Cartographic globes have been in production since for more than 2100 years, though the oldest surviving example is from 1490’s. Though no longer used as a navigational aid, globes remain in high demand as scientific and artistic artefacts. However, the cartographic quality of most modern globes is poor, in part due to a lack of detailed evaluation of the mathematical approach to their creation. This work provides the first detailed evaluation of map projection for manuscript globe production, (those in which paper maps are attached to a sphere) and quantitatively demonstrates the most suitable projection

Huck, Jonny

English

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The University of Manchester ResearchEvaluating Map Projections for Globe ProductionDOI:10.5281/zenodo.10897750Document VersionFinal published versionLink to publication record in Manchester Research ExplorerCitation for published version (APA):Huck, J. (2024). Evaluating Map Projections for Globe Production. Paper presented at 32nd Annual GIS ResearchUK Conference, Leeds, United Kingdom. https://doi.org/10.5281/zenodo.10897750Citing this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:18. May. 2024 Page | 1   Evaluating Map Projections for Globe Production  Huck, J. J.*1  1 MCGIS, Department of Geography, The University of Manchester  GISRUK 2024  Summary Cartographic globes have been in production since for more than 2100 years, though the oldest surviving example is from 1490’s. Though no longer used as a navigational aid, globes remain in high demand as scientific and artistic artefacts. However, the cartographic quality of most modern globes is poor, in part due to a lack of detailed evaluation of the mathematical approach to their creation. This work provides the first detailed evaluation of map projection for manuscript globe production, (those in which paper maps are attached to a sphere) and quantitatively demonstrates the most suitable projection.   KEYWORDS: Cartography, Distortion, Gore, Map Projection, Flex Projector   1. Introduction  The Globemakers: The Curious Story of an Ancient Craft (Bellerby, 2023) highlights the lack of information available in the public domain relating to the production of globes. An investigation into the academic literature reveals a similar lack of engagement with the science of globes, with few articles providing any insight on the topic, particularly with respect to the production of gores: the lens-shaped paper maps that are pasted onto the sphere of globes (Figure 1). Specifically, there appears to be no definitive understanding of which map projections are best suited gore production. Globes constructed using gores are typically called manuscript globes, as opposed to those created using more modern methods (e.g., thermoplastic moulding), which are not considered in this paper.     Figure 1: Nine lens-shaped gores, each representing a 40 section of the earth. These can be cut out and pasted onto a sphere to construct a globe. Reproduced from (Huck, 2016a).   * jonathan.huck@manchester.ac.uk  Page | 2   In the little information that is available (generally in online forums, rather than formal sources), several suggestions are given, the most common of which appear to be the Sinusoidal (Huck, 2016b), Rectangular Polyconic (Strebe, 2002), Transverse Mercator (Huber, 2011) and Cassini (Gede, 2009) projections; though none of these sources offer a rigorous scientific justification for their recommendation.  Part of the reason for this lack of information or accepted ‘best practice’ might relate to the fact that historical globes were not created using formal projections (Adami & Guerra, 2008; Gede, 2009), whereas modern globes appear to simply accept production errors (Bellerby, 2023; e.g., Figure 2). To address this substantial gap in the literature, this research provides the preliminary findings of the first detailed investigation into the relative suitability of map projections for manuscript globe production.   Figure 2: A typical modern manuscript globe. Several production errors are evident, including mis-matched lines of latitude (visible in the northern hemisphere) and gaps along the meridian to enable coastlines and boundaries to align properly (visible in the southern hemisphere). 2. Method  2.1. Map Projection Selection  There are few existing map projections that are suitable for globe production (Strebe, 2002), and none that have been designed for the purpose. The inclusion criteria for projections in this research are: • It should form lens-shaped sections when interrupted at the meridians (e.g., Figure 1) • The spacing of parallels should be even along the central meridian in order to fit the sphere. • Parallels should meet meridians at right angles close to the central meridian in order to ensure alignment between gores. • The spacing of meridians should be even along the equator close to the central meridian, in order to facilitate interruption at regular intervals.  As is frequently the case with map projections selection, it is not possible to attain all of these properties in a single projection (Strebe, 2002), and so the included candidate projections are those that satisfy these criteria as best as possible. The chosen projections are listed in Table 1 and illustrated in Figure 3. In all cases, the spherical form of the projection was used with the Earth radius set to 6371000 m.     