O. Jaibi
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1
This research presents a comprehensive re-evaluation and improvement of map projection distortion analysis, building upon the foundational framework of Goldberg and Gott. The fundamental challenge in cartography lies in representing the Earth’s spherical surface on a flat plane, called map projection, a process that inevitably introduces distortions in area, shape, or distance. This study aims to refine the quantification of these distortions by employing an improved distance metric that accurately measures projected geodesics, utilising high-precision numerical methods with over 10 million sample points to minimise error, and introducing an unbiased total distortion score that eliminates the arbitrary normalisation of previous methods. The results identify the Winkel Tripel projection as the best-performing overall map based on the improved, unbiased total distortion score.
Furthermore, this work critically investigates the viability of using the fractal dimension of coastlines as a novel distortion metric, leading to a shift in analysis towards evaluating distortions specifically along coastal geometries. However, theoretical computations using fractal dimension show that it is unsuitable as a global distortion metric due to its invariance under the bi-Lipschitz transformations that define many map projections. By establishing a normalisation approach based on coastline arc length, this study provides a robust and resolution-independent framework for tailored cartographic evaluation. The findings underscore the sensitivity of distortion scores to globe orientation and establish a foundational methodology for future property-specific distortion analyses, ensuring a more reliable and application-focused approach to selecting map projections than previously done. For the specific application of representing coastlines with minimal distortion, the Azimuthal Equidistant projection is found to be the most fitting.
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Furthermore, this work critically investigates the viability of using the fractal dimension of coastlines as a novel distortion metric, leading to a shift in analysis towards evaluating distortions specifically along coastal geometries. However, theoretical computations using fractal dimension show that it is unsuitable as a global distortion metric due to its invariance under the bi-Lipschitz transformations that define many map projections. By establishing a normalisation approach based on coastline arc length, this study provides a robust and resolution-independent framework for tailored cartographic evaluation. The findings underscore the sensitivity of distortion scores to globe orientation and establish a foundational methodology for future property-specific distortion analyses, ensuring a more reliable and application-focused approach to selecting map projections than previously done. For the specific application of representing coastlines with minimal distortion, the Azimuthal Equidistant projection is found to be the most fitting.
...
This research presents a comprehensive re-evaluation and improvement of map projection distortion analysis, building upon the foundational framework of Goldberg and Gott. The fundamental challenge in cartography lies in representing the Earth’s spherical surface on a flat plane, called map projection, a process that inevitably introduces distortions in area, shape, or distance. This study aims to refine the quantification of these distortions by employing an improved distance metric that accurately measures projected geodesics, utilising high-precision numerical methods with over 10 million sample points to minimise error, and introducing an unbiased total distortion score that eliminates the arbitrary normalisation of previous methods. The results identify the Winkel Tripel projection as the best-performing overall map based on the improved, unbiased total distortion score.
Furthermore, this work critically investigates the viability of using the fractal dimension of coastlines as a novel distortion metric, leading to a shift in analysis towards evaluating distortions specifically along coastal geometries. However, theoretical computations using fractal dimension show that it is unsuitable as a global distortion metric due to its invariance under the bi-Lipschitz transformations that define many map projections. By establishing a normalisation approach based on coastline arc length, this study provides a robust and resolution-independent framework for tailored cartographic evaluation. The findings underscore the sensitivity of distortion scores to globe orientation and establish a foundational methodology for future property-specific distortion analyses, ensuring a more reliable and application-focused approach to selecting map projections than previously done. For the specific application of representing coastlines with minimal distortion, the Azimuthal Equidistant projection is found to be the most fitting.
Furthermore, this work critically investigates the viability of using the fractal dimension of coastlines as a novel distortion metric, leading to a shift in analysis towards evaluating distortions specifically along coastal geometries. However, theoretical computations using fractal dimension show that it is unsuitable as a global distortion metric due to its invariance under the bi-Lipschitz transformations that define many map projections. By establishing a normalisation approach based on coastline arc length, this study provides a robust and resolution-independent framework for tailored cartographic evaluation. The findings underscore the sensitivity of distortion scores to globe orientation and establish a foundational methodology for future property-specific distortion analyses, ensuring a more reliable and application-focused approach to selecting map projections than previously done. For the specific application of representing coastlines with minimal distortion, the Azimuthal Equidistant projection is found to be the most fitting.
