Here again is the sky chart
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.
Then there’s the “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,
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.