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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 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.
... 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 ...
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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). ...
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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. ...
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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: ...
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... 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) ...
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Thesis
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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.
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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.
Chapter
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.
Chapter
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.
Chapter
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).
Chapter
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.
Chapter
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.
Chapter
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).
Chapter
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.
Chapter
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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Every now and then “new best map projections” were presented by amateurs often causing a medial hype, which was of course soon after rapidly fading. Thereinafter technical literature follows and provides a scientific assessment and evaluation. The issue of this article is the “AuthaGraph World Map” of Hajime Narukawa, a Japanese developer and designer. We start with the Dymaxion World Map of Richard Buckminster Fuller. also a polyhedral projection and thus a precursor of Narukowa, followed by Peters’ Projection from the 1970s. The new Japanese world map is a polyhedral map projection. It is referred to as the most precise world map by developer and media. Its outstanding features are an almost equal-area property. its advantageous shape, the possibility of seamless tesselation, the complete representation of the continent of Antarctica and the potential of cropping. This paper discusses both advantages and disadvantages, e. g. shape distortions and deformed graticule. It demonstrates: The new development is no alternative to established equal-area and distortion-compensating planispheres.
Article
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.
Chapter
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.
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