Azequid, and other projections

Here again is the sky chart

Sky chart with azimuthal equidistant projection

from yesterday.

This chart is drawn with a projection technically called “azimuthal equidistant.”  The late beloved George Lovi, who used to write the “Ramblings” column for the middle spread of Sky & Telescope, once sent me copies of some pages he had found about projections; there was a formula for azimuthal equidistant, and he wrote beside it: “A real Kopfschmerz!”  Actually, it isn’t.  All it means is that points (such as stars) are plotted according to their direction and distance from the center of the picture (in this case, the zenith).  The distance of a star from the center of the map is its angular distance multiplied by the scale you’re using.  Directions and distances from the center are as they really are; but any projection of a curved reality onto a flat surface has to have distortion somewhere; for this projection, areas (such as constellations) out toward the edges get flattened – they look too wide in proportion to their height.  But this is the projection I use by default, because it is the simplest and because it can be used for any kind of picture, up to any width.  If you use it to plot the whole celestial sphere, the point opposite to (180 degrees from) the center would become a circle around the edge.

Let’s call it “azequid” for short.  There is a set of other projections that can be thought of as modifications of it.

For instance “azimuthal equal-area.”  Taking the position of a star as found by azequid, you can get its position in a chart on azimuthal equal-area projection by changing the distance, with a formula that you may really think is a headache:

distance = R * sin (distance/2) / sin (R/2)

where R is the radius of the picture.  In this projection, distances of stars farther from the center become decreased, but this allows areas to keep their true relative size.  Azimuthal equidistant could also be used, also with increasing distortion of its own kind, out to 180 degrees.

Sky chart on azimuthal equal-area projection

Then there’s the “stereographic” projection.

Sky chart on stereographic projection

It’s derived from azequid by

distance = R * tan (distance/2) / tan (R/2)

It’s often used, such as in the Sky & Telescope monthly sky maps, because it preserves the shapes of areas, such as constellations.  However, these areas get larger in size, rather rapidly, as you go outward from the center.  (That’s why our 10-degree band of “grass” now looks deeper.)  Stereographic could be used (with gross distortion) for almost the whole sphere, almost out to 180 degrees but not quite.  Why?  Because 180/2 is 90, and if your program tries to find the tangent of 90, which is something divided by zero, it crashes.

And the orthographic,

Sky chart on orthographic projection

derived from azequid rather simply by

distance = R * sin distance / sin R

It’s quite natural for drawing a picture of the Earth globe, because it is as a view of the Earth from infinity: you can see the whole hemisphere.  The dome of the sky is like the hemisphere of a globe, so it works quite well for this too.  You may notice that the sky picture looks rather like a picture of a ball.  But, just as when you see the globe its edges are foreshortened to nothing, so in this picture of the sky the edge, toward the horizon, gets compressed, down to nothing.  And no band of “grass.”  This projection comes to a halt at 90 degrees, after which it starts going backward: distances from the center greater than 90 start becoming less than 90, as if you were seeing through to the hidden side of the globe!


One more: the gnomonic.  I can’t show a whole-sky chart for this because it breaks down before the radius of 90 degrees.  It can show only smaller areas.  It is derived from azequid by

distance = R * tan (distance) / tan (R)

More of those dangerous tangents!

It is essentially the projection used by ordinary cameras: onto a flat plane (or plate).  Points off to left and right rapidly become angled away toward the direct left and right; points exactly (90 degrees) to left and right would be at infinity.

One advantage this projection has for the not-too-dangerously large areas it can map is that in it all great circles (such as the ecliptic, equator, Milky Way, lines of right ascension) show as straight lines.  They can’t curve to meet themselves: they would have to do so way beyond infinity.

It’s been a long time since I learned (from where, I’ve now forgotten) about these projections, and packed them away in a programming subroutine, which is how programming is so convenient: you can put them there and forget about them, as if (as has been said) a subroutine is a little man to whom you can give instructions and then he goes away and keeps carrying them out whenever called on, without your having to look any more into what he does.  Unless something was mal-thought about your instructions, or unless you want to remind yourself, with the comments you should have written, about how it works.  Or unless, as may happen, those who unlike me have had mathematical training can point out to me where I was wrong.

