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# Small-Scale Map Projection Design

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... At the infinitesimal scale, the distortion of scale is equal in all directions and hence local shapes are preserved in conformal projections. In other words, an infinitesimal circle on the globe projects as a circle with a different radius on the plane (Canters, 2002;Snyder, 1987). Although conformal projections provide a good representation of shapes for a small area around every point, the rapid increase in the particular scales away from the points or lines of zero distortion make these projections less suitable for representing large terrestrial features like continents and oceans (Maling, 1992). ...
... The second approach produces a regional shape indicator to analyze shape distortions. Canters (2002) uses the shape analysis method proposed by Boyce and Clark (1964) to measure the shape distortions of spherical hexadecagons after they have been projected on to the plane. He chooses randomly distributed points over the landmasses and creates a multitude of hexadecagons with randomly changing radii (circular radius ≤ 30°) at those points. ...
... Very few attempts have so far been made to analyze and compare shape distortions in world map projections at the finite scale due to the fact that shape is not easy to define (Canters, 2002;Robinson & The Committee on Map Projections, 2017). Previous methods have two key problems. ...
Article
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World maps can have quite different depictions of reality depending on the projection adopted, and this can influence our perception of the world. In this respect, shape is a significant property that needs to be considered, especially when representing large regions in general-purpose world maps. A map projection distorts most geometric properties (area, distance, direction/angle, shape, and specific curves) and usually preserves a single property or provides a compromise between different properties when transforming terrestrial features from globe to plane. The distortions are mainly classified based on area, distance and direction/angle and analyzed with Tissot’s theorem. However, this theorem offers a local (pointwise) solution, so the distortion assessment is valid at infinitesimal scale (i.e. for very small regions). For this reason, different approaches are required to analyze the distortions at finite scale (i.e. for larger regions). However, there are very few attempts at analyzing and comparing shape distortion of landmasses in world map projections owing to the fact that shape measurement is difficult and usually involves measuring different characteristics. Seeking to fill this gap, in this study, compactness and elongation distortion measures are introduced. In this regard, 16 world map projections are analyzed and compared with these distortion measures in a GIS environment, based on map datasets of continents and countries. An analysis of the effect of the levels of detail of the datasets is also presented.
... Each Cartesian coordinate, ܺ and ܻ, can be expressed by two spherical coordinates, ߮ and ߣ. The To simplify the terms in Equation 10, the following considerations given by Canters (2002) for deriving new graticules with polynomials were taken into account: ...
... For the ܺ coordinate, all coefficients of even powers of ߣ and odd powers of ߮ are removed, and for the ܻ coordinate all terms with odd powers of ߣ and all even powers of ߮ are removed. Equation 11 presents this removal for the polynomial of 3 rd degree (Canters, 2002). ...
... The unequal distribution of parallels of the Natural Earth projection has to be expressed as a non-linear function of latitude (Equation 12). A linear expression would result in equally spaced parallels (Canters, 2002). ...
Thesis
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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.
... To find the optimum standard parallels for each mapping scenario, a distortion measure is needed. For the purposes of this research, Canters and Decleir's (1989) weighted mean error in the overall scale distortion D ab after Canters (2002) was used (Eq. (2)). ...
... . P i equals 1 if the sample point is located inside the mapped area; otherwise P i equals 0. The coefficients q Canters and Decleir (1989) introduced this distortion measure for comparing small-scale projections (Canters, 2002). In other research, this index is used for comparing projections, for example, Jenny et al. (2008Jenny et al. ( , 2010 Canters and Decleir (1989) with maximum angular distortion instead of scale distortion for evaluating pixel changes while projecting the global raster data. ...
... In other research, this index is used for comparing projections, for example, Jenny et al. (2008Jenny et al. ( , 2010 Canters and Decleir (1989) with maximum angular distortion instead of scale distortion for evaluating pixel changes while projecting the global raster data. Using the factor P i , one can restrict the distortion measure to an area of interest (Canters, 2002). To compute the weighted mean error in the overall scale distortion, only sample points that are inside of the mapped area in each scenario were used. ...
... , the weighted mean error in areal distortion index !" , and the mean angular deformation index !" of the Natural Earth II projection to other compromise and equal--area projections commonly used for small--scale mapping (for details on how indices are defined, see Canters andDecleir 1989, p. 42-43 andCanters 2002, p. 48). ...
