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A new map projection: Its development and characteristics
... š P 2 projections do not have the closed-form of forward equations, and they require iterative computations [2,34], such as Newton-Raphson iterations. In addition, certain projections may be interrupted (e.g., Hierarchical Equal Area isoLatitude Pixelisation, or HEALPix [35,36], interrupted Goode homolosine [22,23,37] and icosahedral Snyder equal area [38] projections), derivative discontinuous (e.g., uninterrupted Goode homolosine [14] and Robinson [39] projections), have singularity points (Mercator and Peirce Quincuncial [40] projections), or may have an elliptic integral (Adams World in a Square I and II and Peirce Quincuncial [40] projections) in forward equations. Another complicated example is the Chamberlin trimetric projection [41], which has multiple loops and several if-else branches in forward equations in the implementation of PROJ library. ...
... Firstly, establish and solve a set of equations (see Equations (32)- (35) or (40)-(43)) on the projection plane (see Figure 3b) to obtain the coefficients of interpolation depending on the sign of α. Then, the coefficients are used to interpolate the point (see Equations (36)- (39) or (44)-(47)) on the sphere to obtain P i , i = 1, 2, 3, 4 (see Figure 3c). ...
... By substituting three-dimensional points P i (see Equations (36)- (39) or (44)-(47), and Figure 3c), i = 1, 2, 3, 4, into the right-hand side of Equation (15), we obtain threedimensional vectors q i , i = 1, 2. Next, we obtain a forward version of the GCA-based metric (or FWD-GCA metric) according to Equations (16) to (20); it is denoted as ρ f wd . Area distortion in the FWD-GCA metric can be obtained by rewriting Equation (22) as: ...
We studied the numerical approximation problem of distortion in map projections. Most widely used differential methods calculate area distortion and maximum angular distortion using partial derivatives of forward equations of map projections. However, in certain map projections, partial derivatives are difficult to calculate because of the complicated forms of forward equations, e.g., equations with iterations, integrations, or multi-way branches. As an alternative, the spherical great circle arcs–based metric employs the inverse equations of map projections to transform sample points from the projection plane to the spherical surface, and then calculates a differential-independent distortion metric for the map projections. We introduce a novel forward interpolated version of the previous spherical great circle arcs–based metric, solely dependent on the forward equations of map projections. In our proposed numerical solution, a rational function–based regression is also devised and applied to our metric to obtain an approximate metric of angular distortion. The statistical and graphical results indicate that the errors of the proposed metric are fairly low, and a good numerical estimation with high correlation to the differential-based metric can be achieved.
... Projection selection is the cartographic process of determining the most appropriate map projection for a particular geographic application, and is one of the fundamental problems which must be solved by cartographers to determine a map's graphical framework and scale distortion characteristics (Robinson, 1974). The map projection discipline is recognized as one of the most challenging in Geography (Olson, 2006), and choosing a suitable projection continues to be complicated and overwhelming for experienced and novice map-makers alike (De Genst & Canters, 1996;Kimerling, Buckley, Muehrcke, & Muehrcke, 2012;Snyder, 1993). ...
... As footprint area increases a greater proportion of the Earth's curvature is displayed and a higher degree of distortion becomes apparent. Footprint area is therefore directly related to the significance of the selection decision and is especially relevant for maps covering a hemisphere or the entire world (Hsu, 1981;Maling, 1992;Robinson, 1974;Slocum, McMaster, Kessler, & Howard, 2010). Distortion characteristics should be considered alongside projection properties which may be beneficial for a particular requirement together with aesthetic considerations such as the graticule appearance (Bugayevskiy & Snyder, 1995;Canters, 2002;Canters & Decleir, 1989). ...
... Tyner (2010) specifies 'Projections' as one of eight elements of the map design process, and with each of its sub-elements directly related to the other seven elements the significant impact of projection selection on cartographic decision-making is evident. The number and variety of projections available complicates the process (Canters, 2002;Nyerges & Jankowski, 1989;Snyder, 1987), especially for world maps where many were designed to solve mathematical problems rather than addressing particular practical uses (Robinson, 1974). ...
The selection of an appropriate map projection has a fundamental impact on the visualization and analysis of geographic information. Distortion is inevitable and the decision requires simultaneous consideration of several different factors; a process which can be confusing for many cartographers and GIS users. The last few decades have seen numerous attempts to create automated map projection selection solutions based on traditional classification and selection guidelines, but there are no existing tools directly accessible to users of GIS software when making projection selection decisions. This paper outlines key elements of projection selection and distortion theory, critically reviews the previous solutions, and introduces a new tool developed for ESRI's ArcGIS, employing an original selection method tailored to the specific purpose and geographical footprint characteristics of a GIS project. The tool incorporates novel quantitative projection distortion measures which are currently unavailable within existing GIS packages. Parameters are optimized for certain projections to further reduce distortions. A set of candidate projected coordinate systems are generated that can be applied to the GIS project; enabling a qualitative visual assessment to facilitate the final user selection. The proposed tool provides a straightforward application which improves understanding of the projection selection process and assists users in making more effective use of GIS.
... Robinson presented a graphic approach for developing pseudo-cylindrical projections with tabular parameters (Robinson, 1974), all projections were developed by analytical equations. ...
... Arthur H. Robinson proposed the structure of Equation 1 and the associated graphical approach to the design of small-scale map projections when he developed his eponymous projection (Robinson, 1974). ...
... The central meridian is a straight line 0.5072 longer than the equator. Other meridians are equally spaced elliptical arcs, and concave toward the central meridian (Robinson, 1974, Ipbüker, 2004. ...
The Natural Earth projection is a new projection for representing the entire Earth on small-scale maps. It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections. The original Natural Earth projection defines the length and spacing of parallels in a tabular form for every five degrees of increased latitude. It is a true pseudocylindrical projection, and is neither conformal nor equal-area. In the original definition, piece-wise cubic spline interpolation is used to project intermediate values that do not align with the five-degree grid.