The projections I’ve mentioned are “azimuthal,” which in this context means that they are based on angles around a center.  There are many others, on other principles, which are used for geographical maps and which I haven’t used.  The azimuthal projections can be used not only for mapping the sky – or the Earth – but, I realized at some early point, are also what to use in drawing three-dimensional pictures of space.  In all of my family of programs that I call “SF” for sphere (as opposed to “CH” for chart), there is a viewpoint at a certain distance from the center, the center being often the Sun but sometimes the Earth or somewhere else; and everything else is plotted by calculating its position relative to that center, and then its angular position as seen from the viewpoint, in azequid.


8 thoughts on “Azequid, and other projections”

  1. I found your post so interesting, I’ve reread it several times. One thing I believe I have figured out, but wanted to ask you anyway, is the couple of short paragraphs after the sentence, “Let’s call it azequid for short.” I assume that you meant to be talking about azimuthal equal-area, even though you wrote azimuthal equidistant several times? I looked it up to make sure I understood how you got form one to the other, and saw that Wikipedia has a good diagram of how the points are plotted from the sphere to the flat plane in a Lambert equal-area projection.

    1. You’re right, I wrote “For instance “azimuthal equidistant”” where I meant “For instance “azimuthal equal-area””. I’ve corrected it now. I read what I’ve written over and over and over and yet still miss mistakes in it.

      You also raise the point that there are alternative ways of expressing, and constructing, projections, so that when you go to look at descriptions of them in for instance Wikipedia you will find approaches quite different from mine. It’s an endless subject, and would be interesting for me to get into sometime, if I have endless time, but after the stage when I learned these azimuthal projections I left it at that; they draw celestial charts efficiently for me. I don’t know whether conical and various other completely different projections would even work for the celestial sphere or for 3-D space.

  2. Thank you for these “behind the scenes” details. I’ve wondered which method you used! This is another aspect of astronomy that has fascinated me for a long time; I love looking at ground maps of the globe and comparing the many different methods of projection to the systems used for star charts. It’s all very fascinating.

  3. There oughta be (or is there one already?) a device with an astronomical program and a hemispherical screen; a hand-held planetarium; not a 2D distortion. From eyeballing an online planisphere and adjusting the size on the screen, and imagining a hemisphere, I think the diameter would have to be about 7 inches minimum to be useful.

    1. You’re jumping my gun (as it were). I’m saving up (as it were) to write a post about a small physical – not virtual – device that shows the whole celestial hemisphere, any celestial hemisphere.

      1. Patent that idea right now before you do anything else LOL! I really enjoyed your posts about projections ~ very interesting. I remember that you wrote once about George Lovi encouraging you to switch from your original “flat” monthly star charts to something like S&T used, which I think you did around 1994 or 1995? I’d be curious to know how S&T drew their charts, since I know they were using those from at least the early 1970’s onward. Surely they were not using computers to create the charts, were they? Did they calculate the position of each star in their monthly maps and essentially plot the locations by hand, and then draw the lines? One thing I always loved about those charts accompanying the Ramblings column was the multi-latitude views.

        1. The early way I did it was (if I’d known) also azimuthal equidistant, but with the north celestial pole as the center. In other words, each month’s sky chart was the same chart – the same piece of paper – rotated to a different position and showing through a paper window. Only by computer could I switch to the differently shaped charts for each month.

          I could only guess how things were done at Sky & Telescope and by other chart-drawers. Yes, I think they did, at least up to and perhaps after the time I computerized, calculate and draw the positions of stars by hand. After all, that had to be the way star atlases were drawn, including such wonders as the Skalnate Pleso atlas from Czechoslovakia.

          Roger Sinnott would be the one to know how charts and diagrams were created for Sky & Telescope. He made interesting visuals himself, such as his February world maps of the best locations for Young Moons, but he must have turned in a sort of raw material: he told me that the final form was never by him, it was always created in the art department. I’d have liked to see that art department, in various decades!

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