... In 1932, the German cartographer Karl Heinrich Wagner suggested a transformation technique for the development of new map projections, which he referred to as Umbeziffern, meaning renumbering (Wagner 1931(Wagner , 1932(Wagner , 1941(Wagner , 1949(Wagner , 1962(Wagner , 1982Canters 2002). Tobler (1964) translated the term to 're--labeling. ...
... The result is a new map projection (Step 4 on Figure 4.3). Equation (4.2) below shows the general formula of Karl Siemon (1936Siemon ( , 1937Siemon ( , 1938 formalized, expanded, and presented Wagner's ideas as a general theory of map projection transformation (Canters 2002). He also showed that Aitoff's transformation method is a special case of Wagner's area preserving transformation (Siemon 1937). ...
Thesis
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... The disadvantage of MaPKBS was that it did not always return the best projection because users could not always answer all of the questions about specifying projection characteristics unambiguously (De Genst and Canters, 1996). Additionally, MaPKBS did not include all of the selection criteria, and it was limited to equal-area and conformal projections (Canters, 2002). ...
... The system then proposed a projection with the minimum distortion out of 50 map projections included in the system (Smith and Snyder, 1989). Smith and Snyder (1989) did not discuss the algorithmic details of the system (Canters, 2002). ...
... Mekenkamp (1990) presented the Integrated Projection Design System (IPDS). He constrained the selection to only 11 map projections (Canters, 2002;De Genst and Canters, 1996). He based the selection on the purpose of the map and the shape of the mapped region. ...
Article
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The selection of map projections is difficult and confusing for many. This article introduces Projection Wizard, an online map projection selection tool available at projectionwizard.org that helps mapmakers select projections. The user selects the desired distortion property, and the area to be mapped on an interactive web map. Projection Wizard then proposes a projection, along with projection parameters (such as standard parallels). The tool also creates a preview map with the proposed projection, and provides the corresponding projection code in PROJ.4 format, if applicable. The automated selection process is based on John P. Snyder’s selection guideline with a few adjustments. This article discusses the automated selection process, and the map projections suggested. Projection Wizard solves the problem of map projection selection for many applications and helps cartographers and GIS users choose appropriate map projections.
... Canters states that distortions identifiable at the local level based on Tissot s indicatrix are three; angle, area and scale (Canters and Decleir, 1989;Canters, 2002). It is the most widely used method to analyze projections based on Tissot s indicatrix, a representative measurement tool of local distortions. ...
... If integral calculus on the distortion calculated by m-number of grid poin find out its average, we can figure out the global distortion for each el which brings the following formula (Canters and Decleir, 1989). When illustrating the distortion generated by such scale factor (SF), the standard line is the reference line of map weighted mean error in the overall scale distortion, D ab and the following indicates the formula (Canters and Decleir, 1989;Canters, 2002). ...
... Here, indicates the distance on the latitude between grid points, while indicates the distance on the longitude between grid points. Paying attention to the fact that the focus of distortion does not concentrate on the entire globe but on the land area, Canters additionally defines the distortion index, calculated only for the land area; D anc , D arc , D abc (Canters and Decleir, 1989;Canters, 2002). 6 global distortion indexes were calculated for the total of 54 projection methods appropriate to use to make the global map. ...
Article
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.
... This method of presentation is the "Plate Carré" or "Rectangular" projection that uses the simplest possible map transformation formulae x= λ, y= ϕ [16,29]. Neglecting to select a map projection is an elementary mistake, made not only in ECDIS system but also in some GIS applications [30,31]. ...
... The methodology for the selection of map projection for a particular application comprises two distinct processes [30,31]: ...
... According to the conclusions of other studies on the selection of map projection in GIS and computer-based applications, the number and variety of candidate projections complicates the selection process. Consequently, limiting the number of candidate projections provides simple and straightforward process for election of map projections [30,31]. ...
Article
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Electronic navigational charts (ENCs) are geospatial databases, compiled for the operational use of Electronic Chart Display and Information systems (ECDIS) according to strict technical specifications of the International Hydrographic Organization (IHO). ECDIS is a GIS system designed for marine navigation according to the relevant standards of the International Maritime Organization (IMO). The international standards for ENCs and ECDIS, issued by the IHO and IMO, cover many aspects of the portrayal of ENCs in ECDIS but do not specify or recommend map projections. Consequently, in some cases, the unjustified employment of map projections by the manufacturers has caused certain functional drawbacks and inadequacies. This article reviews, evaluates and supplements the results of earlier studies on the selection of map projections for the depiction of ENCs in ECDIS and proposes a reasonable set of suitable projections with pertinent selection/implementation rules. These proposals took into consideration that ECDIS users (navigators) are not GIS experts or professional cartographers and consequently, the proposed election/implementation rules have to be simple and straightforward.