This graduation thesis introduces alternative polynomial equations that are considerably simpler to compute. The polynomial expression also improves the smoothness of the rounded corners where the meridians meet the horizontal pole lines, a distinguished mark of the Natural Earth projection which suggests to readers that the Earth is spherical in shape. An inverse projection is presented. The formulas are simple to implement in cartographic software and libraries. Distortion values of this new graticule are not significantly different from the original piece-wise projection.
The development of the polynomial equations was inspired by a similar study of the Robinson projection. The polynomial coefficients were determined with least square adjustment of indirect observations with additional constraints to preserve the height and width of the graticule. The inverse procedure uses the Newton-Raphson method and converges in a few iterations.
... In particular, the spatial linkage of the datasets at a global scale can be easily noticed when properly projected. In commonly used landmass-oriented global map projections (e.g., the Robinson global projection 4 Fig. 1b), they often split one of two biggest oceans, i.e., the Pacific Ocean and the Atlantic Ocean, or have incomplete or discontinuous representations of water masses. Otherwise, oceans in two polar regions are significantly distorted in their shapes and scales, giving some erroneous conceptions of their distribution and extent. ...
... The Adams square II projection was created in 1925 25 , which is similarly conformal and square (Fig. 1e). Its equation is referred to Snyder 3 4 Spilhaus square projection for heat flow (a; data from ref. 15 ), sediment thickness (b; data from ref. 16 ), and gravity (c; data from ref. 17 ) and magnetic (d; data from ref. 18 ) anomalies. ...
The ocean, as a vast interconnected body of water on Earth, plays an essential role in Earth’s planetary dynamics, climate change, and the evolution of human society and decision-making processes. An ocean-focused global map is necessary to visually capture numerous phenomena within the world’s ocean and seafloor. Here we present the power of the Spilhaus square projection with various geological and geophysical datasets, including bathymetry, teleseismicity, seafloor geography, and seafloor spreading parameters. The Spilhaus projection, compared to widely-used map projections (e.g., Mercator and Robinson), emphasizes the seamless connection of water masses surrounded by continents. This projection has recently garnered attention for presenting ocean-oriented data, although it is not extensively used and currently supported by the ArcGIS software. Maps presented here provide not only a novel geological perspective on the world ocean as a whole body, but also new insights/questions to be addressed regarding features and processes of the water body, the seafloor, and ocean-atmosphere dynamics, which can be used for research, education, media, and policy decisions, and promote similar approaches.
... To visualize the color histograms the binned 2D color sphere was rotated so that the black-white direction would align with the vertical axis. The sphere was then flattened using a Robinson-like projection 160 , and each bin was assigned a reference color using the closest RGB color from the bin centroid. Finally, for each color of the histogram, the reference color b ∈[ 0,1] ³ in each bin was faded to gray using the histogram values, ...
V4 is a key area within the visual processing hierarchy, and it represents features of intermediate complexity. However, no current computational model explains V4 responses under natural conditions. To address this, we developed a new hierarchical convolutional energy (HCE) model reflecting computations thought to occur in areas V1, V2, and V4, but which consists entirely of simple- and complex-like units like those found in V1. In contrast to prior models, the HCE model is trained end-to-end on neurophysiology data, without relying on pre-trained network features. We recorded 313 V4 neurons during full-color nature video stimulation and fit the HCE model to each neuron. The model's predicted optimal patterns (POPs) revealed complex spatiotemporal pattern selectivity in V4, supporting its role in representing space, time, and color. These findings indicate that area V4 is crucial for image segmentation and grouping operations that are essential for complex vision. Thus, responses of V4 neurons under naturalistic conditions can be explained by a hierarchical three-stage model where each stage consists entirely of units like those found in area V1.
... Interactive visualization can be obtained via, for example, the elate package [66]. Alternatively, via a spherical coordinates transformation: n = [sin θ · cos φ, sin θ · sin φ, cos θ] (Fig. 4c), it can be represented in two dimensions (Fig. 4d, with a Robinson map projection [67]). Such plots make it easier to visually investigate the anisotropic characteristics of Young's modulus. ...
The elasticity tensor is a fundamental material property that describes the elastic response of a material to external force. The availability of full elasticity tensors for inorganic crystalline compounds, however, is limited due to experimental and computational challenges. Here, we report the materials tensor (MatTen) model for rapid and accurate prediction of the full fourth-rank elasticity tensors of crystals. Based on equivariant graph neural networks, MatTen satisfies two essential requirements for elasticity tensors: independence of the frame of reference and preservation of material symmetry. Consequently, it provides a unified treatment of elasticity tensors for all seven crystal systems across diverse chemical spaces, without the need to deal with each separately. MatTen was trained on a dataset of first-principles elasticity tensors garnered by the Materials Project over the past several years (we are releasing the data herein) and has broad applications in predicting the isotropic elastic properties of polycrystalline materials, examining the anisotropic behavior of single crystals, and discovering materials with exceptional mechanical properties. Using MatTen, we have found a hundred crystals with extremely large maximum directional Young's modulus and eleven polymorphs of elemental cubic metals with unconventional spatial orientation of Young's modulus.
... Interactive visualization can be obtained via, for example, the elate package [54]. Alternatively, via a spherical coordinates transformation: n = [sin θ · cos φ, sin θ · sin φ, cos θ] (Fig. 4c), it can be represented in two dimensions (Fig. 4d, with a Robinson map projection [55]). Such plots make it easier to visually investigate the anisotropic characteristics ...
The elasticity tensor that describes the elastic response of a material to external forces is among the most fundamental properties of materials. The availability of full elasticity tensors for inorganic crystalline compounds, however, is limited due to experimental and computational challenges. Here, we report the materials tensor (MatTen) model for rapid and accurate estimation of the full fourth-rank elasticity tensors of crystals. Based on equivariant graph neural networks, MatTen satisfies the two essential requirements for elasticity tensors: independence of the frame of reference and preservation of material symmetry. Consequently, it provides a universal treatment of elasticity tensors for all crystal systems across diverse chemical spaces. MatTen was trained on a dataset of first-principles elasticity tensors garnered by the Materials Project over the past several years (we are releasing the data herein) and has broad applications in predicting the isotropic elastic properties of polycrystalline materials, examining the anisotropic behavior of single crystals, and discovering new materials with exceptional mechanical properties. Using MatTen, we have discovered a hundred new crystals with extremely large maximum directional Young's modulus and eleven polymorphs of elemental cubic metals with unconventional spatial orientation of Young's modulus.
