Preface
Consider a plane curve defined by parametric equations (for $t_1\le t \le t_2$):
$$x=x\left(t\right)$$ $$y=y\left(t\right)$$
In addition, there is a scalar function $f\left(x,y\right)$ defined for every value of $x$ and $y$.
Giving an example to illustrate (in this example, $t$ is the central angle in the $x,y$ plane): $$x\left(t\right)=\cos\left(t\right)$$ $$y\left(t\right)=\sqrt{1-\cos^2\left(t\right)}$$ $$f\left(x,y\right)=\cos^2\left(\pi * 6 * x * y\right)$$
The parametric curve, and the value of $f\left(x,y\right)$ along that curve, can be depicted in a 3D plot (in this example, for the range $0\leq t\leq 2\pi$), expressed something like: $$ParametricPlot3D\left[x\left(t\right), y\left(t\right), f\left(x,y\right) \right]$$
and depicted graphically as:
I'll choose one segment $s\left(t\right)$ of this curve to simplify the example (e.g. for the range $\frac{\pi}{4}\leq t\leq \frac{3\pi}{4}$):
Objective
My objective is to plot the values of $f\left(x,y\right)$ along the curve segment $s\left(t\right)$ as a 2D parametric plot. That is, with the curve $s\left(t\right)$ "straightened out" to be the horizontal axis of the 2D plot, and with the vertical axis being the value of $f\left(x,y\right)$ -- for example, something like this:
Question
My question is, how do I express the curve segment $s\left(t\right)$ in parametric form as the horizontal axis (clumsy description, but not sure how else to state the concept)? I am guessing the form may be related to the differential for the arc length of $s\left(t\right)$: $$ds=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt$$
But that may be way off the mark.
I think that the length of $s$ would, for any curve, be $\ge$ the length of the parallel projection of $s$ onto the $x-f\left(x,y\right)$ plane, and so I'd expect the length of $s$ as the horizontal axis in a 2D plot to be $\ge$ the length of that parallel projection of $s$.
I am expecting there is an expression $S\left(t\right)$ for displacement on the horizontal axis such that each point on the horizontal axis represents a corresponding point on the curve $s$, so that a 2D parametric plot might look something like: $$ParametricPlot2D\left[S\left(t\right), f\left(x,y\right) \right]$$
A further constraint on $S\left(t\right)$ would be, I believe, that the following would hold true for both the arc length of $s\left(t\right)$ and $S\left(t\right)$ over the interval $t1\leq t\leq t2$: $$S\left(t2\right) - S\left(t1\right) = \int_{t1}^{t2}{s\left(t\right)dt}$$
Maybe this suggests that the following is true? $$S\left(t\right)=\int{s\left(t\right)dt}$$
I can certainly provide any additional clarification if needed.
For your parametric plot, your $x$ coordinate as a function of $t$ should be the length traversed along the curve between time $t_1$ and $t$. This is:
$$\int_{w = t_1}^{w = t} ds = \int_{w = t_1}^{w = t} \sqrt{\left(x'(w)\right)^2+\left(y'(w)\right)^2}dw$$