Page | 3   Table 1 Candidate Projections Name Proj String Class Sinusoidal +proj=sinu +R=6371000 Equal Area Cassini +proj=cass +R=6371000 Equidistant Transverse Mercator +proj=tmerc +R=6371000 Conformal Rectangular Polyconic +proj=rpoly +R=6371000 Compromise American Polyconic +proj=poly +R=6371000 Compromise    Figure 3: Illustration of the seven candidate map projections, each with a grid of 6 geodesic circles † to illustrate the distortion across the gore. Each is shown as a 90 gore (i.e., for a 4-gore globe) to maximise the apparent differences. Land and Graticule data from Natural Earth.  2.2. Distortion Assessment  It is important to recognise that there are two fundamental challenges in the creation of a suitable projection for globe gores: mathematical (i.e., those relating to the properties of the projection itself) and mechanical (i.e., those relating to the properties of the medium – such as how the paper on which the gore is printed might stretch when applied to the sphere) (Strebe, 2002). For the purposes of this paper, I am concerned only with the mathematical challenges, though I will briefly respond to mechanical challenges at the end.  The distortion metrics used herein are drawn from the existing literature into small-scale map design, permitting the evaluation of areal, shape and distance distortion. The metric for area distortion (𝐸𝐴) is based on the method given by Canters et al. (2005), but with the values converted to scale factor prior to summing to facilitate calibration later on (Equation 1).  𝐸𝐴 =  1𝑚 ∑|𝑆𝑖 − 𝑆𝑖′||𝑆𝑖 + 𝑆𝑖′|𝑚𝑖=1 (1)  Where: 𝐸𝐴 is the finite area distortion metric, m is the number of randomly generated circles (in this case 10,000 circles of random radius between 10,000 and 100,000 m), and 𝑆𝑖 and 𝑆𝑖′are geodesic and planar area of those circles respectively. The finite distance distortion metric (𝐸𝑃) is calculated using the approach described by Gosling and Symeonakis (2020), which uses the same equation (Equation 1), changing 𝐸𝐴 for 𝐸𝑃 and where: m is the number pairs of random points (again 10,000), and 𝑆𝑖 and 𝑆𝑖′ are the geodesic and planar distances between those pairs of points. The finite shape distortion metric  † This is not a true Tissot Indicatrix but rather a visual approximation that is sufficient for this purpose.  Page | 4   (𝐸𝑆) is calculated as described by (Canters et al., 2005).   To solve the problem of unequal units between these three metrics and hence facilitate comparison and combination, all three metrics are ‘calibrated’ using Equation 2 (Canters et al., 2005).  𝐸𝐴𝑐 =1𝐸𝐴,𝑚𝑎𝑥 − 𝐸𝐴,𝑚𝑖𝑛 (𝐸𝐴 − 𝐸𝐴,𝑚𝑖𝑛) (2)  Where 𝐸𝐴𝑐 is the calibrated finite area distortion metric. For shape and distance, 𝐸𝐴 is directly replaced by 𝐸𝑆 and 𝐸𝑃 respectively. Canters et al. (2005) described difficulty in adequately determining values for 𝐸𝐴,𝑚𝑎𝑥 and 𝐸𝐴,𝑚𝑖𝑛. Unlike Canters et al. (2005), I determine 𝐸𝐴,𝑚𝑎𝑥 and 𝐸𝐴,𝑚𝑖𝑛 as the largest and smallest individual values that were calculated across all projections prior to the summing step in Equation 1, providing a robust and objective relative measure between the projections under consideration. Also unlike Canters et al. (2005), an ‘overall’ distortion value 𝐸𝑇 here is calculated as the mean (rather than sum) of these three values, which has the advantage of maintaining the 0-1 range and gives the option of weighting the components according to their relative importance for the map in production. Note that this is a stochastic metric, so there will be small variations in results between runs unless a seed is specified for the random number generator.  2.3. Gore ‘Fit’ Error Assessment  Unlike some other applications of the selection of a suitable map projection, distortion is not the only consideration for gore production. Perhaps more important is the precision with which the gores will fit around the spherical globe (i.e., whether they will cause gaps or overlap between gores at some latitudes). This can be evaluated by simply measuring the width of a gore at a certain latitude and calculating the difference between comparing this and the same width on the sphere using the Haversine equation. This permits an evaluation of the ‘fit error’ of the gores, as illustrated in Figure 3. Both distortion and gore fit will be assessed for 4, 6, 12, 24, 36 and 48 gores, which are numbers commonly used in globe production (e.g., Bellerby, 2023).  3. Results  3.1. Distortion Evaluation  The distortion values for each candidate projection for a 12-gore (30) globe are given in Table 2. In addition to the overall scores, it is also instructive to examine the distribution of values that went into the calculation of the scores. These are illustrated (in their un-calibrated form) for a 12-gore (30) globe in Figure 2.   Table 2 Candidate Projection Distortion Results, ordered by 𝐸𝑇. The best performing projection in each column is shown in bold.  