... The original data format is World Geodesic System 1984 (WGS 84), which we transform by using the Robinson projection, cp. [28], to reduce the distortion and the projection angles. From the data set we randomly select 10%, resulting in N = 1 245 888 data points. ...
We consider kernel-based learning in samplet coordinates with l1-regularization. The application of an l1-regularization term enforces sparsity of the coefficients with respect to the samplet basis. Therefore, we call this approach samplet basis pursuit. Samplets are wavelet-type signed measures, which are tailored to scattered data. They provide similar properties as wavelets in terms of localization, multiresolution analysis, and data compression. The class of signals that can sparsely be represented in a samplet basis is considerably larger than the class of signals which exhibit a sparse representation in the single-scale basis. In particular, every signal that can be represented by the superposition of only a few features of the canonical feature map is also sparse in samplet coordinates. We propose the efficient solution of the problem under consideration by combining soft-shrinkage with the semi-smooth Newton method and compare the approach to the fast iterative shrinkage thresholding algorithm. We present numerical benchmarks as well as applications to surface reconstruction from noisy data and to the reconstruction of temperature data using a dictionary of multiple kernels.
... Thus, we refer to the globe as the root node and the four quadrants as four child nodes called tiles. The Quadtree implementation for a global map requires a suitable projection, of which many are available, such as sinusoidal (Snyder, 1987), Equirectangular (Snyder, 1997), Mercator (Snyder, 1987), and Robinson (Robinson, 1974). Of these, the Mercator projection has two main properties that make it a suitable choice for its use with the Quadtree. ...
The Collaboratory for the Study of Earthquake Predictability (CSEP) is an international effort to evaluate probabilistic earthquake forecasting models. CSEP provides the cyberinfrastructure and testing methods needed to evaluate earthquake forecasts. The most common way to represent a probabilistic earthquake forecast involves specifying the average rate of earthquakes within discrete spatial cells, subdivided into magnitude bins. Typically, the spatial component uses a single-resolution Cartesian grid with spatial cell dimensions of 0.1° × 0.1° in latitude and longitude, leading to 6.48 million spatial cells for the global testing region. However, the quantity of data (e.g., number of earthquakes) available to generate and test a forecast model is usually several orders of magnitude less than the millions of spatial cells, leading to a huge disparity in the number of earthquakes and the number of cells in the grid. In this study, we propose the Quadtree to create multi-resolution grid, locally adjusted mirroring the available data for forecast generation and testing, thus providing a data-driven resolution of forecasts. The Quadtree is a hierarchical tree-based data structure used in combination with the Mercator projection to generate spatial grids. It is easy to implement and has numerous scientific and technological applications. To facilitate its application to end users, we integrated codes handling Quadtrees into pyCSEP, an open-source Python package containing tools for evaluating earthquake forecasts. Using a sample model, we demonstrate how forecast model generation can be improved significantly in terms of information gain if constrained on a multi-resolution grid instead of a high-resolution uniform grid. In addition, we demonstrate that multi-resolution Quadtree grids lead to reduced computational costs. Thus, we anitcipate that Quadtree grids will be useful for developing and evaluating earthquake forecasts.
... • Lambert azimuthal equal-area projection is an azimuthal projection with three aspects, i.e., polar, equatorial and oblique; it is best suited for individual territories that are symmetrically proportioned, either round or square (Snyder, 1987;Snyder and Voxland, 1989). • Van der Grinten projection is a polyconic projection, and is used for whole world maps (Snyder, 1987;Snyder and Voxland, 1989), and • Robinson projection (compromised distortion, not equal-area) is a pseudocylindrical map projection for plotting world maps (Robinson, 1974;Snyder, 1987Snyder, , 1990Snyder, , 1993Snyder and Voxland, 1989). ...
This Manual presents, a comprehensive overview of the standardised methods to be employed across the land surface of the Earth to map the distribution of chemical elements in rock, soil, sediment and water
... There is little benefit in maintaining this separation when explicating his thoughts on maps and map-users at a general level. 40 As part of his project to create an objective science of cartography, Robinson also developed a new projection (Robinson, 1974) in an attempt to depoliticise representations of the world (Wood and Krygier, 2009). users did not enter Robinson's framework -his focus lay on the production of maps as media artefacts. ...
This thesis provides a theoretical contribution towards understanding how, and to what extent, people’s engagements with digital maps feature in the constitution of their social practices. Existing theory tends not to focus on people as active interpreters that engage with digital maps across a variety of contexts, or on the influence of their map use on wider sets of social practices. Addressing this, the thesis draws on practice theory, media studies, and internet studies to develop a conceptual framework, applying it to empirical findings to address three research questions: (1) How do people engage with digital maps; (2) How do people engage with the web-based affordances of digital maps, such as those for collaboration, sharing, and end-user amendment/generation of content; and (3) What influence does people’s engagement with digital maps have on the way they perform wider sets of social practices? The research provides insights from three contexts, each operating at a different temporal scale: home choice covers longer-term processes of selecting and viewing properties before buying or renting; countryside leisure-walking covers mid-term processes of route-planning and assessment; University orientation covers shorter-term processes of navigation and gaining orientation around campus. Those insights are gathered through: a scoping survey (N=260) to identify relevant contexts; 32 semi-structured interviews to initiate data analysis; and 3 focus groups to gather participant feedback (member validation) on the emerging analysis. The approach to data analysis borrows heavily from constructivist grounded theory (albeit sensitised by practice theory ontology) to generate seven concepts. Together, the concepts constitute a practicetheory oriented digital sociology of map use. Overall, this thesis argues that digital maps are engaged with as mundane technologies that partially anchor people’s senses of place and security (physical and ontological), their performance of practices and social positions, and more broadly, the movement and distribution of bodies in space.
... Surface temperature data has been excluded and marked by gray color. The projection method followed was of Robinson [25] with 1200 km smoothing radius. Base period of mean global annual temperature has been referenced to the 14ºC estimated average of 1951-1980 [13]. ...
