How can a sphere be put on a flat surface or why do star maps have distortions?

March 6, 2013

Soviet celestial globe

Celestial globe made for navigation shows accurate positions of bright stars. Courtesy Powerhouse Museum

Stars spread out in all directions from us and are at very different distances. However, to define the positions of stars and other bodies in space astronomers regard the sky as lying on a giant sphere called the celestial sphere. On this sphere there is a coordinate system similar to the system of longitude and latitude on Earth. The celestial equivalent of longitude is called right ascension, which is usually measured in hours, minutes and seconds, while the equivalent of latitude is called declination, which is measured in degrees north or south of the equator.

It is relatively easy to plot star positions on a sphere to form a celestial globe. Such globes show the positions of stars and their relative distances and angles from each other accurately. However, when my colleagues and I would like to indicate to readers of this blog the position of a comet or other object of interest in the sky it is only possible to show this on a flat surface. Similarly, published star maps and those produced by computer programs and apps all have to output their maps on a flat surface.

equal area_cylindrical_perspective

The area around the south pole in the sky with the Southern Cross and the Pointers on the left or east and the bright star Achernar on the right or west. The distortions using three different projections are shown: equal-area, cylindrical and perspective. Illustration Nick Lomb using Stellarium software

To achieve a star map the spherical celestial surface has to be projected on to a flat surface. The same techniques can be used for this purpose as have been developed over thousands of years to map the Earth. There are a large numbers of ways of putting a sphere on a flat surface with each having its advantages and disadvantages. Some methods preserve angles between objects, others preserve the area of particular regions while in others the scale is constant along one direction though not in others. There is some distortion in all methods and an appropriate method has to be chosen in each case.

An example is the perspective or gnomic projection. In this technique one hemisphere of the sky is projected on to a flat plane that is tangent to it with the projection point being the centre of the Earth. The advantage of this method is that great circles – the shortest distances between two points on the celestial sphere – map as straight lines. The disadvantage is that there is severe distortion in shape, area and direction.

Another example is equal-area projection, one version of which glories in the name ‘Lambert azimuthal equal-area projection’, in which the areas of regions in the sky such as constellations are preserved. Directions from the poles are also maintained but the scale changes as the distance from the pole increases.

A final example is cylindrical equidistant projection. This is a simple projection of a sphere onto a tangent cylinder which is then unwrapped. It is possible to arrange the projection such that the scale remains constant along all parallels, that is lines of constant declination, so that there is no distortion in the east-west direction. Of course, there are changes of scale or distortions along the north-south direction.

Unfortunately, all star maps on a flat surface necessarily distort the view of the sky in some way. In each case the person producing the map has to decide on the best compromise for the particular purpose for which the map is drawn or the particular purposes for which a computer program or app is likely to be used. Maybe in the near future we will all have pocket projectors that will be able to produce star maps on virtual spheres, but as yet these do not exist.

Reference Richard Kippers of the Faculty of Geo-Information Science and Earth Observation at the University of Twente in the Netherlands provides an excellent discussion with illustrations of various map projections here.

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