Map projections
Exploring and analysing different types of distortions caused by map projections
Maps, which have been in used for centuries, are constructed by map projections. A map projection is a transformation of the latitudes and longitudes positions on the globe to the x and y coordinates on a flat map. There are probably over a hundred of different map projections, some more practical than others. A perfect map is impossible to create, so it is essential to determine the next best alternative. The different maps can be classified in five different categories: Conformal, Equal-area, compromise, equidistant and true-direction. To evaluate this, various distortions are considered. The six distortions defined in this thesis are: area (A), isotropy (I), flexion (F), skewness (S), distance (D), and boundary cut (B). These distortions together form a comprehensive set of all distortions that occur due to map projections. Since these distortions are quite difficult to compute directly, they are numerically approximated. To achieve this, the data points of the coordinates need to be sampled, which can be done through various methods. This thesis examines systematic generation and random sampling of points. Additionally, it investigates how many points are needed to achieve a sufficiently homogeneous covering of the globe and a stable solution for the distortion, this is achieved from around 20.000 point on the globe. These methods are then applied to different map projections, including the Mercator, Equirectangular, Winkel Tripel, Gott-Wagner, and Azimuthal Equidistant and the Azimuthal Equidistant split into two-hemispheres.. Among these map projections, the Gott-Wagner projection has the lowest total distortion value and is therefore considered the best overall map projection. However, each distortion has its own advantages and disadvantages, so it is not possible to definitively declare one map projection as the best in all scenarios. Selecting the best map projection depends on minimising the distorting specific to its needs.
...
Maps, which have been in used for centuries, are constructed by map projections. A map projection is a transformation of the latitudes and longitudes positions on the globe to the x and y coordinates on a flat map. There are probably over a hundred of different map projections, some more practical than others. A perfect map is impossible to create, so it is essential to determine the next best alternative. The different maps can be classified in five different categories: Conformal, Equal-area, compromise, equidistant and true-direction. To evaluate this, various distortions are considered. The six distortions defined in this thesis are: area (A), isotropy (I), flexion (F), skewness (S), distance (D), and boundary cut (B). These distortions together form a comprehensive set of all distortions that occur due to map projections. Since these distortions are quite difficult to compute directly, they are numerically approximated. To achieve this, the data points of the coordinates need to be sampled, which can be done through various methods. This thesis examines systematic generation and random sampling of points. Additionally, it investigates how many points are needed to achieve a sufficiently homogeneous covering of the globe and a stable solution for the distortion, this is achieved from around 20.000 point on the globe. These methods are then applied to different map projections, including the Mercator, Equirectangular, Winkel Tripel, Gott-Wagner, and Azimuthal Equidistant and the Azimuthal Equidistant split into two-hemispheres.. Among these map projections, the Gott-Wagner projection has the lowest total distortion value and is therefore considered the best overall map projection. However, each distortion has its own advantages and disadvantages, so it is not possible to definitively declare one map projection as the best in all scenarios. Selecting the best map projection depends on minimising the distorting specific to its needs.
This bachelor's thesis explores the realm of map projections, aiming to establish a systematic approach for evaluating and ranking them based on angular and area distortions. Beginning with a definition of map projections, the study progresses to quantify the distortions that occur when projecting a spherical surface onto a flat one. Using these distortions, various maps are assessed; for each, figures of the map are displayed and a short summary of the map's features and the angular and area distortion are given. These distortions are also visualised in figures. To rank the map projections, distortion numbers are defined and refined throughout the study. The findings reveal a ranked list of map projections, providing a structured basis for selecting appropriate projections in cartographic endeavours. Through mathematical analysis, this thesis contributes to the understanding and evaluation of map projections, offering valuable insights for cartographers and researchers in the field.
...
This bachelor's thesis explores the realm of map projections, aiming to establish a systematic approach for evaluating and ranking them based on angular and area distortions. Beginning with a definition of map projections, the study progresses to quantify the distortions that occur when projecting a spherical surface onto a flat one. Using these distortions, various maps are assessed; for each, figures of the map are displayed and a short summary of the map's features and the angular and area distortion are given. These distortions are also visualised in figures. To rank the map projections, distortion numbers are defined and refined throughout the study. The findings reveal a ranked list of map projections, providing a structured basis for selecting appropriate projections in cartographic endeavours. Through mathematical analysis, this thesis contributes to the understanding and evaluation of map projections, offering valuable insights for cartographers and researchers in the field.