Global warming and associated sea level rise due to melting of major ice reserves is one of the most important and debated issues of present time. It has unequivocally been proved that mean global surface and ocean temperature is increasing due to anthropogenic input of greenhouse gases like carbon dioxide which has reached its maximum concentration up to 399.35 ppm since last 650,000 years. Due to greenhouse effect global ice caps, sheets and sea ice are melting. But there is an ambiguity of rate of melting of ice and rate of global sea level rise. This can be best exemplified by melting rate of Arctic sea ice with nonequivalent response to the rate of global sea level rise. Conceptual model proposed in this paper differentiates the role of continental glacier, ice sheets and sea ice in the process of global or eustatic sea level rise. It shows that melting of sea ice is not responsible for global or eustatic sea level change. However convoluted effect of melting of continental glaciers and ice sheets and thermal expansion of ocean water due to increasing ocean heat budget is responsible for recent eustatic sea level rise.
... To compare the relative positions of connection sites of AZM and non-AZM macromolecules on the cytosolic surface of a vesicle membrane to the nub connection sites on the luminal surface of the membrane, we converted the 3D spherical plots of the connection sites to 2D Robinson projections (Robinson, 1974). The positions of the spatial coordinates (x, y, z) of the centroids of connection sites were first plotted onto an idealized sphere using IDL software. ...
Active zone material is an organelle that is common to active zones along the presynaptic membrane of chemical synapses. Electron tomography on active zones at frog neuromuscular junctions has provided evidence that active zone material directs the docking of synaptic vesicles (SVs) on the presynaptic membrane at this synapse. Certain active zone material macromolecules connect to stereotypically arranged macromolecules in the membrane of undocked SVs, stably orienting a predetermined fusion domain of the vesicle membrane toward the presynaptic membrane while bringing and holding the two membranes together. Docking of the vesicles is required for the impulse-triggered vesicle membrane-presynaptic membrane fusion that releases the vesicles’ neurotransmitter into the synaptic cleft. As at other synapses, axon terminals at frog neuromuscular junctions contain, in addition to SVs, vesicles that are larger, are much less frequent and, when viewed by electron microscopy, have a distinctive electron dense core. Dense core vesicles at neuromuscular junctions are likely to contain peptides that are released into the synaptic cleft to regulate formation, maintenance and behavior of cellular apparatus essential for synaptic impulse transmission. We show by electron tomography on axon terminals of frog neuromuscular junctions fixed at rest and during repetitive impulse transmission that dense core vesicles selectively dock on and fuse with the presynaptic membrane alongside SVs at active zones, and that active zone material connects to the dense core vesicles undergoing these processes in the same way it connects to SVs. We conclude that undocked dense core vesicles have a predetermined fusion domain, as do undocked SVs, and that active zone material directs oriented docking and fusion of these different vesicle types at active zones of the presynaptic membrane by similar macromolecular interactions.
... comm., October 2014 and April 2018 2 ). The Robinson (1974) projection meets all of our criteria: it is a pseudocylindrical projection with pole lines and meridians that do not excessively bulge outwards, has regularly spaced meridians, and a height-to-width ratio close to 1:2. However, it is not an equal-area projection. ...
The Equal Earth map projection is a new equal-area pseudocylindrical projection for world maps. It is inspired by the widely used Robinson projection, but unlike the Robinson projection, retains the relative size of areas. The projection equations are simple to implement and fast to evaluate. Continental outlines are shown in a visually pleasing and balanced way.
... For example, the American (or ordinary) polyconic projection was developed by Hassler (1825) for large-scale topographic mapping of the United States. On the other hand, the Robinson projection, developed by Arthur Robinson (1974), was designed specifically to show the appearance of the world's landmasses as if looking on a globe. ...
The field of map projections
can be described as mathematical, static, and challenging. However, this description is evolving in concert with the development of the Internet. The Internet has enabled new outlets for software applications, learning, and interaction with and about map projections
. This chapter examines specific ways in which the Internet has moved map projections from a relatively obscure paper-based setting to a more engaging and accessible online environment. After a brief overview of map projections, this chapter discusses four perspectives on how map projections have been integrated into the Internet. First, map projections and their role in web maps
and mapping services is examined. Second, an overview of online atlases and the map projections chosen for their maps is presented. Third, new programming languages and code libraries that enable map projections to be included in mapping applications are reviewed. Fourth, the Internet has facilitated map projection education and research especially with the map reader’s comprehension and understanding of complex topics like map projection distortion
is discussed.
... Flex Projector, a freeware application for the interactive design and evaluation of map projections (Jenny & Patterson, 2014), was used to design Natural Earth II. For developing projections, Flex Projector takes a graphic approach that was first introduced by Arthur H. Robinson during the design of his well-known, eponymous projection (Jenny & Patterson, 2013;Jenny et al., 2008;Jenny, Patterson, & Hurni, 2010;Robinson, 1974). In Flex Projector, the user adjusts the length, shape, and spacing of parallels and meridians for every 5°of latitude and longitude. ...
The Natural Earth II projection is a new compromise pseudocylindrical projection for world maps. The Natural Earth II projection has a unique shape compared to most other pseudocylindrical projections. At high latitudes, meridians bend steeply toward short pole lines resulting in a map with highly rounded corners that resembles an elongated globe. Its distortion properties are similar to most other established world map projections. Equations consist of simple polynomials. A user study evaluated whether map-readers prefer Natural Earth II to similar compromise projections. The 355 participating general map-readers rated the Natural Earth II projection lower than the Robinson and Natural Earth projections, but higher than the Wagner VI, Kavrayskiy VII and Wagner II projections.
... There exists, however, a complementary visual approach for the definition of map projections, where graticules are defined in a graphical, nonmathematical way. This hitherto seldom-used approach was pioneered by Arthur Robinson in 1961, while commissioned by the Rand McNally publishing house to create a world map projection (Robinson, 1974). Robinson proceeded through an iterative process to create his pseudocylindrical projection, graphically evaluating the appearance and relative relationships of landmasses. ...