𝐸𝑇 𝐾𝐴𝑐 𝐸𝑆𝑐 𝐸𝑃𝑐 American Polyconic  0.003587 0.00424 0.003371 0.003151 Cassini 0.003657 0.004334 0.00343 0.003207 Rectangular Polyconic 0.004473 0.00532 0.004933 0.003167 Transverse Mercator 0.004514 0.0075 0.002695 0.003346 Sinusoidal 0.010707 0.002034 0.026522 0.003566  From the perspective of distortion, it is clear that the American Polyconic and Cassini projections perform best for a 12-gore globe. However, there is some variation across other numbers of gores, as is illustrated in Figure 3. For 4- and 6-gore globes, American Polyconic is superior, but for 12- and 24- Page | 5   gore globes the performance of Cassini is near identical. For 48-gore globes (which is unusual and would only be used for the largest globes), Transverse Mercator also has a similar performance. Sinusoidal consistently performs worst due to the substantial shape distortion inherent in this projection.    Figure 2 Distortion of Distance, Shape and Area for a 12-gore (30) globe.   Figure 3 Overall distortion by number of gores.   3.2. Gore ‘Fit’ Evaluation  An assessment of the ‘fit’ of 30 gores is given in Figure 4. A negative value suggests that there will be a gap of that size and a positive value suggests an overlap of that size on a 300 mm globe. Though this has not previously been formally assessed in the literature, a similar analysis was undertaken by (Ciura, 2011) and the existence of the issue is highlighted by (Strebe, 2002), who recommended that  Page | 6   each gore was printed beyond the seam meridian (called ‘bleed’) to cover the gaps when using the Rectangular Polyconic projection. From the perspective of ‘fit’ to the globe, the Cassini projection performs best throughout, though the American Polyconic achieves a similar performance from 12 gores upwards. Transverse Mercator and the Sinusoidal projection have a far worse performance than the others, though even this becomes negligible (sub-millimetre) from 24 gores upwards.   Figure 4 Gore ‘fit’ errors, given by latitude for a. a 4-gore (90) and b. a 12-gore (30) globe of 400 mm diameter. 4. Conclusion Based on the above assessment, it is clear that there is not a universal ‘best’ choice of projection for the production of a globe. Nevertheless, the candidate choices for a general-purpose globe are fewer than expected, with the Cassini projection providing the overall best compromise between ‘fit’ and distortion for most purposes. American Polyconic has the best distortion performance, but a poor fit for globes with the fewest (largest) gores, and Transverse Mercator is also acceptable for globes with very large numbers of gores.   Figure 5 A comparison of the proposed Globe projection with the Sinusoidal projection (whole earth shown with 30 graticule).  Page | 7    This research has only accounted for ‘out of the box’ projections and did not investigate the creation of new projections or the adjustment of existing ones. For example, Figure 5 demonstrates a modified Sinusoidal projection that removes all of the gore ‘fit’ error (Figure 4) by altering the length of the parallels using Flex Projector (Jenny et al., 2008), thereby making it a better candidate. The same approach can also be applied to any of the other projections using trial-and-error to eliminate any gaps or overlaps caused by the mechanical properties of the medium, in order to ensure an optimal fit to the globe. All Python scripts and data used in this research are available at https://github.com/jonnyhuck/globe-gore-evaluation .  References   Adami, A., & Guerra, F. (2008). Coronelli’s virtual globe. e-Perimetron, 3(4), 243-250.  Bellerby, P. (2023). The Globemakers: The Curious Story of an Ancient Craft. Bloomsbury.  Canters, F., Deknopper, R., & De Genst, W. (2005). A new approach for designing orthophanic world maps. Proceedings of the 22nd International Cartographic Conference,  Ciura, M. (2011). What projections should I use to make my own Globe? https://gis.stackexchange.com/questions/15639/what-projections-should-i-use-to-make-my-own-globe Gede, M. (2009). The projection aspects of digitising globes. ICC 2009.  Gosling, P. C., & Symeonakis, E. (2020). Automated map projection selection for GIS. Cartography and Geographic Information Science, 47(3), 261-276.  Huber, W. (2011). What projections should I use to make my own Globe? https://gis.stackexchange.com/questions/15639/what-projections-should-i-use-to-make-my-own-globe Huck, J. J. (2016a). GlobeMaker. https://github.com/jonnyhuck/GlobeMaker Huck, J. J. (2016b). Globemaking for Beginners. https://jonnyhuckblog.wordpress.com/2016/06/29/globemaking-for-beginners/ Jenny, B., Patterson, T., & Hurni, L. (2008). Flex Projector–interactive software for designing world map projections. Cartographic Perspectives(59), 12-27.  Strebe, D. (2002). The design of globe gores. Cartographic Perspectives(42), 46-54.   Acknowledgements  The author gratefully acknowledges the open source software that enabled this research to take place, particularly the Proj library and the Flex Projector software.  Biographies  Jonny Huck is a Senior Lecturer in Geographical Information Science at the University of Manchester and Chair of the GISRUK National Steering Committee. He got a copy of “The Globemakers: The Curious Story of an Ancient Craft” from his parents for Christmas.  

Evaluating Map Projections for Globe Production

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Evaluating Map Projections for Globe Production

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