Suppose we have a parametric curve $x = x(s)$, $y = y(s)$, and $z = z(s)$. We define the unit tangential vector of the curve by $(\cos\theta_x, \cos\theta_y, \cos\theta_z)$, which we assume to be known at every point on the curve. Then, from a point $s_A$ to $s_B$, the $x$ component of their distance is $\Delta x=\int_{s_A}^{s_B} \cos\theta_x \,ds$, where $\theta_x = \theta_x(s)$ is a function of $s$ which does not have an explicit expression (even an implicit relation seems impossible because $\theta_x$ and $s$ are related through some PDEs that can only be solved numerically).
I want to re-write $\Delta x=\int_{s_A}^{s_B} \cos\theta_x \,ds$ to the form of $\Delta x=\int_{\theta_{xA}}^{\theta_{xB}} f(\theta_x) \,d\theta_x$, trying to derive some general closed-from equations about $\Delta x$ and $\theta_x$. However, I don't know what the $f(\theta_x)$ should be. Can anyone help me?
p.s., the physical background of this question is that I know the distribution of the inclination angle along a chain, and I want to determine the chain's configuration accordingly.