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I've always appreciated the Bresenham's line algorithm. It's a very simple algorithm and has been "superseded" by Xiaolin Wu's line which can also do antialiasing, but ever since I learnt about it, I've been very fond of it. Also the fact that it's an algorithm for use in computer graphics helps, because for me it's very easy to get excited about visual stuff as opposed to the more abstract CS stuff, which I have deep appreciation for but can't understand most of or kick myself enough to try.

> It's a very simple algorithm and has been "superseded" by Xiaolin Wu's line which can also do antialiasing

Actually, not really! Bresenham's Line Algorithm secretly has a very useful generic low-level algorithm at its core. It just happened to be created for drawing lines first.

The thing most people miss is that it describes a way to do error-free repeated addition of a fraction using only integers and addition. In fact, you can add many different fractions, as long as they all share a denominator.

That is hugely powerful in the right context: imagine an embedded context where you have to add fractions to a counter, but the counter should be an integer value at all times. Maybe you must add different fractions in different situations. For example, an embedded device with two non-matching frequencies to keep track of over longer periods of time. With this method you're guaranteed that no matter how long you keep adding, the sum will always be correct.

Yep; it's one-dimentional variant of Error Diffusion (which itself is a notably beautiful dithering algorithm), where "input" = slope of line and output is quantized dy=0 or dy=1.

Dithering also appears in audio. The old PC Speaker had merely 1/0 states. It's supposed to be limited to square waves, limited and unpleasant, right? But at some point people figured out if you flip 1/0 fast enough you can approximate continuous values: http://bespin.org/~qz/pc-gpe/speaker.txt

I once played with using "error diffusion" like dithering to play wav files. In theory that should produce better audio than the simple fixed-table and PWM dithering suggesting in above article, but I did this on a much newer computer (Pentium?) by which time PC Speaker was irrelevant (only for hack value) and only reached 30-40 bits per input sample at 100% CPU. Plus as that article explains, simple PWM might(?) actually produce less noise from timing irregularity.

But the technique is not merely an obsolete hack! Every CD player that visibly brags it has "1-bit DAC" uses a closely related technique, though in fast hardware. See: https://en.wikipedia.org/wiki/Delta_modulation https://en.wikipedia.org/wiki/Delta-sigma_modulation Take me with a grain of salt, I always get confused trying to grok these... Full explanations of the noise shaping properties that make Delta-Sigma highly useful involve quite a bit of signal-processing, but the core accumulate-compare loop is a very simple algorithm...

The Bresenham Circle Algorithm is cool too. You can draw a circle with it in just a few lines of code: https://en.wikipedia.org/wiki/Midpoint_circle_algorithm

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