Years back I spent a lot of time starting at the spline art on the front cover of PostScript Language Tutorial and Cookbook. It wasn't until years later that I would understand enough math or have good enough explanations to understand.
But I now have enough math to understand. And like all hobbies I get pleasure from talking about it and fiddling with it.
The splines that are most common these days are cubic Bézier splines (the wikipedia page Bézier curve has better pictures). They find use in CAD (a lot of the original work was done for the auto industry), fonts, and animation. Basically they're the goto solution where smooth curves are desired.
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