Here is a picture of a Möbius strip, made out of some thick green paper:
I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as it appears in the picture. Now before you jump up and declare, "That's easy! It's just $\left(\left[1+u \cos \frac \theta2\right]\cos \theta,\left[1+u\cos\frac \theta2\right]\sin \theta,u\sin \frac \theta2\right)$ for $u\in\left[-\frac12\!,\frac12\right]$, $\theta\in[0,2\pi)$!" Understand that this parametrization misses some features of the picture; specifically, if you draw a line down the center of the strip, you get a circle, but the one in the picture is a kidney-bean shape, and non-planar. What equations would I need to solve to get a "minimum-energy" curve of a piece of planar paper which is being topologically constrained like this? Is it even true that the surface has zero curvature? (When I "reasonably" bend a piece of paper into a smooth shape, will it have zero curvature across the entire surface, or does some of paper's resistance correspond to my imparting non-zero curvature to the surface?)
This question is thus primarily concerned with the equilibrium shapes formed by paper and paper-like objects (analogous to minimal surface theory in relation to soap-bubble models). Anyone know references for this topic?