... Over the last two thousand years, several map projections have been proposed. Map construction methods involve geometry projections, mathematical constructions [6][7][8][9], transformation and combinations [10][11][12][13][14][15], and approximation and optimization [16][17][18][19]. Modern computer software and libraries are also available for cartography and coordinate transformations, e.g., NASA's G.Projector and Open Source Geospatial Foundation's PROJ library [20,21]. ...
... As spheres and ellipsoids are not developable surfaces, distortions in map projections are inevitable [22,23]. The measurement and analyses of distortions [24,25] for different types of map projections are critical and still a challenging task for the design [16], categorization, evaluation, comparison, selection [26,27], and optimization [19] of map projections. ...
... 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: ...
Article
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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.
... Tobler (1964) generalized the method by using a huge number of randomly chosen spherical elements in the area of interest. Later publications (Canters, 2002;Gott III, Mugnolo, & Colley, 2007;Laskowski, 1997;Peters, 1975) tried to fine-tune the method by describing different functions of the spherical and planar quantities to correctly compute the finite distortion. A common point in these research papers is that their results consistently show a big difference between the infinitesimal and finite measures. ...
... In this paper, it is showed that the areal and angular distortions of the Kavrayskiy type used by Frančula (1971) contain all information about linear scales. Canters (2002) and Peters (1975) both state that infinitesimal measures should not be used for world maps, as the distribution of the distortions is captured only by using measurements on the finite scale. Here, a theoretical explanation will show that the finite distance method differs from the infinitesimal counterpart not by considering the 'pattern of distortion' but by capturing the effects of flexion and skewness. ...
... A scale-independent measure may be developed by changing the nominal scale. This change in the scale is selected by minimizing the corresponding measure (Canters, 2002;Gott III et al., 2007). In case of the areal distortion and the Airy-Kavrayskiy criterion, this scale correction is ...
Article
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The traditional way to calculate the global distortion of a given area in a map projection is to create what we call a local distortion criterion that is a function of the infinitesimal semi-axes of the Tissot's indicatrix. Some contemporary scholars criticize this method, saying that the map readers face distortion of the finite type. These researchers suggest taking plenty of simple random spherical elements (line sections, triangles) and average the distortion on them. Although the aforementioned researchers all state that their approach is something fundamentally different from the traditional method, the major disadvantage is that this method is irreproducible. Therefore, it has to be investigated whether the difference is really significant between these methods and if it is, what its nature is. At first, different distortion values are evaluated on a huge number of various projections showing the whole Earth. Correlation analysis shows that there exists a strong linear dependence between the corresponding infinitesimal and finite measures. A considerable difference can be observed if the examined area is not the whole globe rather a part of it. After optimizing a projection for different distortion measures, the isolines of equal distortion follow the boundary lines significantly closer using the traditional approach.
... Distortion in map projections is inevitable [1,2], and it remains a challenging subject in cartography [3]. Distortions in map projections should be measured [4,5] to evaluate [6], compare, select [7,8], improve, and optimize [9] existing map projections, as well as to devise new map projections [3,[9][10][11]. ...
... Distortion in map projections is inevitable [1,2], and it remains a challenging subject in cartography [3]. Distortions in map projections should be measured [4,5] to evaluate [6], compare, select [7,8], improve, and optimize [9] existing map projections, as well as to devise new map projections [3,[9][10][11]. ...
Article
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Measuring, analyzing, reducing, and optimizing distortions in map projections is important in cartography. In this study, we introduced a novel image-based angular distortion metric based on the previous spherical great circle arcs-based metric. Images with predefined patterns were used to generate distorted images using mapping software. The generated distorted images with known patterns were then exploited to calculate the proposed angular distortion metric. The mapping software performed the underlying transformation of map projections. Therefore, there was no direct explicit dependence on the forward equations of the map projections in our proposed metric. However, there were fairly large computation errors in the ordinary image-based approach without special correction. To reduce the error, we introduced surface-fitting-based noise reduction in our approach. We established and solved systems of linear equations based on bivariate polynomial functions in the process of noise reduction. Sufficient experiments were made to validate the proposed image-based metric and the accompanying noise reduction approach. In the experiment, the NASA G.Projector was employed as the mapping software for evaluating more than 200 map projections. Experimental results demonstrated that the proposed image-based approach and surface fitting-based noise reduction are feasible and practical for the evaluation of the angular distortion of map projections.
