Given a generalized cylinder1 whose axis is defined by $r(s) = (x(s), y(s), z(s))$, we want to derive a lower bound of the lateral surface area between the cross sections of $r(a)$ and $r(b)$, where $a < b$ and their cross sections don't intersect. Let $L(s)$ denote the arc length of the cross section curve at $r(s)$, is there a surface area lower bound be based on $L(a)$, $L(b)$, and $(b-a)$? What assumptions (such as surface curvature or minimum Hausdorff distance) do we need? The literature on generalized cylinders focuses on approximating the volume, and we are unable to find publications about approximating its lateral surface area.
Our current idea is that the true surface area is bounded by an integral that sums up all the cross section arc lengths, so: $$ SA \geq \int_{a}^{b} L(s) \,ds $$ This is because the integral does not take slants on the lateral surface into account, as detailed in this case about circular cylinders. The exact error is hard to approximate since the slant is hard to be quantified when every cross section curve can take different shapes as long as it's simple and closed.
The above bound is not expressed in terms of $L(a)$, $L(b)$, and $(b-a)$, so we can simplify it by defining $L_{\min}$ as the minimum arc length of all cross section curves between $r(a)$ and $r(b)$. Formally, $\forall s$ where $a \leq s \leq b$, $L(s) \geq L_{\min}$ and $\exists c$ where $L(c) = L_{\min}$ and $a \leq c \leq b$. Thus, $$ SA \geq \int_{a}^{b} L(s) \,ds \geq \int_{a}^{b} L_{\min} \,ds = L_{\min}(b-a) $$
The problem is that $L_{\min}$ has to be bounded by $L(a)$ and $L(b)$, in a format such as $L_{\min} \geq c(L(a) + L(b))$. Making an assumption about the curvature of the surface seems to allow a bound by limiting how much the cross section curve can shrink. But, cross section curves can take any shape. It's possible that at $r(a)$, the cross section curve makes lots of slow turns, resulting in a large $L(a)$, while at $r(c)$ where $L(c)=L_{\min}$, the cross section curve is a circle. Thus, the curvature is still small since the surface of the cylinder transitions gradually, but the arc length of the cross section curve is reduced dramatically. Specifying a maximum Hausdorff distance would not help either because arc length can become smaller without significantly altering the radius of each point on the cross section curve.
In conclusion, we are looking for literature in this area as well as possible assumptions that are (1) broad enough to include most generalized cylinders and (2) general enough to provide a desired lower bound on the surface area.
[1]: We define a generalized cylinder as a solid with a 3d curve as its axis (also called skeleton). Any point on the axis is associated with a cross section curve situated on a plane normal to the tangent vector of the axis. This cross section curve is simple and closed, and it defines the surface of this generalized cylinder. Furthermore, we assume that for any two points on the axis, their cross section curves do not intersect. This assumption will become useful when calculating the surface area. For a picture and the source of this definition, see the third page of this document.
