I'm trying to give shapes in my physics engine roundness/ curvature.
I am aware of various methods for mathematically defining curvature such as bezier-curves, ellipses, etc; but I'm not sure which methods are most appropriate for use in a physics engine. I must consider the speed, flexibility, and difficulty of constructing/joining the curves when choosing my approach. The ideal is a system where a user could easily construct and collide a large amount of curved, aesthetic polygon bodies together, yet I could still maintain a satisfactory framerate. However it's inevitable that ease-of-use and other attributes will suffer.
NOTE: my physics engine is continuous, which means I must pre-calculate the times at which polygons will collide. Keep in mind that the curve's route my not be linear.
For each suggestion please give the positives/negatives!
What techniques are there for mathematically defining a 2D curve?
Advantages/Disadvantages (speed, flexibility, ease of construction/use)?
Is the technique feasible for an engine where predictability of collisions is crucial?