... The pseudocylindrical graticule has straight parallels that are unequally spaced. This means that y coordinates solely depend on the latitude ϕ and are independent of the longitude λ (Snyder 1985, p. 37;Canters 2002, p. 141, Werenskiold 1945. In Equation (3), f y ϕ ð Þ is a function depending on the latitude ϕ. ...
... To avoid representing poles as points (for details see Snyder 1985, p. 124), latitudes in Equation (6) are renumbered to parametric latitudes θ. We use the approach applied by Wagner for his Umbeziffern transformation (Wagner 1931, 1932, 1941, 1949, 1962, 1982, Canters 2002) and by Putniņš for his P4ʹ projection (Putniņš 1934, Snyder 1993: sin θ ¼ m Á sin ϕ, where m is a factor between 0 and 1 (see Šavrič and Jenny (2014) for a recent detailed description of Wagner's transformation method). The spacing between parallels is approximated with a new function dependent on the parametric latitude θ: ...
Article
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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.
... The range of methodological concepts to construct the different types of cartograms is widespread, similar to the range of map projections that have been developed over time. As argued before, in principle cartograms are a special form of a map transformation (Canters 2002), and some approaches even aim to combine the principles of value-area and other geographical map transformations (Carroll & Moore 2008, Kadman & Shlomi 1978, Panse et al. 2006. ...
Chapter
This chapter introduces the research context of the thesis within the fields of cartography and geography with a special consideration of the role of globalisation and their interrelation. Globalisation has not only changed the way the world works, but also the way we, as those affecting and being affected by globalisation processes, see and perceive the planet. Graphic displays have a long history in translating the complexity of our environment into understandable visual representation, with maps being the most fundamental image that we have in our minds when we reflect on the spaces that we are living in. Maps and visualisations shape our view of the world, and how they do so in the context of a globalised world will be outlined and discussed in this chapter.
... For more information about the projections used in the tests refer to Canters (2002), Snyder (1993Snyder ( , 1987, Bugayevskiy and Snyder (1995), Maling (1992), and Pearson (1990), for formulas refer to Appendix 2. ...
Article
Map projections are given by forward transformation equations. Inverse transformation is derived from forward transformation analytically or numerically. In this paper, a numerical approach for inverse transformation of map projections is proposed, which is based on numerical differentiation and Newton–Raphson root finding method. This approach can facilitate the program developments for map projections when inverse transformation is needed. Numerical differentiation is tested with three map projections. It is seen that seven-digit precision or more can be reached. Boundary conditions and initial guess problem in inverse transformation are discussed. In terms of initial guess, map projections are divided into three categories, and appropriate initial guess values for cylindrical, pseudocylindrical, azimuthal, and conical projections in normal aspect are suggested. Newton–Raphson method with numerical differentiation is tested with 20 different map projections by using test data sets. The results show that the proposed approach is applicable if appropriate initial guess is available.
... The designer of new map projections is not limited to the techniques discussed in this chapter. There are alternative methods for creating a new projection from scratch, deriving it from existing ones, or adjusting projection parameters to create a new one (Canters 2002;Snyder 1993). Some of these techniques are used in the adaptive composite projections for Web maps, a new field of map projection research (Jenny 2012). ...
Chapter
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Cartographers have developed various techniques for deriving new projections from existing projections. The goal of these techniques is to substitute a disadvantageous trait of one of the source projections with the second source projection. This chapter discusses creating new projections by the juxtaposition and blending of two existing projections. It also presents a new approach for selectively combining projection characteristics. The emphasis in this chapter is on projections for world maps , as the described techniques are most useful for this scale.
... Snyder (1988) gives a transformation that can be applied to Lambert azimuthal equal-area and repeatedly thereafter in order to coax the angular isocols toward desired paths. Canters (2002) gives polynomial transformations for the same purpose that can be applied to any equal-area map and optimized via, for example, simplex minimization against specified constraints. Neither technique appears obviously adaptable to generating homotopies. ...
Article
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Equivalence (the equal-area property of a map projection) is important to some categories of maps. However, unlike for conformal projections, completely general techniques have not been developed 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 develops a tractable technique for generating a continuum of equal-area projections between two chosen equal-area projections. The technique gives map projection designers unlimited choice in tailoring the projection to the need. The technique is particularly suited to maps that dynamically adapt optimally to changing scale and region of interest, such as required for online maps.
... where a and b are the maximal and minimal local linear scales (Canters, 2002), given by the formulae ...