The idea of combining existing map projections to create new projections has been applied in the past by various authors. Many of the commonly used world map projections are such hybrid projections, for example, the Goode Homolosine projection or the Winkel Tripel projection. This paper discusses three digital methods for blending existing projections to create new projections for world maps at small scales. The three methods are implemented in Flex Projector, a software application specifically designed for the creation and evaluation of new world maps projections.
... Surface temperature data has been excluded and marked by gray color. The projection method followed was of Robinson [25] with 1200 km smoothing radius. Base period of mean global annual temperature has been referenced to the 14ºC estimated average of 1951-1980 [13]. ...
... Robinson's projection, adopted by the National Geographic Society is a key example (Robinson, 1974), and so is the sinusoidal equal-area ( Figure 5). ...
There has been a dramatic increase in the handling of geospatial information, and also in the production of maps. However, because the Earth is three-dimensional, geo-referenced data must be projected on a two-dimensional surface. Depending on the area being mapped, the projection process generates a varying amount of distortion, especially for continental and world maps. Geospatial users have a wide variety of projections too choose from; it is therefore important to understand distortion characteristics for each of them. This chapter reviews foundations of map projection, such as map projection families, distortion characteristics (areal, angular, shape and distance), geometric features and special properties. The chapter ends by a discussion on projection selection and current research trends.
... The inspiration for developing Flex Projector was Arthur Robinson's graphical approach to projection design. In 1961, working on a commission for the Rand McNally publishing house, Robinson created his eponymous world map projection, originally dubbed the orthophanic, meaning correct-looking (Robinson 1974). Robinson proceeded through an iterative process to create his pseudocylindrical projection, graphically evaluating the appearance and relative relationships of landmasses. ...
The idea of designing a new map projection via combination of two projections is well established. Some of the most popular world map projections in use today were devised in this manner. One construction method is to combine two source projections along a common parallel; a second method calculates the arithmetic means of two projections. These two methods for creating new world map projections are included in the latest version of Flex Projector. Flex Projector, afreeware mapping application, offers a graphical approach for customizing existing projections and creating new projections. The Mixer is a new feature in the latest version that allows the user to blend two existing projections to create a new hybrid projection. In addition to the two established combination methods, the software includes a new method for blending projections specific to its visual design approach. With this new method, a unique trait of one projection is transferable to a second projection. Flex Projector allows for the blending of four different projection traits separately or in combination: (1) the horizontal length of parallels, (2) the vertical distance of parallels from the equator, (3) the distribution of meridians, and (4) the bending of parallels. This article briefly describes the main characteristics of Flex Projector and then documents the new approaches to projection blending. The integration of the three methods into Flex Projector makes creating new projections simple and easy to control and allows the user to evaluate distortion characteristics of new projections. As an applied example, the article also introduces the new Pacific projection that is a blend of the Ginzburg VIII and Mollweide projections.
... We related each species to one single floristic region. All maps presented are in the Robinson projection (Robinson, 1974). The mean latitude for each species was estimated from published and unpublished data (Appendix S1). ...
AimGymnosperms are often described as a marginal and threatened group, members of which tend to be out-competed by angiosperms and which therefore preferentially persist at higher latitudes and elevations. The aim of our synthesis was to test these statements by investigating the global latitudinal and elevational distribution of gymnosperms, as well as their conservation status, using all extant gymnosperm groups (cycads, gnetophytes, ginkgophytes and conifers).LocationWorldwide.Methods
We developed a database of 1014 species of gymnosperms containing latitudinal and elevational distribution data, as well as their global conservation status, as described in the literature. The 1014 species comprised 305 cycads, 101 gnetophytes, the only living representative of ginkgophytes, and 607 conifers. Generalized additive models, frequency histograms, kernel density estimations and distribution maps based on Takhtajan's floristic regions were used.ResultsAlthough the diversity of gymnosperms decreases at equatorial latitudes, approximately 50% of the extant species occur primarily between the tropics. More than 43% of gymnosperms can occur at very low elevations (≤ 200 m a.s.l.). Gymnosperms, considering all species together as well as their main taxonomic groups separately, do not exhibit a latitudinal diversity gradient as commonly observed for many other taxa. Gymnosperms, and especially conifers, are on average less threatened at higher and equatorial latitudes.Main conclusionsGymnosperms display an unusual latitudinal diversity gradient, which we suggest cannot fully be accounted for by angiosperm dominance and competitive superiority. We hypothesize that other factors explain their present distribution, such as the development of centres of endemism in several regions and the adaptation of certain taxa to cold and arid climates.
... Flex Projector uses a graphical approach to map projection design similar to that used by Arthur H. Robinson for devising the famous projection that shares his name. In 1961, Robinson was commissioned by Rand Mc-Nally to design a world map projection that, among other criteria, was uninterrupted, had limited distortion, and was pleasing to the eye of general viewers (Robinson, 1974). He came up with a very simple idea: instead of devising a mathematical formula that relates longitude and latitude intersections on the sphere to X/Y coordinates on the map, he developed two sets of tabular parameters by trial and error. ...
Flex Projector is a free, open-source, and cross-platform software application that allows cartographers to interactively design custom projections for small-scale world maps. It specializes in cylindrical, and pseudocylindrical projections, as well as polyconical projections with curved parallels. Giving meridians non-uniform spacing is an option for all classes of projections. The interface of Flex Projector enables cartographers to shape the projection graticule, and provides visual and numerical feedback to judge its distortion properties. The intended users of Flex Projector are those without specialized mathematical expertise, including practicing mapmakers and cartography students. The pages that follow discuss why the authors developed Flex Projector, give an overview of its features, and introduce two new map projections created by the authors with this new software: the A4 and the Natural Earth projection.
We consider scattered data approximation in samplet coordinates with
-regularization. The application of an
-regularization term enforces sparsity of the coefficients with respect to the samplet basis. Samplets are wavelet-type signed measures, which are tailored to scattered data. They provide similar properties as wavelets in terms of localization, multiresolution analysis, and data compression. By using the Riesz isometry, we embed samplets into reproducing kernel Hilbert spaces and discuss the properties of the resulting functions. We argue that the class of signals that are sparse with respect to the embedded samplet basis is considerably larger than the class of signals that are sparse with respect to the basis of kernel translates. Vice versa, every signal that is a linear combination of only a few kernel translates is sparse in samplet coordinates. Therefore, samplets enable the use of well-established multiresolution techniques on general scattered data sets.
