radiation load on the patient can also be accordingly reduced with preservation of the necessary contrast and spatial resolution of the 3D image of the patient. The new heuristic iterative algorithm can be used as an alternative to the known 3D Feldkamp algorithm.

Custom Map Projections for Regional Groundwater Models

For regional groundwater flow models (areas greater than 100,000 km2 ), improper choice of map projection parameters can result in model error for boundary conditions dependent on area (recharge or evapotranspiration simulated by application of a rate using cell area from model discretization) and length (rivers simulated with head-dependent flux boundary). Smaller model areas can use local map coordinates, such as State Plane (United States) or Universal Transverse Mercator (correct zone) without introducing large errors. Map projections vary in order to preserve one or more of the following properties: area, shape, distance (length), or direction. Numerous map projections are developed for different purposes as all four properties cannot be preserved simultaneously. Preservation of area and length are most critical for groundwater models. The Albers equal-area conic projection with custom standard parallels, selected by dividing the length north to south by 6 and selecting standard parallels 1/6th above or below the southern and northern extent, preserves both area and length for continental areas in mid latitudes oriented east-west. Custom map projection parameters can also minimize area and length error in non-ideal projections. Additionally, one must also use consistent vertical and horizontal datums for all geographic data. The generalized polygon for the Floridan aquifer system study area (306,247.59 km2 ) is used to provide quantitative examples of the effect of map projections on length and area with different projections and parameter choices. Use of improper map projection is one model construction problem easily avoided.

Small Scale Map Projection and Coordinate System Improvement in Consideration of Usability and Compatibility

Small-scale maps currently used are made by scanning and editing printed maps and its shortcoming is accumulated errors at the time of editing and low accuracy. TM projection method is used but its accuracy varies. In addition, small-scale maps are made without consideration of usability and compatibility with other scale maps. Therefore, it is necessary to suggest projection and coordinates system improvement methods in consideration of usability and compatibility between data. The results of this study reveal that in order to make the optimum small-scale map, projection that fits the purpose of map usage in each scale, coordinate system and neat line composition should be selected in consideration of interrelation and compatibility with other maps. Conic projection should be used to accurately illustrate the entire country, but considering usability and compatibility with other maps, traversing cylindrical projection should be used instead of conic projection. For coordinates system of the small-scale map, Universal Transverse Mercator (UTM-K) based on the World Geodetic System should be used instead of conventional longitude and latitude coordinate system or Transverse Mercator.

GPU-accelerated iterative reconstruction from Compton scattered data using a matched pair of conic projector and backprojector

Background and ObjectiveIterative reconstruction from Compton scattered data is known to be computationally more challenging than that from conventional line-projection based emission data in that the gamma rays that undergo Compton scattering are modeled as conic projections rather than line projections. In conventional tomographic reconstruction, to parallelize the projection and backprojection operations using the graphics processing unit (GPU), approximated methods that use an unmatched pair of ray-tracing forward projector and voxel-driven backprojector have been widely used. In this work, we propose a new GPU-accelerated method for Compton camera reconstruction which is more accurate by using exactly matched pair of projector and backprojector. MethodsTo calculate conic forward projection, we first sample the cone surface into conic rays and accumulate the intersecting chord lengths of the conic rays passing through voxels using a fast ray-tracing method (RTM). For conic backprojection, to obtain the true adjoint of the conic forward projection, while retaining the computational efficiency of the GPU, we use a voxel-driven RTM which is essentially the same as the standard RTM used for the conic forward projector. ResultsOur simulation results show that, while the new method is about 3 times slower than the approximated method, it is still about 16 times faster than the CPU-based method without any loss of accuracy. ConclusionsThe net conclusion is that our proposed method is guaranteed to retain the reconstruction accuracy regardless of the number of iterations by providing a perfectly matched projector–backprojector pair, which makes iterative reconstruction methods for Compton imaging faster and more accurate.

Automating the selection of standard parallels for conic map projections

Conic map projections are appropriate for mapping regions at medium and large scales with east–west extents at intermediate latitudes. Conic projections are appropriate for these cases because they show the mapped area with less distortion than other projections. In order to minimize the distortion of the mapped area, the two standard parallels of conic projections need to be selected carefully. Rules of thumb exist for placing the standard parallels based on the width-to-height ratio of the map. These rules of thumb are simple to apply, but do not result in maps with minimum distortion. There also exist more sophisticated methods that determine standard parallels such that distortion in the mapped area is minimized. These methods are computationally expensive and cannot be used for real-time web mapping and GIS applications where the projection is adjusted automatically to the displayed area. This article presents a polynomial model that quickly provides the standard parallels for the three most common conic map projections: the Albers equal-area, the Lambert conformal, and the equidistant conic projection. The model defines the standard parallels with polynomial expressions based on the spatial extent of the mapped area. The spatial extent is defined by the length of the mapped central meridian segment, the central latitude of the displayed area, and the width-to-height ratio of the map. The polynomial model was derived from 3825 maps—each with a different spatial extent and computationally determined standard parallels that minimize the mean scale distortion index. The resulting model is computationally simple and can be used for the automatic selection of the standard parallels of conic map projections in GIS software and web mapping applications.