Article
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Both the media and the geosciences often use small-scale world maps for demonstrating global phenomena. The most important demands on the projection of these maps are: (1) the map distortions have to be reduced as much as possible; (2) the outline shape of the mapped Earth must remind the viewer of the Globe. If the map theme to be illustrated requires neither equivalency (nor, which rarely happens, conformality) nor prescriptions for the map graticule, an aphylactic non-conical projection with simultaneously minimized angular and area distortions is advisable. In this paper, a graticule transformation by a parameterizable function helps to convert minimum distortion pointed-polar pseudocylindrical projections for world maps into general non-conical projections with further minimized distortions. The maximum curvature of the outline shape will be moderated at the same time in order to obtain a definitely pointed-polar character.
... Because the previous renumbering shrank the longitude and latitude values on the sphere, the area of the resulting map is too small, and an enlarging scaling factor is applied to retain the correct area. Wagner finally adjusts the height-to-width ratio of the graticule by multiplying horizontal x-coordinates by a chosen factor, and dividing y-coordinates by the same factor (Canters 2002). In adaptive composite projections, the Wagner transformation is used to transform the Lambert azimuthal projection used for continental-scale maps toward various equal-area world map projections (Šavrič & Jenny, 2014). ...
Article
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The 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.
... Comments related to the Interrupted Goode Homolosine projection varied in the survey from supportive (9) to noting it was unfamiliar (9) to suggesting it should be changed (7) to expressing dislike because it was unfamiliar or splits (5). Responses to the projection were concurrent with differing views on using non-continuous map projections, and the tension between reducing distortion and introducing discontinuities (Canters, 2002). Despite the projection's focus on ocean regions, participants were able to identify and compare countries and regions split by the map. ...
Article
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Small-scale thematic maps help to visualize world-wide data, yet small nations can be difficult to discern or are omitted completely. This occurs for small island developing states (SIDS), a group of more than fifty states recognized by the United Nations for their social, economic and environmental vulnerabilities. Through this study we proposed and evaluated alternative maps to increase the perceptibility of SIDS using indicator data of the Sustainable Development Goals (SDGs). These goals link social, economic and environmental objectives to achieve globally by 2030. Five cartographic solutions were refined to one based on input from two focus groups of geoinformation scientists and cartographers as well as an interview with a SIDS resident. The selected map was evaluated by a larger audience in an online survey. Most survey participants had some experience with SIDS, worked in international organizations and/or had graduate-level degrees in a geographic-related science. While recommendations for improvement were provided, nearly seventy percent of the participants agreed the presented design was appropriate to represent SIDS in choropleth world maps.
... To allow for consistent global calculations of area, we first projected all spatial data to Eckert IV. For global-level maps, Eckert IV is the equal-area projection system with the lowest weighted mean error of scale distortion (Canters, 2002;Šavrič et al., 2016;Jenny et al., 2017). We clipped all initiative layers to the Natural Earth 10 m Ocean polygon prior to analysis (Ocean Version 4.1.0: ...
Article
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To safeguard biodiversity effectively, marine protected areas (MPAs) should be sited using the best available science. There are numerous ongoing United Nations and non-governmental initiatives to map globally important marine areas. The criteria used by these initiatives vary, resulting in contradictions in the areas identified as important. Our analysis is the first to overlay these initiatives, quantify consensus, and conduct gap analyses at the global scale. We found that 55% of the ocean has been identified as important by one or more initiatives, and that individual areas have been identified by as many as seven overlapping initiatives. Using our overlay map and data on current MPA coverage, we highlight gaps in protection of important areas of the ocean. We considered any area identified by two to four initiatives to be of moderate consensus. Over 14% of the ocean fell under this category and most of this area (88%) is not yet protected. The largest concentrations of medium-consensus areas without protection were found in the Caribbean Sea, Madagascar and the southern tip of Africa, the Mediterranean Sea, and the Coral Triangle. Areas of high consensus (identified by five to seven initiatives) were almost always within MPAs, but their no-take status was often unreported. We found that nearly every marine province and nearly every exclusive economic zone contained area that has been identified as important but is not yet protected. Much of the identified area lies within contiguous stretches of >100,000 km2; it is unrealistic to expect that all this area be protected. Nonetheless, our results on areas of consensus provide initial insight into opportunities for further ocean protection.
... Global positioning system (GPS) units used for site investigation usually collects the borehole location in geographical coordinate (latitude/longitude) system with respect to an assumed datum and spheroid. However, for a smaller area of interest, representation in planar coordinate systems is convenient and appropriate [21]. The code is developed to consider the datum and projections corresponding to the geographic location of the study area, and project into planar coordinates. ...
