# Comparison of FOSS4G Supported Equal-Area Projections Using Discrete Distortion Indicatrices

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## Abstract

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## 1. Introduction

`GRASS`[1]. These areas equate to the number of cells in a global raster with a cell side of 1 km. The Marinus of Tyre and Mercator projections, even though arguably the most popular projections in Earth Sciences today, impose massive overheads in storage space and computation time with the number of extra raster cells they require to discretise the surface of the globe. In order to minimise the size of the datasets and render them spatially representative of the surface of the Earth, big spatial data researchers often undertake as first task a re-projection with an equal-area projection.

`geographiclib`[3],

`PROJ`[4] and

`GDAL`[5]. Moreover, distortions that may result from the algorithmic implementations in these programmes are not simple to assess. Therefore, the choice of map projection in a big spatial data context supported by FOSS4G may not be straightforward.

## 2. Materials and Methods

#### 2.1. Selected Projections

`PROJ`and

`GDAL`libraries, plus the correct display of raster layers in cartography programmes (e.g.,

`QGis`[10],

`gvSIG`[11]). In general, the newer the projection the less likely it is to be supported. The pseudo-cylindrical projections developed around the turn of the XX century, like the series proposed by Eckert [12], are supported but many other relevant projections developed later, like the Eumorphic [13] or the polyhedral proposed by Snyder [2], are not implemented by any of the OSGeo sanctioned programmes.

`PROJ`strings. Other equal-area projections are supported by FOSS4G but do not present enough unique characteristics to set them markedly apart from the quintet analysed. The list of projections supported by

`PROJ`can be accessed at https://proj.org/operations/projections/index.html (note that many of these are not supported by

`GDAL`and/or

`GeoTools`). Possibly avoiding mathematical complexity (and computation burden), software developers have favoured projections in the vein of those proposed around the turn of the XX century, with a single interruption and approachable formulations.

#### 2.2. Discrete Indicatrices

`geographiclib`library, a cornerstone of FOSS4G, underlying many important packages (like

`PROJ`). To compute a geodesic,

`geographiclib`takes as inputs: (i) the coordinates of a starting point, (ii) an azimuth and (iii) a distance. The coordinates of the end point of the geodesic are the output. This computation is also referred to as the direct geodesic. For each indicatrix four geodesics are thus computed, all with the same starting point (the centre) and distance (1000 m) but with four different azimuths: ${0}^{\circ}$, ${90}^{\circ}$, ${180}^{\circ}$ and ${270}^{\circ}$.

#### 2.3. Discrete Global Grid

`dggridR`[20] for the R programming language was used to create the ISEAG. This package wraps the ISEAG creation functions originally developed by Sahr in the C language. The grid was created with an aperture number of 3 and a resolution of 7, resulting in a cell area of 23 322 km${}^{2}$. This translated into a total of 21,872 indicatrices positioned at an average distance of 165 km from their immediate neighbours.

#### 2.4. Computation of Distortions

## 3. Results

`dggridR`library). The Python implementation of the

`geographiclib`library greatly simplifies its use. The resulting programme is available at Codeberg (https://codeberg.org/ldesousa/projections-compare) under an open source licence.

## 4. Discussion

#### 4.1. Projection Performance

#### 4.2. Difficulties with FOSS4G

`GDAL`supports the projection but the correct application of its inverse requires explicit parametrisation (https://github.com/OSGeo/gdal/issues/959). Since programmes using

`GDAL`are not aware of this parametrisation, they generally apply the projection incorrectly in re-projections between different coordinate systems. Therefore any re-projections to and from a coordinate system including the Homolosine projection must always be carried out directly with

`GDAL`.

`QGis`or

`gvSIG`portray vector objects spanning over areas that do not have correspondence on the surface of the ellipsoid. To address this problem, a discrete co-domain in vector form was developed that can be used to clean cartograms in such programmes [21].

`QGis`). The Sinusoidal is markedly off, even if the difference is little more than 0.3% of the total area of the geometry. The exact cause has not been identified so far, but this is another reason to avoid using the Sinusoidal projection.

`MapServer`[22], for instance, do not admit any coordinate system that is not indexed by the EPSG. Even if this issue can be partially addressed by introducing “fake” ESPG codes into the

`PROJ`database, this dependence by OSGeo sanctioned software on the Petroleum industry is something to reflect upon.

`GeoTools`[23], for instance. And as it relies on

`GeoTools`, the popular cloud computing platform Google Earth Engine [24] does not allow the retrieval or computation of datasets in the Homolosine projection.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Graphical sketch of a projected discrete indicatrix. ${N}_{p}$, ${E}_{p}$, ${S}_{p}$ and ${W}_{p}$ represent the end points of the discrete indicatrix stems after projection; N, E and S, W represent their positioning if no distortion took place. The red segments and the red arc represent the distance and angular distortions computed.