We propose the rapid solution of the problem under consideration by combining soft-shrinkage with the semi-smooth Newton method. Leveraging on the sparse representation of kernel matrices in samplet coordinates, this approach converges faster than the fast iterative shrinkage thresholding algorithm and is feasible for large-scale data. Numerical benchmarks are presented and demonstrate the superiority of the multiresolution approach over the single-scale approach. As large-scale applications, the surface reconstruction from scattered data and the reconstruction of scattered temperature data using a dictionary of multiple kernels are considered.</p
Despite the ubiquitous and persistent presence of microplastic (MP) in marine ecosystems, knowledge of its potential harmful ecological effects is low. In this work, we assessed the risk of floating MP (1 μm – 5 mm) to marine ecosystems by comparing ambient concentrations in the global ocean with available ecotoxicity data. The integration of twenty-three species-specific effect threshold concentration data in a species sensitivity distribution yielded a median unacceptable level of 1.21 * 10⁵ MP m⁻³ (95% CI: 7.99 * 10³ – 1.49 * 10⁶ MP m⁻³). We found that in 2010 for 0.17% of the surface layer (0 – 5 m) of the global ocean a threatening risk would occur. By 2050 and 2100, this fraction increases to 0.52% and 1.62%, respectively, according to the worst-case predicted future plastic discharge into the ocean. Our results reveal a spatial and multidecadal variability of MP-related risk at the global ocean surface. For example, we have identified the Mediterranean Sea and the Yellow Sea as hotspots of marine microplastic risks already now and even more pronounced in future decades.
The selection of an appropriate map projection has a fundamental impact on the visualization and analysis of geographic information. Distortion is inevitable and the decision requires simultaneous consideration of several different factors; a process which can be confusing for many cartographers and GIS users. The last few decades have seen numerous attempts to create automated map projection selection solutions based on traditional classification and selection guidelines, but there are no existing tools directly accessible to users of GIS software when making projection selection decisions. This paper outlines key elements of projection selection and distortion theory, critically reviews the previous solutions, and introduces a new tool developed for ESRI’s ArcGIS, employing an original selection method tailored to the specific purpose and geographical footprint characteristics of a GIS project. The tool incorporates novel quantitative projection distortion measures which are currently unavailable within existing GIS packages. Parameters are optimized for certain projections to further reduce distortions. A set of candidate projected coordinate systems are generated that can be applied to the GIS project; enabling a qualitative visual assessment to facilitate the final user selection. The proposed tool provides a straightforward application which improves understanding of the projection selection process and assists users in making more effective use of GIS.
Harmonic maps are a certain kind of an optimal map projection which has been developed for map projections of the sphere. Here we generalize it to the “ellipsoid of revolution”. The subject of an optimization of a map projection is not new for a cartographer. For instance, in Sect. 5-25, we compute the minimum distortion energy for mapping the “sphere-to-plane”.
Conventionally, conformal coordinates, also called conformal charts, representing the surface of the Earth or any other Planet as an ellipsoid-of-revolution, also called the Geodetic Reference Figure, are generated by a two-step procedure. First, conformal coordinates (isometric coordinates, isothermal coordinates) of type UMP (Universal Mercator Projection, compare with Example 15.1) or of type UPS (Universal Polar Stereographic Projection, compare with Example 15.2) are derived from geodetic coordinates such as surface normal ellipsoidal longitude/ellipsoidal latitude. UMP is classified as a conformal mapping on a circular cylinder, while UPS refers to a conformal mapping onto a polar tangential plane with respect to an ellipsoid-of-revolution, an azimuthal mapping.
In this chapter, we present a collection of most widely used map projections in the polar aspect in which meridians are shown as a set of equidistant parallel straight lines and parallel circles (parallels) by a system of parallel straight lines orthogonally crossing the images of the meridians. As a specialty, the poles are not displayed as points but straight lines as long as the equator. First, we derive the general mapping equations for both cases of (i) a tangent cylinder and (ii) a secant cylinder and describe the construction principle.
At the beginning of this chapter, let us briefly refer to Chap. 8, where the data of the best fitting “ellipsoid-of-revolution to Earth” are derived in form of a table. Here, we specialize on the mapping equations and the distortion measures for mapping an ellipsoid-of-revolution to a cylinder, equidistant on the equator. Section 14-1 concentrates on the structure of the mapping equations, while Sect. 14-2 gives special cylindric mappings of the ellipsoid-of-revolution, equidistant on the equator. At the end, we shortly review in Sect. 14-3 the general mapping equations of a rotationally symmetric figure different from an ellipsoid-of-revolution, namely the torus.
A special mapping, which was invented by Gauss (1822, 1844), is the double projection of the ellipsoid-of-revolution to the sphere and from the sphere to the plane. These are conformal mappings. A very efficient compiler version of the Gauss double projection was presented by Rosenmund (1903) (ROM mapping equations) and applied for mapping Switzerland and the Netherlands, for example. An alternative mapping, called “authalic”, is equal area, first ellipsoid-of-revolution to sphere, and second sphere to plane.
In the world of conformal mappings of the Earth or other celestial bodies, the Mercator projection plays a central role. The Mercator projection of the sphere or of the ellipsoid-of-revolution beside conformality is characterized by the equidistant mapping of the equator. In contrast, the transverse Mercator projection is conformal and maps the transverse meta-equator, the meridian of reference, equidistantly. Accordingly, the Mercator projection is very well suited for regions which extend East–West around the equator, while the transverse Mercator projection fits well to those regions which have a South–North extension. Obviously, several geographical regions are centered along lines which are neither equatorial, parallel circles, or meridians, but may be taken as central intersection of a plane and the reference figure of the Earth or other celestial bodies, the ellipsoid-of-revolution (spheroid).
Up to now, we treated various mappings of the ellipsoid and the sphere, for instance of type conformal, equidistant, or equal areal or perspective and geodetic.