Chapter
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Application of geostatistical techniques for in situ site characterization has received much attention in the last couple of decades. Kriging is one of the popular and accurate geostatistical techniques for the interpolation of spatially varied random variables. This study considers the development of a generalized ordinary kriging algorithm for use with site characterization. Ordinary kriging with a constant mean of stationary variable was applied to generate contour map and the error variance map to infer on the spatial variation of the parameter under consideration. In this study, the clay content parameter from a refinery project area in Orissa is interpolated using ordinary kriging technique. A generalized MATLAB code is developed to select the best fit semi-variogram for the sample data, to apply ordinary kriging technique, and to generate the surface profile. The spatial distribution of clay content values across the region is studied using prediction surface, and accuracy is checked using error variance profiles. Results of the analysis are also compared with simulation using ArcGIS based geostatistical analyst® and cross-validated using statistical parameters. Our results conclude that the proposed algorithm can be extended to predict other in situ soil properties in the field of geotechnical engineering.
... Pictures can be found at https://at-a-lanta.nl/weia/cupola.html. (Böhm, 2006;Canters, 2002;Eckert-Greifendorff, 1935;Frančula, 1971;Gall, 1885;Arno Peters, 1967;Aribert Peters 1978) ...
Article
In this article a new equal-area projection for small scale world maps is presented, with north-south asymmetry. The equations used form a system including well-known equivalent projections designed by Lambert, Hammer and Wagner. A new criterion is used to compare the quality of projections, concentrating on angular distortion on continents on the far side of the map. For Hammer's and Wagner's projection a version is presented which is having less distorted continents than the original versions. The final projection is named the Cupola Projection. It is having the Pacific Ocean at the far sides of the map and therefore less land close to the border of the map than would be the case with other central meridians. Due to its asymmetry landmasses near the borders of the map are less distorted than is the case with other frequently used equal-area world maps.
... Tripel, Wagner VI and Natural Earth projections. Table 2 compares the weighted mean error in the overall scale distortion index D ab , the weighted mean error in areal distortion index D ar , and the mean angular deformation index D an of the Natural Earth II projection to other compromise and equal-area projections commonly used for small-scale mapping (for details on how indices are defined, see Canters &Decleir, 1989, pp. 42-43 andCanters, 2002, p. 48). ...
Article
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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.
... While the first and last stages are certainly common to other similar projects, the mid stage is highly specific to ours and takes advantage of the metric characteristics of the map, which are somewhat encompassed in the underlying triangulation network. The transformation of a map without losing its original metric properties is of particular importance, and it is for this reason that special attention must be paid to the preservation of geometric properties when modifying or transforming the map [12]. In [10], the authors computed the global transformation of the primary triangulation of the 1929 map to UTM-ETRS89 and it was suggested that the rigorous geometric transformation of the 1929 map should undergo local transformations using ground truth data points. ...
Article
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The use of geographic data from early maps is a common approach to understanding urban geography as well as to study the evolution of cities over time. The specific goal of this paper is to provide a means for the integration of the first 1:500 urban map of the city of València (Spain) on a tile-based geospatial system. We developed a workflow consisting of three stages: the digitization of the original 421 map sheets, the transformation to the European Terrestrial Reference System of 1989 (ETRS89), and the conversion to a tile-based file format, where the second stage is clearly the most mathematically involved. The second stage actually consists of two steps, one transformation from the pixel reference system to the 1929 local reference system followed by a second transformation from the 1929 local to the ETRS89 system. The last stage comprises a map reprojection to adapt to tile-based geospatial standards. The paper describes a pilot study of one map sheet and results showed that the affine and bilinear transformations performed well in both transformations with average residuals under 6 and 3 cm respectively. The online viewer developed in this study shows that the derived tile-based map conforms to common standards and lines up well with other raster and vector datasets.
Chapter
This chapter reports on the merging of geospatial data transformation, high-performance computing (HPC), and cyberinfrastructure (CI) domains for map projection transformation through performance profiling and tuning of pRasterBlaster, a parallel map projection transformation program. pRasterBlaster is built on the desktop version of mapIMG. Profiling was employed in an effort to identify and resolve computational bottlenecks that could prevent the program from scaling to thousands of processors for map projection on large raster datasets. Performance evaluation of a parallel program is critical to achieving projection transformation as factors such as the number of processors, overhead of communications, and input/output (I/O) all contribute to efficiency in an HPC environment. Flaws in the workload distribution algorithm, in this reported work, could hardly be observed when the number of processors was small. Without being exposed to large-scale supercomputers through software integration efforts, such flaws might remain unidentified. Overall, the two computational bottlenecks highlighted in this chapter, workload distribution and data-dependent load balancing, showed that in order to produce scalable code, profiling is an important process and scaling tests are necessary to identify bottlenecks that are otherwise difficult to discover.
