Reddit reviews The Mathematics of Juggling

It's funny that you say exactly "The mathematics of juggling", because there's a book by that name that may be exactly what you're looking for. A little pricy, so if you're near a juggling club, perhaps ask if you can borrow a copy or check your library. The math inside it is mid-late university complexity.

This page is a bit less advanced, and has a nice further reading section.

Claude E Shannon is a pretty big deal in the information science world, and he developed Shannon's Juggling Theorem.

Burkard Polster (aka Mathologer) has a book all about the mathematics of juggling. It's quite interesting, but can get advanced.

Interesting. I'd be curious to see what he says that this book doesn't say. I found that book in my college's math department library and found it really complete although obtuse in places.

Thanks for the comments. Yes that is an interesting thing to watch for, I often scan for and select those freeze-frames for my video thumbnails (e.g. here is one of my favorites, includes equidistant ball distances and symmetric hand positions). It's quite frustrating when I know that there is a perfect line moment in a pattern I've recorded, but there's no screenshot to be captured because the line happens between frames because a) unlucky, b) very fast pattern and 30fps sampling just wasn't adequate, c) didn't run it for long enough, d) some combination of all of the above. A couple side notes: 1) The Mathematics of Juggling book has a page or two (edit found it! section 5.3.2 "Why Clubs and Balls Line Up") about alignment, 2) I really enjoy watching expert 5 club jugglers with all clubs spinning perfectly in phase, there's something deeply satisfying about that.

A similar aspect I frequently consider is the minimum distance between any two balls at any time. I usually try to maximize that variable for the pattern (without increasing the overall height/width bounding box perimeter of the whole pattern) giving each ball as much personal space as possible, especially in the animations where I can easily fine tune the tiny adjustments to the timing and throw/catch locations. It's not just aesthetic, makes the pattern paths much easier to follow visually (less of shell game), and the obvious practical purpose is maximizing your margin of error, i.e. minimizing the risk of two balls colliding midair.