So, I finally made a procedural star pattern adjacent to the big American holiday. It works nicely but relies on converting uv coordinates from cartesian to polar, manipulating the angle, and then converting back, similarly to how one would make a paper star by folding the paper, making a single cut and then unfolding. For the conversion to polar coordinates, I used the arctan2 function provided by the Cycles math node because I did not find an arctan function in the "regular" math node. This means, among other things, that the stars don't show up in the preview. So, my question is whether there's an arctan (or arcsin, arccos etc.) function hidden somewhere in the jungle of pre-Cycles material nodes, or some other straightforward* way to convert cartesian into polar.

* As in, more straightforward than using something like the Taylor expansion of the arctan.

(... coming out of lurk mode ...)

No. In fact I found a thread from 2006 where anxcon wrote "wish they wouldve added arctan though :("

I'm almost positive I posted some very efficient matmatic scripts (using series expansions with some personal tweaks) for arcsin and arctan but they were lost with the closure of RDNA.

(... returns to lurk mode ...)

Thanks, Mr. Baggins! Well, if you ever happen to come across forgotten backups of those scripts, let us know! :-)

I did an arctan approximation in the material room 7 years ago - it's available as a PP2014 compound node over at ShareCG -  PP2014 MT5 Atan_Approx1_FIX1

Here's a screenshot of it in use  (the 'Image_Map' node and the unconnected 'Image_Map_2' node both use the same image).

Peril-sensitive sunglasses advised if you wish to look inside the compound node... ;o)

Cool! I'll have a look.

Well, it does definitely work for my little stars, but I don't feel particularly inclined to "read" that compound node. :-)

That's the beauty of compound nodes - they hide all the spaghetti! 😁

So, I took that as a challenge and tried my hand at an atan2 approximation myself:

The rightmost row in the bottom part produces min(abs(x) / abs(y), abs(y) / abs(x)) so that we get an argument between 0 and 1. The center row then computes the approximation 1.5707 * t / (0.62 + sqrt(0.9035 + t^2)) for arctan(t) which I found on Stack Exchange and tweaked a little bit in Desmos. Finally, the row on the left plus the node labelled atan2 above it do all the necessary "unfolding" to get from 8 copies of the angle range 0 through pi / 4 to the full function.

A quick spreadsheet check indicates a max error of around 0.001° for your approximation, compared to 0.1° for the one I used. Nice one ! :o)