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Grab some trigonometry and functions, for a start. Learning some Physics (yeah, not only of Maths the world is made) also helps A LOT. The rest is basically your programming logic. For example, you can easily see a Sine Wave (the graphic of a Sine function) on a pendulum's movement. You see how I'm relationing both trigonometry and functions here?

You can probably scrape by without any trig knowledge but if you really want to polish your game and make things move in more realistic and unique ways then trig is the way to go.

From what I can tell, trigonometry is paramount to wizardry for 2D games, particularly Sonic ones. So yeah, learn some of that.

Whoa, the feedback was a lot more positive than I imagined. Thanks everyone! I'm glad that it's going in the right direction. Virt - I hear you there. I'm wondering if it would help if I shrunk the player's sprite a little bit as they're moving into it... Maybe I can look at how Sonic Advance does it again. Rawr/Seraph - Ristar is definitely an inspiration for me. Rocket Knight was more of a coincidence, but I really like how Sash's boost ended up similar to Sparkster's. Kindof a melting pot of various Sega Genesis games, no? I just realized that I've only been working on this for two weeks, so with that in mind, I'm pretty satisfied with the progress so far. With my hiatus starting tomorrow, it's time for me slow down and take a break before I burn myself out. While I'm away, I'll put some more thought into the story and setting since that's something I've been struggling with for the past few days. When I've got my real life sorted out and I'm ready to get back to work, I'll hopefully have a fresh set of ideas to work with. See you in a few weeks!