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There are undeniable practical consequences to consider when choosing an appropriate map projection for a specific region. The surface of a globe covered by global, continental, and regional maps are so singular that each type distinctively affects the amount of distortion incurred during a projection transformation because of the an assortment of effects caused by distance, direction, scale , and area. A Decision Support System (DSS) for Map Projections of Small Scale Data was previously developed to help select an appropriate projection. This paper reports on a tutorial to accompany that DSS. The DSS poses questions interactively, allowing the user to decide on the parameters, which in turn determines the logic path to a solution. The objective of including a tutorial to accompany the DSS is achieved by visually representing the path of logic that is taken to a recommended map projection derived from the parameters the user selects. The tutorial informs the DSS user about the pedigree of the projection and provides a basic explanation of the specific projection design. This information is provided by informational pop-ups and other aids.
Thesis
Nel dettaglio, il lavoro si compone di due capitoli: • il primo introduce i concetti chiave della Business Intelligence e come l’integrazione con i Geographical Information System abbia dato vita alla Location Intelligence, evidenziandone i vantaggi e i principali ambiti di applicazione. • il secondo capitolo illustrata il contesto in cui verrà sviluppato un sistema di Location Intelligence, focalizzato sul Piano Coordinato di Controllo del Territorio (con particolare riferimento al settore degli appalti pubblici), e le caratteristiche del progetto preliminare, evidenziando il possibile supporto strategico. In conclusione sono esposte alcune considerazioni sugli obiettivi conseguibili e vengono proposti altri ambiti di utilizzo per l’attuazione di un’efficace Governance del Territorio.
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Polyconic map projections are rarely used for world maps because traditional polyconic mappings do not have a favourable distortion pattern. However, this paper demonstrates that the generalized versions of these projections can be adjusted well for global maps. The formulae are optimized numerically to ensure that the projections will have as low distortion as possible. Furthermore, optimal polyconic mappings with further restrictions (e. g. rectangular graticule, equal-area) are also developed.
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In map projection theory, it is usual to utilize numerical quadrature rules to estimate the overall map distortion. However, it is not known which method is the most efficient to approximate this integral. In this paper, overall map distortion is calculated analytically by a computer algebra system. Various integration methods are compared to the exact results. Some calculations are also performed on irregular spherical polygons. Considering the experiments, the author suggests utilizing the first-order Gaussian quadrature as it always gave reasonable results, although it is not the best for all cases.
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The chorographic world maps cannot go missing from the atlases for the general public and for schools. Among other requirements, their map distortions are usually expected to be minimal. In the aphylactic projections both the angular and area distortions can be reduced by the principle of “balance of errors”. Equations for the mapping functions help us to search pseudocylindrical projections showing minimum distortions according to the Kavrayskiy criterion, while the outline shape of the mapped Earth is monitored. Some of the best solutions are demonstrated in this paper.
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Map projections are an area of cartography with a firm mathematical foundation for their creation and display providing a basis for a knowledge representation. Using only variations on a single equation set, an infinite number of projections can be created, but less than 100 are in active use. Because each projection preserves specific characteristics, such as area, angles, global look, or a compromise of properties, classifications of map projections have been developed to aid in knowledge representation. These classifications are used for decision-making. They help select the correct projection for the map use. They assist users with determining the correct orientation, standard parallels and meridians. The classifications also inform the user how to adjust the selection based on size, extent, and latitude. Semantics can be used to automate map projections knowledge into a knowledge base that can be accessed by humans and machines. This work details a semantic representation of map projections knowledge and provides a simple example of a use case that exploits the knowledge base.
Article
A projection aspect is usually defined in references as the relation to the so-called auxiliary surface. However, such surfaces do not usually exist in map projection theory, which raises the issue of defining projection aspects without reference to auxiliary surfaces. This paper explains how projection aspects can be defined in two ways which are not mutually exclusive. According to the first definition, the aspect is the position of a projection axis in relation to the axis of geographic parameterization of a sphere. The projection axis is the axis of the pseudogeographic parameterization of a sphere, based on which the basic equations of map projection are defined. The basic equations of map projection are selected according to agreement and/or custom. According to this definition, aspects can be normal, transverse or oblique. According to the second definition, an aspect is the representation of the area in the central part of a map, and can be polar, equatorial or oblique. Therefore, it is possible for a map projection to have a normal and polar aspect, but it can also have a normal and equatorial aspect. The second definition is not recommended for use, due to its ambiguity.
