
this looks as if you would draw concentric rings with an increasing radius. How can you prove, that discretisation does not produce small "holes" in your ring?
Addendum: I'm currently reading the french Wikipedia entry (well, after translation into german  my french is not good enough for maths) and it seems to be a characteristic of that algorithm that no holes are left

Yes, that is exactly how I came to that algorithm. I had been trying to create a solid ring. At one point in desperation I started with the same polygon approach you tried, but I stopped when I hit the point limit.
It isn't described on the English wikipedia, I don't think I saw it on the German one either even though it lists many more circle drawing algorithms than the English one.
I haven't been able to read the original paper by Andres, it is paywalled, but I have read some of his later papers* and others that expand on the work. If you want to see the proof (or just out of interest) they can be found online.
 I say read, but I don't understand most of it. It took me a long time before I even remembered how the notation for sets worked.
 I say read, but I don't understand most of it. It took me a long time before I even remembered how the notation for sets worked.
@Andreas_Rozek  this is my function, implementing an Andres circle. Heavily based on example implementations on French Wikipedia .
I think it can probably be optimised more, and ultimately I also want to draw this a bit at a time. My best approach to that so far is having an array per octant and pushing/unshifting on each iteration for each circle so that I can step through later, but I think I can improve on that.
I spent a lot of thinking about allocating a fixed length array for the coordinates, but I've come to the conclusion that it isn't possible to know the length a priori (at least, I think that is the case).