I've been looking at elementary cubic equations for curves and seem to understand them well enough. Going the other way and driving parametric equations has been mystifying. For example: given a simple cubic equation with 3 real roots, I'd like to derive a pair of parametric equations f(t), g(t) for the curve such that the velocity of a particle along the curve (in the direction of the curve) is a constant.
Perhaps a line integral will help? I'm not sure how to approach this.