Among cylindrical projections, mappings in the transverse aspect play the most important role. Although many worldwide adopted legal map projections use the ellipsoid-of-revolution as the reference figure for the Earth, the spherical variant forms the basis for the Universal Transverse Mercator (UTM) grid and projection. In the subsequent chapter, we first introduce the general concept of a cylindrical projection in the transverse aspect. Following this, three special map projections are presented: (i) the equidistant mapping (transverse Plate Carrée projection), (ii) the conformal mapping (transverse Mercator projection), and (iii) the equal area mapping (transverse Lambert projection). The transverse Mercator projection is especially appropriate for regions with a predominant North-South extent. As in previous chapters, the two possible cases of a tangent and a secant cylinder are treated simultaneously by introducing the meta-latitude B = ±B1 of a meta-parallel circle which is mapped equidistantly. For a first impression, have a look at Fig. 11.1.
In Chap. 21, we already transformed from a global three- dimensional geodetic network into a regional or local geodetic network. We aimed at the analysis of datum parameters, namely seven parameters of type translation, rotation and scale, as elements of the global conformal group C7(3).
Pseudo-cylindrical projections have, in the normal aspect, straight parallel lines for parallels. The meridians are most often equally spaced along parallels, as they are on a cylindrical projection, but on which the meridians are curved. Meridians may be mapped as straight lines or general curves.
Cylindrical projections in the oblique aspect are mainly used to display regions which have a predominant extent in the oblique direction, neither East-West nor North-South. In addition, they form the most general cylindrical projections because mapping equations for projections in the polar and the transverse aspect can easily be derived from it. This is done by setting the corresponding latitude of the meta-North Pole Φ
0 to a specific value: Φ
0 = 90∘ generates cylindrical projections in the polar aspect, Φ
0 = 0∘ result in cylindrical projections in the transverse aspect. As an introductory part, we present the equations for general cylindrical mappings together with the equations for the principal stretches, before derivations for specific cylindrical map projections of the sphere (oblique equidistant projection, oblique conformal projection and oblique equal area projection) are given. For a first impression, have a look at Fig. 12.1.
Mapping the ellipsoid-of-revolution to a tangential plane. Azimuthal projections in the normal aspect (polar aspect): equidistant, conformal, equiareal, and perspective mapping.
Mapping the sphere to a tangential plane: meta-azimuthal projections in the oblique aspect. Equidistant, conformal (oblique UPS), and equal area (oblique Lambert) mappings.
That net plant production plays a key role in ecologic, environmental, and planning considerations requires little supporting discussion. In addition to its implication for the upper limit to the Earth’s sustainable human population, knowledge of the Earth’s production and its spatial distribution permits us to estimate such characteristics of our planet as
1.
Geographic distribution of the Earth’s potential food resources
2.
Sizes and geographic distribution of the various reservoirs in the Earth’s carbon and oxygen cycles
3.
Limits toward which we might be able to increase regional productivity levels artificially, assuming that such grandiose projects were environmentally desirable
4.
Effects of the destruction of major vegetation formations, such as the Amazon rain forest, on the Earth’s atmospheric composition and climate
5.
Potential and actual productivity levels of individual countries, hence the maximum carrying capacities of their national and regional ecosystems
Any system that uses maps must take into account three properties: datums, coordinate systems, and projections. These properties specify our model of the Earth and the way in which we specify locations upon it. In the world of geodesy there are a number of different options for each of these properties with no one “best fit” choice for all applications. While it is often not necessary to use more than one datum or coordinate system, it is important to understand them in case there is a need for interoperability with another system or data source which uses a different datum or coordinate system. As such, we will provide a general introduction to these two properties.
The selection process for map projections is a mystery to many mapmakers and GIS users. Map projections ought to be selected based on the map’s geographic extent and the required distortion properties, with the goal of minimizing the distortion of the mapped area. Despite some available selection guidelines, the selection of map projections is not yet automated. Automated selection would help mapmakers and GIS users to better select a projection for their map. The overall goal of this dissertation is to take a step towards this automation and explore user preferences with an objective to provide additional criteria for selecting world map projections. An additional goal is to optimize automatic map projection selection for web maps. The results presented in this work are mathematical models (new map projections for world maps, polynomial equations for selecting standard parallels) and new selection criteria for world maps. They improve our knowledge about map projection selection for world maps and web maps. As a result of the research presented in this doctoral dissertation, we know more about the map projection preferences of map-readers and have improved techniques for adapting map projections for scalable web maps and GIS software. Altogether, four concrete research questions were addressed.
The first research question explores user preferences for world map projections. Many small-scale map projections exist and they have different shapes and distortion characteristics. World map projections are mainly chosen based on their distortion properties and the personal preferences of cartographers. Very little is known about the map projection preferences of map-readers; only two studies have addressed this question so far. This dissertation presents a user study among map-readers and trained cartographers that tests their preferences for world map projections. The paired comparison test of nine commonly used map projections reveals that the map-readers in our study prefer the Robinson and Plate Carrée projections, followed by the Winkel Tripel, Eckert IV, and Mollweide projections. The Mercator and Wagner VII projections come in sixth and seventh place, and the least preferred are two interrupted projections, the interrupted Mollweide and the interrupted Goode Homolosine. Separate binominal tests indicate that map-readers involved in the study seem to like projections with straight rather than curved parallels, and meridians with elliptical rather than sinusoidal shapes. The results indicate that map-readers prefer projections that represent poles as lines to projections that show poles as protruding edges, but there is no clear preference for pole lines in general. The trained cartographers involved in this study have similar preferences, but they prefer pole lines to represent the poles, and they select the Plate Carrée and Mercator projections less frequently than the other participants.
The second research question introduces the polynomial equations for the Natural Earth II projection and tests user preferences for its graticule characteristics. The Natural Earth II projection is a new compromise pseudocylindrical projection for world maps. It has a unique shape compared to most other pseudocylindrical projections. At high latitudes, the meridians bend steeply toward short pole lines resulting in a map with highly rounded corners that resembles an elongated globe. Its distortion properties are similar to most other established world map projections. The projection equation consists of simple polynomials. A user study evaluated whether map-readers prefer Natural Earth II to similar compromise projections. The 355 participating general map-readers rated the Natural Earth II projection lower than the Robinson and Natural Earth projections, but higher than the Wagner VI, Kavrayskiy VII, and Wagner II projections.
