I have a set of $(t, f(t))$ points that describe a function of time. I need to interpolate those with a spline that goes through all the given points, and doesn't have any loops, so it creates a single-valued C2-continuous function of t. All the spline types I've looked at, primarily centripetal Catmull-Rom and interpolating b-splines, don't have the single-valued property. If I place two knot points very close to each other on the t axis, the spline may loop around (go backwards in t) because with those splines they're really in $(x,y)$ and the $t$ is along the curve length.
I've looked at Bézier control points for drawing an interpolating cubic spline *function* which says it's just a matter of reparamaterization where $t=(x-x_{min})/(x_{max}-x_{min})$, but I don't see how that removes the looping. Could someone explain it simply?