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In map projections theory, various criteria have been proposed to evaluate the mean distortion of a map projection over a given area. Reports of studies are not comparable because researchers use different methods for estimating the deviation from the undistorted state. In this paper, statistical methods are extended to be used for averaging map projection distortions over an area. It turns out that the measure known as the Airy–Kavrayskiy criterion stands out as a simple statistical quantity making it a good candidate for standardization. The theoretical arguments are strengthened by a practical map projection optimization exercise.
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The apparent “distance” between two configurations of a system and the “length” of trajectories through its configuration space can be significantly distorted by plots that use “natural” or intuitively selected coordinates. This effect is similar to the way that a latitude–longitude plot of the Earth distorts the size and shape of the continents. In this paper, we explore how ideas from cartography can be used to identify system parameterizations that better reflect the effort costs of changing configuration. We then apply these new parameters to provide geometric insight about two aspects of moving in dissipative environments such as low Reynolds number fluids: The shape of the optimal gait cycle for a three-link swimmer and the fundamentally superior efficiency of a serpenoid swimmer as compared to the classic three-link system.
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Selecting the most suitable projection can be challenging, but it is as essential a part of cartographic design as color and symbol selection and should be given the same degree of consideration. A poorly chosen projection can result in misinterpreted information and impact the effectiveness of a map. This chapter provides guidance in selecting projections for world and hemisphere maps .
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Sometimes map projection designers need to create equal-area projections to best fill the projections’ purposes. However, unlike for conformal projections, few transformations have been described that can be applied to equal-area projections to develop new equal-area projections. Here, I survey area-preserving transformations, giving examples of their applications and proposing an efficient way of deploying an equal-area system for raster-based Web mapping. Together, these transformations provide a toolbox for the map projection designer working in the area-preserving domain.
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Although the theory of the minimum-distortion projections is well known, there were only a few attempts to develop such maps. This is mostly due to the fact that a solution of this problem usually connects to differential equations, which are difficult to solve. In this paper, the author shows how to approximate the best projection even for irregular areas using numeric methods by the example of equal-area and pseudo-conic projections for smallscale maps. Distortion values of optimal solutions are displayed in a table for several regions. Furthermore, the paper suggests various applications of their usage.
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Mapping the sphere to a tangential plane: meta-azimuthal projections in the oblique aspect. Equidistant, conformal (oblique UPS), and equal area (oblique Lambert) mappings.
Article
In the study of map projections, it is relatively simple to obtain meaningful estimators of distortion for a small area. The definition and especially the evaluation of global distortion measures (i.e., estimators representing the distortion worldwide or in a continent-like area) are undoubtedly more troublesome. Therefore, it is relatively common to find that recommendations for the parameters to use in a particular map projection, be it devised for a continent or a country, are based on simple rules (like the one-sixth rule of thumb for conic projections), with no possibility of further improvement in terms of resulting distortions and sometimes even with no knowledge at all of the sizes of these distortions. Although the choice of map defining parameters is normally made for reasons other than distortion minimization, such as ease of use (e.g., integer or half-integer numbers may be preferable), preservation of conventional or traditional definitions, and uniformity of parameters between neighboring regions, it is always worthwhile to know the optimal set of parameters in terms of minimal distortion. Then, the cartographer may mindfully deviate from this optimal set, documenting the differences in defining parameters and in the resulting distortions. The present research provides a means to do this by extending a related work presented in a previous contribution, where the evaluation and optimization of distortions were studied for a single map projection and only two areas of interest. To this end, a new tool has been developed and presented in this paper. This tool allows users to evaluate several measures of distortion for the most common conformal and equal-area projections within user-defined geographic boundaries of interest. Also embedded in the tool and transparent to users are global optimization techniques operating on Fibonacci grids, which permit the optimization of parameters for the particular map projection and area of interest under two possible criteria: minimization of typical distortion or minimization of extreme distortions. This tool and the associated techniques are applied to several official projections to analyze their original performance and to propose new parameters that significantly improve the resulting distortions while leaving room for users to easily evaluate and optimize the tool for the lowest distortions of these projections within their regions of interest.
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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.
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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).
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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.
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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.
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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 \(\mathbb{S}_{r}^{2}\) or of the ellipsoid-of-revolution \(\mathbb{E}_{A_{1},A_{2}}^{2}\) 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).
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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.
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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”.
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