基于极点到坐标原点的距离函数的常规地图投影分类体系

地球椭球面上的经线交于极点而纬线则相互平行。地图上的经纬网是地球椭球面上的经纬网的拓扑映射。投影网中极点到坐标原点的距离函数关联着7类常规地图投影的分类。距离等于零时对应着方位投影；距离为有限值时对应着圆锥投影和多圆锥投影;距离为无穷大时对应着圆柱投影。方位和伪方位投影的纬线为同心圆；圆锥和伪圆锥投影的纬线为同心圆弧；多圆锥投影的纬线为同轴圆弧；圆柱和伪圆柱投影的纬线为赤道的平行线。所有正常投影的经线均为直线，而伪型投影只有作为投影网对称轴的中央经线是直线，其余经线为中央经线的轴对称曲线。多圆锥投影的经线与伪型投影相同。极点到坐标原点的距离函数，与常规地图投影的分类具有等价关系，从而可以推出常规地图投影的一般定义。 The longitudes of earth ellipsoid intersect at the poles and the latitudes are parallel to each other. The grid of longitudes and latitudes on the map is the topological map of the grid of longitudes and latitudes on earth ellipsoid. The distance function of the poles to the coordinate origin in the pro-jection grid is related to the classification of seven types of routine map projection. When the dis-tance is equal to zero, it is azimuthal projection; when the distance is finite, it is conic projection or polyconic projection; when the distance is infinite, it is cylindrical projection. The latitude of azi-muthal projection and pseudo-azimuthal projection are homocentric circles. The latitude of conic projection and pseudo-conic projection are homocentric circular arc. The latitude of polyconic projection is coaxial arc. The latitude of cylindrical projection and pseudo-cylindrical projection are parallel lines of the equator. All the longitudes of normal projection are straight, while for the pseudo-projection, only the central meridian of symmetry axis of a projection grid is a straight line and the remaining longitudes is the axisymmetric curve of the central longitudes. Meridian of polyconic projection is identical with the pseudo type projection. The distance function of the pole to the coordinate origin is equivalent to the classification of the routine map projection, so that the general definition of routine map projection can be deduced.

An adaptable equal-area pseudoconic map projection

ABSTRACTEquivalence (the equal-area property of a map projection) is important to some categories of maps. However, unlike conformal projections, completely general techniques do not exist for creating new, computationally reasonable equal-area projections. The literature describes many specific equal-area projections and a few equal-area projections that are more or less configurable, but flexibility is still sparse. This work describes a new, highly configurable equal-area projection system consisting of arcs of concentric circles, placing it in the pseudoconic class. The system uses a novel technique to hybridize the Bonne pseudoconic projection and the Albers conic projection, subsuming many existing projections as degenerate cases. With the resulting system and the technique used to develop it, map projection designers will have greater choice in tailoring the projection to the need. The system may be particularly suited to maps that dynamically adapt to changing scale and region of interest, such as required for online maps.

Identification of sand and dust storm source areas in Iran

Sand and dust storms (SDS) are common phenomena in arid and semi-arid areas. In recent years, SDS frequencies and intensities have increased significantly in Iran. A research on SDS sources is important for understanding the mechanisms of dust generation and assessing its socio-economic and environmental impacts. In this paper, we developed a new approach to identify SDS source areas in Iran using a combination of nine related datasets, namely drought events, temperature, precipitation, location of sandy soils, SDS frequency, human-induced soil degradation (HISD), human influence index (HII), rain use efficiency (RUE) and net primary productivity (NPP) loss. To identify SDS source areas, we firstly normalized these datasets under uniform criteria including layer reprojection using Lambert conformal conic projection, data conversion from shapefile to raster, Min-Max Normalization with data range from 0 to 1, and data interpolation by Kriging and images resampling (resolution of 1 km). After that, a score map for the possibility of SDS sources was generated through overlaying multiple datasets under average weight allocation criterion, in which each item obtained weight equally. In the score map, the higher the score, the more possible a specific area could be regarded as SDS source area. Exceptions mostly came from large cities, like Tehran and Isfahan. As a result, final SDS source areas were mapped out, and Al-Howizeh/Al-Azim marshes and Sistan Basin were identified as main SDS source areas in Iran. The SDS source area in Al-Howizeh/Al-Azim marshes still keeps expanding. In addition, Al-Howizeh/Al-Azim marshes are now suffering rapid land degradation due to natural and human-induced factors and might totally vanish in the near future. Sistan Basin also demonstrates the impacts of soil degradation and wind erosion. With appropriate intensity, duration, wind speed and altitude of the dust storms, sand particles uplifting from this area might have developed into extreme dust storms, especially during the summer.

