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Question: Mathematics of Ritardando
Answer: "...is there a rule conductors use for ritardando in terms of (a) its rate, (b) its change in rate, and/or (c) the relationship between the final tempo and the tempo of the piece?"

Not that I'm aware of. Such a rule would be of little value, because -- unless you're practicing with a drum machine, or other device that permits varying tempo -- there's generally no good way (other than feel) to for conductors or performers to accurately measure their tempo during a performance. They're certainly not likely to be thinking "Now this beat has to be played 10% longer than the previous one, and the following beat has to be 17.4% longer than that..."

However, if I had to guess a general shape of a tempo curve in a ritard, they generally seem to be monotonically decreasing (you don't speed up again in the middle of a ritard), and concave downward (i.e., the most ritard'ing comes just before the end of the ritard). But to what degree this is true is up to the conductor or performers to decide.

On the other hand, about the most prescriptive that I've usually seen music notation get is poco rit. (just a little bit of ritardando), or molto rit. (much ritardando). And of course, the dashed line that Richard mentions, to indicate the duration of the ritard.

That said, I am interested in what the tempo curves of actual performances would look like. I know that approximate tempo maps can be created for real performances, and I'd be surprised if someone, somewhere, hasn't researched what these curves look like.