I'm designing software for generating animation curves. I'd like the curves to be based on those found in Blender 3D, which they call "F-Curves." According to the page on the Blender Wiki, they are similar to Bezier curves but they are a function of time rather than a parametric function. This is necessary because the property that the curve represents cannot have more than one value at any given time.
So far, I haven't been able to find a mathematical definition for an F-Curve. Does anyone here know they are defined? The screenshots on the wiki page might give some clues. Are they just Bezier curves with a restricted set of inputs? Or are they something fundamentally different?
Edit: Here's a screenshot of the curve. It's important that the control points have the ability to influence the curve more or less than an adjacent control point. This is possible with Bezier curves, but once the influence of one control point is strong enough, you can get loops in the curve or the curve will turn back on itself, both of which are undesirable for an F-Curve.
My guess is that they are cubic Hermite polynomials. These are indeed closely related to Bézier curves. They are sometimes called F-curves because they were popularized in CAD by someone named Jim Ferguson.
The idea is that you can construct a cubic curve by interpolating given values and slopes at its two end-points. Specifically, suppose you are given two values $p_0$ and $p_1$, and two end-slopes $q_0$ and $q_1$. Define a function $f(t)$ by $$ f(t)= (2t^3-3t^2+1)p_0 + (3t^2 - 2t^3)p_1 + (t^3-2t^2+t)q_0 + (t^3-t^2)q_1 $$ Then it's easy to verify that $f(0)=p_0$, $f(1)=p_1$, $f'(0)=q_0$, and $f'(1)=q_1$.
More details here.