The third question examines how Wagner’s transformation method can be used for improving map projections for scalable web maps, and its integration into the adaptive composite map projections schema. The adaptive composite map projections schema, invented by Bernhard Jenny, changes the projection to the geographic area shown on a map. It is meant as a replacement for the commonly used web Mercator projection, which grossly distorts areas when representing the entire world. The original equal-area version of the adaptive composite map projections schema uses the Lambert azimuthal projection for regional maps, and three alternative projections for world maps. In this dissertation, it is explored how the adaptive composite map projections schema can include a variety of other equal-area projections when the transformation between the Lambert azimuthal and the world projections uses Wagner’s method. In order to select the most suitable pseudocylindrical projection, the distortion characteristics of a pseudocylindrical projection family were analyzed, and a user study among experts in the area of map projections was carried out. Based on the results of the distortion analysis and the user study, a new pseudocylindrical projection is recommended for extending the adaptive composite map projections schema. The new projection is equal-area throughout the transformation to the Lambert azimuthal projection, has better distortion characteristics than small-scale projections currently included in the original adaptive composite map projections schema, and aligns with map-readers’ preferences for world map projections.
The last research question explores how the selection of the standard parallels of conic projections can be automated. 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.
Four approaches to describe the global distortion of small scale maps are presented. These are the quantities of Peters, Capek, Canters and a quantity which is based on the first mean value theorem for integration. Some common projections are chosen to conduct an analysis. First the analysis of each approach is performed according to its original definition and second the quantities are slightly modified to adapt them to one another. The analysis delivered a ranking of the projections. In the next step the rankings - each based on one of the quantities - are compared mutually. The comparison revealed significant differences in the rankings.
The design of new map projections has up until to now required specialized mathematical knowledge and was therefore reserved to a small group of experts. To change this situation, Institute of Cartography, ETH Zurich, has developed a new digital method offering cartographers a simple way to design new world map projections. A new projection is developed iteratively within an interactive graphical environment. The representation of the continents and the shape of the graticule can be verified and improved using graphical and numerical tools in Flex Projector. This software application is open-source and available for free (www.flexprojector.com). The design tools in Flex Projector work in tandem with specialized visualizations and numerical indices that help optimize the graticule by illustrating the inevitable angular and areal distortions.
Der Entwurf neuer Kartennetze hat bis anhin Fachwissen in spezialisierter Mathematik vorausgesetzt und war deshalb einer kleinen Gruppe von Experten vorbehalten. Am Institut für Kartografie der ETH Zürich wurde eine neue digitale Methode entwickelt, die es Kartographen erlaubt, auf einfache Weise neue Netze für Weltkarten zu gestalten. Ein neuer Netzentwurf wird in einer graphischen Umgebung interaktiv in einem iterativen Prozess hergeleitet. Die Darstellung der Kontinente und des Kartennetzes kann laufend graphisch und numerisch überprüft und optimiert werden. Dazu wurde Flex Projector entwickelt, eine freie und quelloffene Software, die auf den Entwurf von neuen Netzen für Weltkarten ausgerichtet ist (www.flexprojector.com). Die spezialisierten Werkzeuge in Flex Projector werden durch Visualisierungen und numerische Indikatoren vervollständigt, welche die unvermeidlichen Winkel- und Flächenverzerrungen aufzeigen und helfen, das Kartennetz zu optimieren.
The Natural Earth projection is a new projection for representing the entire Earth on small-scale maps. It was designed in Flex Projector, a specialized software application that offers a graphical approach for the creation of new projections. The original Natural Earth projection defines the length and spacing of parallels in tabular form for every five degrees of increasing latitude. It is a pseudocylindrical projection, and is neither conformal nor equal-area. In the original definition, piece-wise cubic spline interpolation is used to project intermediate values that do not align with the five-degree grid. This paper introduces alternative polynomial equations that closely approximate the original projection. The polynomial equations are considerably simpler to compute and program, and require fewer parameters, which should facilitate the implementation of the Natural Earth projection in geospatial software. The polynomial expression also improves the smoothness of the rounded corners where the meridians meet the horizontal pole lines, a distinguishing trait of the Natural Earth projection that suggests to readers that the Earth is spherical in shape. Details on the least squares adjustment for obtaining the polynomial formulas are provided, including constraints for preserving the geometry of the graticule. This technique is applicable to similar projections that are defined by tabular parameters. For inverting the polynomial projection the Newton-Raphson root finding algorithm is suggested.
We characterized the spectro-polarimetric emission properties of random
lasers in the regime of strong scattering. The study involved the
preparation of microstructured samples, which were shown to support very
stable random laser modes in spectral location and intensity. We show
that random lasing modes from such samples are highly polarized in
statically random, but well-defined states that can be used as a unique
sample identifier. Our findings reveal a strong dependency of the
emission spectrum on the pump polarization and demonstrate how the
spectro-polarimetric emission can be efficiently manipulated.
By the 1990s, Geographic Information Systems (GIS) have been widely used in almost every profession. Map projection is however the mathematical foundation for the spatial information system in a GIS. An impressive number and a large variety of map projections are available in cartography. The agreement to the selection of a grid reference system from the available projection alternatives is also the main problem in designing world-scale geographic information systems. Because projections distort angles, areas and distances on maps. Distortion misleads users in the way they visualize, cognize or locate geographic features. An appropriate choice of the projection framework can maximize the communication of the map. Distortion, in terms of visual appearance, is less apparent on a larger scale map because the curvature of the Earth is less pronounced and it is unlikely that the map-reader notices it. But in spite of this distortion on small-scale maps is more perceptible, while less significant on a larger scale map. Different approaches and criterions have been presented to study distortion on maps in order to decide for a suitable projection. Even all those approaches are assumed too mathematical. This paper contributes to give considerations and a sequential index of some suitable map projections before interacting with GIS software's. Ten popular world projections are selected as candidate and classified under a new equivalency criterion. Projections are analyzed according to the standard deviations calculated using the ratios of the continental areas to the total map area in each latitude zone with 10 degrees interval.
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