I would like to draw geometric shapes and curves in 3D in order to get a better spatial intuition for the math describing such a system.

If you know about Sketchup, that's how easy to use I'd like it to be. But with the ability to draw smooth and exactly specified curves, to specify and measure shapes mathematically and with other analytical tools. Also, the ability to specify how objects move over time would be very interesting.

Maybe there is an application like that in Matlab or Mathematica? Or some serious CAD software? A 3D graphics development tool? It be odd if it isn't out there already.