**Figure 3.**Mean absolute angular distortion at each indicatrix for the Sinusoidal, Mollweide and Hammer projections.

**Figure 4.**Mean absolute angular distortion at each indicatrix for the Eckert IV and Homolosine projections.

**Figure 6.**Mean absolute distance distortion at each indicatrix for the Sinusoidal, Mollweide and Hammer projections.

**Figure 7.**Mean absolute distance distortion at each indicatrix for the Eckert IV and Homolosine projections.

Projection | Co-Domain (km${}^{2}$) | Land Masses (km${}^{2}$) |
---|---|---|

Equal-area | 5.10 × 10^{8} | 1.47 × 10^{8} |

Robinson | 5.12 × 10^{8} | 1.56 × 10^{8} |

Marinus of Tyre | 8.00 × 10^{8} | 2.65 × 10^{8} |

Mercator ^{1} | 2.77 × 10^{9} | 1.20 × 10^{9} |

^{1}Domain limited to the latitude interval [$-{89.5}^{\circ}$, ${89.5}^{\circ}$].

Sinusoidal | +proj=sinu +lat_0=0 +lon_0=0 +datum=WGS84 +units=m+no_defs |

Mollweide | +proj=moll +lat_0=0 +lon_0=0 +datum=WGS84 +units=m+no_defs |

Hammer | +proj=hammer +lat_0=0 +lon_0=0 +datum=WGS84 +units=m+no_defs +wktext |

Eckert IV | +proj=eck4 +lat_0=0 +lon_0=0 +datum=WGS84 +units=m+no_defs |

Homolosine | +proj=igh +lat_0=0 +lon_0=0 +datum=WGS84 +units=m+no_defs |

Global | Land Masses | |||||
---|---|---|---|---|---|---|

(Degrees) | Mean | RMSE | MAE | Mean | RMSE | MAE |

Sinusoidal | 0 | 26.9 | 15.8 | −1.4 | 26.7 | 15.5 |

Mollweide | 0 | 22.8 | 12.3 | −0.7 | 24.1 | 12.9 |

Hammer | 0 | 27.1 | 17.7 | −1.2 | 27.9 | 17.8 |

Eckert IV | 0 | 17.4 | 8.7 | −0.2 | 19.0 | 9.4 |

Homolosine | −1.2 | 15.4 | 7.9 | −3.7 | 16.4 | 7.9 |

Global | Land Masses | |||||
---|---|---|---|---|---|---|

(Degrees) | Mean | RMSE | MAE | Mean | RMSE | MAE |

Sinusoidal | 380 | 40,838 | 380 | 178 | 401 | 178 |

Mollweide | 439 | 64,168 | 493 | 150 | 348 | 199 |

Hammer | 298 | 36,653 | 415 | 126 | 357 | 197 |

Eckert IV | 590 | 102,019 | 693 | 132 | 618 | 237 |

Homolosine | 383 | 64,153 | 383 | 93 | 200 | 93 |

**Table 5.**Differences in the area of five large countries, respective to the Homolosine projection, computed with

`QGis`.

Area (km${}^{2}$) | Sinusoidal | Mollweide | Hammer | Eckert IV |
---|---|---|---|---|

Russia | 56,471.34 | 282.16 | 352.88 | 292.54 |

Canada | 28,974.75 | −11.11 | −7.88 | −10.49 |

China | −19,174.73 | 1.10 | −2.39 | 5.19 |

Brasil | −51,327.38 | 1.23 | −0.57 | 2.74 |

Australia | −31,999.04 | 5.06 | 2.79 | 7.79 |

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**MDPI and ACS Style**

Moreira de Sousa, L.; Poggio, L.; Kempen, B. Comparison of FOSS4G Supported Equal-Area Projections Using Discrete Distortion Indicatrices. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 351.
https://doi.org/10.3390/ijgi8080351

**AMA Style**

Moreira de Sousa L, Poggio L, Kempen B. Comparison of FOSS4G Supported Equal-Area Projections Using Discrete Distortion Indicatrices. *ISPRS International Journal of Geo-Information*. 2019; 8(8):351.
https://doi.org/10.3390/ijgi8080351

**Chicago/Turabian Style**

Moreira de Sousa, Luís, Laura Poggio, and Bas Kempen. 2019. "Comparison of FOSS4G Supported Equal-Area Projections Using Discrete Distortion Indicatrices" *ISPRS International Journal of Geo-Information* 8, no. 8: 351.
https://doi.org/10.3390/ijgi8080351