From a sphere to an ellipsoid
The Earth is not perferctly spherical. While its shape is ultimately irregular on a small enough scale, it can be better approximated as an ellipsoid rather than a sphere. In other words, it looks like a "sphere squished on the poles", due to the Earth's rotation which gives rise to centrifugal forces. When people speak of spherical Earth, it is meant that the surface approximately looks like a sphere of average radius around 6 371 km long. In the most common ellipsoid model, WGS84, the length of the "radius" on the equator (semi major axis) is 6 378 137 m, while the "radius" at the poles (semi minor axis) is approximately 6 356 372 m long. The difference is around 22 km, or about 0.33%, which is not a big difference, hence the Earth as a sphere is often a good enough approximation. However for precise calculations, this more precise shape needs to be taken into consideration. In particular, to calculate the distance between two points on Earth. The calculator below asks to enter the coordinates of two points (or clicking on the 3D sphere below) and gives back two distances. The first distance is for a spherical Earth model, and the second for the ellipsoid WGS84 model (using Vincenty's algorithm). The difference between the two distances varies depending on how the shortest path between the points looks like. For example a spherical Earth underestimates the distance between Paris (France) and Beijing (China) by 22 km , but it is off by only 300 meters for the much longer distance Marseille (France) to Nouméa (France) .
Distance calculator
Distance on Earth as a sphere: km
Distance on Earth as an ellipsoid: km
Graphical representations
Below is a 3D representation of the Earth as a sphere. Were it an ellipsoid, its height would be around 498.3 pixels for 500 pixels width. On a modern screen, this difference is not perceptible, hence the choice of spherical representation, which also greatly simplifies the 3D computations. When the above calculator contains valid coordinates, the two corresponding points are shown as red dots on the representation below. The shortest path between the two points (called "geodesic"), whose distance is computed above, is represented as a red line.
How to use: click and drag the Earth to rotate it. The mouse wheel controls the zoom level (on mobile: zoom with 2 fingers). CTRL + click puts the corresponding point's coordinates in the calculator above (on mobile: long press). Successive CTRL + clicks alternates between points 1 and 2 in the calculator.
It is possible to choose the image quality. There are 3 available qualities: low, medium, and high, which apply a texture of, respectively, 2048x1024, 4096x2048, and 8192x4096 pixels. The default quality is low, since the difference can only be seen when zooming in. The high quality texture requires downloading a 4 MB file, which may take some time depending on your internet connection. These images come from NASA.
Image quality:
infiniteIt is also possible to change the altitude at which we view the Earth using the above slider. Changing the altitude is a different effect from changing the zoom. Changing the zoom does not change how much we see of the Earth, it just makes the picture bigger (or smaller), with proportions unchanged. The view altitude change what we can see of the Earth. When we are on Earth, at altitude 0, we barely see a few (hundred) meters around us, or slightly more depending on the local topography. But as we go higher, we see more and more of the Earth. By default, with the "Infinite altitude" box checked, the view is that of the Earth as seen from infinitely far away. It is possible to choose an altitude (after unchecking the infinite altitude box) from 400 km, which corresponds approximately to the altitude of the international space station, and 63781 km which corresponds to 10 times the radius of the Earth. The zoom is automatically adjusted such that the Earth appears the same size for all altitude views. Using this slider and by rotating the Earht over the correct spot, we can recreate the famous photo with fisheye lens over the Nile delta and the Sinai, up to lens distortions and imprecisions for the altitude (note that at such view altitude it is best to use the High quality image).
Below is another representation of the surface of the Earth, projected on flat map. By default, the equirectangular projection (or plate carrée) is used. It is a very simple projection in that the latitude and longitude are directly mapped as cartesian coordinates. In a sense, there is no calculation to be done to obtain this projection. The map below also shows the same red dots and line corresponding to the points in the calculator above. Like all Earth projections, it has some drawbacks. In particular here, the longitudinal ("horizontal") distance becomes more and more exaggerated as we go near the poles. This shows very well when looking at geodesics, which are exactly the same as those displayed on the spherical Earth above, which are not straight lines on the representation below, especially if the trajectory goes near the poles. See for example the geodesic between Bucharest (Romania) and Vancouver (Canada): , this beautiful geodesic going from near the north pole to near the south pole , or the shortest path between Ulaanbaatar (Mongolia) and Santiago (Chile) which looks perfectly normal on the spherical Earth representation, but may be surprising on the plate carrée projection: . On this map, we could say that the geodesic is "attracted" to the poles because the horizontal distance is lesser there.
