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I've done a broad range of animations over the last ten years. Wherever possible I like to animate elements with code rather than on the timeline. This has been made much easier of the last few years with the increasing powers of tweening classes. For basic tweens I use caurina and greensock's Tweenlite, but I also have written a lot of my own particle generator classes, and often use circular trigonometry for when things need to get a bit more intricate...