Bachelor thesis
(2023)
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P.A. Jager, H.M. Schuttelaars, C.R. Kleijn, O. Jaibi, B. Bera, D. den Ouden-van der Horst
In dit verslag is een eolisch sedimenttransport model geformuleerd en het model van De Vries geïmplementeerd en geanalyseerd. Na de afleiding van de eendimensionale advectie-diffusievergelijking vanuit de driedimensionale continuïteitsvergelijking is eerst het model van De Vries zonder diffusie onderzocht. In het model kan er onderscheid gemaakt worden tussen 3 verschillende situaties: Geen (toevoer van) sediment, Sediment-gelimiteerd en Erosie-gelimiteerd. In elke situatie zijn voor lange tijd\-schalen de toestanden van het model afgeleid vanuit de advectievergelijking. In de resultaten kwam naar boven dat de theoretische toestanden sterk overeenkwamen met de toestanden in de numerieke simulatie. Verder is het effect van numerieke diffusie van het model van De Vries geconstateerd en de grootte van dit effect afgeleid. Om het effect van numerieke diffusie te verminderen is diffusie in het model geïntroduceerd samen met het Lax-Wendroff numerieke schema. Dit schema gebruikt de gewogen upwind methode samen met de gewogen centrale Euler methode om de nauwkeurigheid van het model te verbeteren. In de resultaten is aangetoond dat numerieke diffusie kan worden gecompenseerd door het juist kiezen van dit gewicht. Uit een stabiliteitsanalyse blijkt dat de Lax-Wendorff methode een betere stabiliteitcriterium heeft dan de upwind methode. Hierdoor kan een grotere tijdstap gekozen in een numerieke simulatie. De maximale tijdstap is getest door het model te runnen met verschillende tijdstappen rondom het theoretische maximum. Uit de resultaten bleek dat het model op tijdstap die een fractie kleiner was dan de theoretische maximale tijdstap, stabiele oplossingen simuleerde. Zodra de tijdstap de maximale waarde behaalde werden de oplossingen instabiel. Door het toevoegen van diffusie aan het model is een extra voorgeschreven randvoorwaarde nodig. Door implementatie van verschillende randvoorwaarden aan het einde van het domein is gebleken dat de randvoorwaarde 'extrapoleren tweede orde' het best is. De dimensie-analyse van de advectie-diffusievergelijking wijst uit de eolischsedimenttransport gedomineerd wordt door advectief transport en het effect van diffusie verwaarloosd mag worden.
...
In dit verslag is een eolisch sedimenttransport model geformuleerd en het model van De Vries geïmplementeerd en geanalyseerd. Na de afleiding van de eendimensionale advectie-diffusievergelijking vanuit de driedimensionale continuïteitsvergelijking is eerst het model van De Vries zonder diffusie onderzocht. In het model kan er onderscheid gemaakt worden tussen 3 verschillende situaties: Geen (toevoer van) sediment, Sediment-gelimiteerd en Erosie-gelimiteerd. In elke situatie zijn voor lange tijd\-schalen de toestanden van het model afgeleid vanuit de advectievergelijking. In de resultaten kwam naar boven dat de theoretische toestanden sterk overeenkwamen met de toestanden in de numerieke simulatie. Verder is het effect van numerieke diffusie van het model van De Vries geconstateerd en de grootte van dit effect afgeleid. Om het effect van numerieke diffusie te verminderen is diffusie in het model geïntroduceerd samen met het Lax-Wendroff numerieke schema. Dit schema gebruikt de gewogen upwind methode samen met de gewogen centrale Euler methode om de nauwkeurigheid van het model te verbeteren. In de resultaten is aangetoond dat numerieke diffusie kan worden gecompenseerd door het juist kiezen van dit gewicht. Uit een stabiliteitsanalyse blijkt dat de Lax-Wendorff methode een betere stabiliteitcriterium heeft dan de upwind methode. Hierdoor kan een grotere tijdstap gekozen in een numerieke simulatie. De maximale tijdstap is getest door het model te runnen met verschillende tijdstappen rondom het theoretische maximum. Uit de resultaten bleek dat het model op tijdstap die een fractie kleiner was dan de theoretische maximale tijdstap, stabiele oplossingen simuleerde. Zodra de tijdstap de maximale waarde behaalde werden de oplossingen instabiel. Door het toevoegen van diffusie aan het model is een extra voorgeschreven randvoorwaarde nodig. Door implementatie van verschillende randvoorwaarden aan het einde van het domein is gebleken dat de randvoorwaarde 'extrapoleren tweede orde' het best is. De dimensie-analyse van de advectie-diffusievergelijking wijst uit de eolischsedimenttransport gedomineerd wordt door advectief transport en het effect van diffusie verwaarloosd mag worden.