Smooth, seamless, and structured grid generation with flexibility in resolution distribution on a sphere based on conformal mapping and the spring dynamics method

A generation method for smooth, seamless, and structured triangular grids on a sphere with flexibility in resolution distribution is proposed. This method is applicable to many fields that deal with a sphere on which the required resolution is not uniform. The grids were generated using the spring dynamics method, and adjustments were made using analytical functions. The mesh topology determined its resolution distribution, derived from a combination of conformal mapping factors: polar stereographic projection (PSP), Lambert conformal conic projection (LCCP), and Mercator projection (MP). Their combination generated, for example, a tropically fine grid that had a nearly constant high-resolution belt around the equator, with a gradual decrease in resolution distribution outside of the belt. This grid can be applied to boundary-less simulations of tropical meteorology. The other example involves a regionally fine grid with a nearly constant high-resolution circular region and a gradually decreasing resolution distribution outside of the region. This is applicable to regional atmospheric simulations without grid nesting. The proposed grids are compatible with computer architecture because they possess a structured form. Each triangle of the proposed grids was highly regular, implying a high local isotropy in resolution. Finally, the proposed grids were examined by advection and shallow water simulations.

The Mapping of Lancashire by William Smith

Recently an unreported manuscript draft of a map of Lancashire from about 1604 by William Smith was located in the collections of the John Rylands Library in Manchester (Latin MS 509). In this note the new manuscript is compared with Smith’s printed map of Lancashire, for many years referred to as ‘Anonymous’, which was engraved from the 1604 manuscript, with the previously known 1598 manuscript map of Lancashire by Smith (Harley MS 6159) and with the four other known drafts for the Smith series in the British Library (Maps C. 2. cc. 2 [12‒15]). The comparison highlights Smith’s ability as a mapmaker and reveals some of the construction methods he employed. An underlying pencilled grid was used to transfer details from the 1598 draft and to convert from a cylindrical to a conic projection. Smith seems to be the first to have used such a projection for a county map, a novelty similar to his introduction of true north, and a key to map signs.

Determination of definitive standard parallels of normal aspect conic projections equidistant along meridians on an old map

ABSTRACTIn previous research we proved that meridians are straight lines and parallels are arcs of concentric circles on the map Nuova carta geografica dello Stato Ecclesiastico by J. R. Bošković and Ch. Maire, published in 1755. We also proved that the map was produced in a normal aspect conic projection equidistant along meridians. Knowing the map projection type is not enough. We need to know its parameters, too. In the first part of the paper, we described the procedure of determining standard parallels of the normal aspect conic projection equidistant along meridians. We proposed a procedure for determining projection parameters of an old map, especially its standard parallels, if the radius of the sphere in which the map was produced in is unknown. Constant C is introduced as a ratio of constant K and sphere radius R, both of which are unknown. The procedure is illustrated on the map Nuova carta … . Using the procedure, we obtained standard parallels φ1 = 41°50'N and φ2 = 44°30'N. In addition, we tested whether the obtained standard parallels are actually standard parallels in which the map was produced and determined the definitive standard parallels in which the map Nuova carta … has been produced.

Optimal map conic projection - a case study for the geographic territory of Serbia

Optimal map conic projection - a case study for the geographic territory of Serbia

Investigating a solar influence on cloud cover using the North American Regional Reanalysis data

The controversial connection between cosmic rays, solar activity, and cloud cover is investigated using a climatological reconstructed reanalysis product: the North American Regional Reanalysis which provides high-resolution, low, mid-level, high, and total cloud cover data over a Lambert conformal conic projection permitting land/ocean discrimination. Pearson’s product-moment regional correlations were obtained between monthly cloud cover data and solar variability indicators, cosmic ray neutron monitors, several climatological indices, including the Atlantic Multidecadal Oscillation (AMO), and between cloud layers. Regions of the mid-latitude oceans exhibited a positive correlation with cosmic ray flux. Additionally, this maritime low cloud cover exhibits the only failed correlation significance with other altitudes. The cross correlation reveals that cloud cover is positively correlated everywhere but for ocean low cloud cover, supporting the unique response of the marine layer. The results of this investigation suggest that with the assumption that solar forcing does impact cloud cover, measurements of solar activity exhibits a slightly higher correlation than GCRs. The only instance where GCRs exhibit a positive regional correlation with cloud cover is for maritime low clouds. The AMO exerts the greatest control of cloud cover in the NARR domain.