There exists other projections than the equirectangular one. Each projection has some advantages, but also some drawbacks. Due to the curved shape of the surface of the Earth, no projection on a flat map is perfect, for the same reason one cannot flatten an orange's peel perfectly on the table without tearing it apart. Some projections preserve angles, some preserve area, and so on, but none is perfect. Below, it is possible to choose different projections in the menu, as well as the origin's longitude. There are also two checkboxes to choose to display meridians ("vertical" lines) and latitudes ("horizontal" lines).
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Some details about the projections other than equirectangular projection:
- Mercator: this projection was mostly useful for navigation in the past, as it preserves angles. This allows to draw a trajectory with a constant bearing as a simple line. The obvious drawback is that the areas are not preserved. The closer we approach the poles, the greater the projected area is, which makes, for example, Greenland appear much bigger than it really is. In fact, the projected coordinate on the vertical axis tends to infinity as we get to the pole. This means that if we want to draw the map with a finite height, we have to cut the projection at a given latitude. Above, there are two choices of Mercator projections with cuts at 66°N and 66°S, and at 85°N and 85°S.
- Gall-Peters. In some ways, it is the opposite as the Mercator projection. It preserves areas, but not angles, which gives it a distorted look.
- Mollweide: this projection conserves areas, but by making the meridians close at two points on the projection, at the north and south poles, it introduces less deformations than, for example, the Gall-Peters projection. The cost to pay being that the projected map is not rectangular. This projection is often used to represent some data on the sky, since the sky looks like a sphere just like the surface of the Earth (the Celestial sphere). For example, see the projected data of the Cosmological Microwave Background.
- Sinusoidal: It is similar to the Mollweide projection, in that it preserves angles and that the meridians meet at two points, but it does so such that the meridians are sinusoidal.
- Eckert II: very angular, the meridians are straight lines on each hemisphere.
- Peirce quincuncial: this projection is centered on the North pole, and has many interesting properties. First, a drawback: four points are very deformed on this projection. They are at the middle of each edges of the map. By default, by "centering" on a 20° longitude, these 4 points happen to be (mostly) in oceans, thus hiding the deformations introduced by this map. We can see this easily by changing the center longitude such that land is projection on these points, see the . Now, an interesting property of this projection is that it is periodic, in that we can glue copies of the projection on its edges, rotating every other copy by 180°. This reconstitutes Antartica. This allows to pave the plane, see the exemple.
This list is of course incomplete, there are many other projections, see for example. See also xkcd/977.
These projections are used in other areas than terrestrial cartography. For example, in cosmology or astrophysics, it is common to observe and represent the celestial sphere. What we see when we look at the sky: stars, planets, galaxies, are so far away that we do not perceive any distance. Instead, we feel like all these objects are the same distance away from us, so that they all look like shiny dots on a sphere that would be around the Earth. Since this is a sphere, we can use the same algorithms to project it on a plannar surface than those we used above to represent the surface of the Earth. In short, we can make sky maps. The point of view is rather different in both cases however, since we as observers are inside the celestial sphere, while the surface of the Earth is (usually) under our feet. In cosmology it is common to use the Mollweide projection to represent the sky. The most notable map being that of the Cosmic Microwave Background (CMB).
We can play with different projections for these sky maps, just as we can with the surface of the Earth. Below you can choose three spheres: that of the Earth (selected by default), that of the CMB, and also one of the sky in true colors, where we can mostly see our galaxy, as observed by Gaia. Of course, for the sky maps, the distance calculator at the top of the page does not make any sense.