My bachelor thesis in mathematics is about Discrete Elastic Rods as described in the paper of the same title by Miklós Bergou et al 1.
My task was to understand the paper and its sources, and to retrace all steps thoroughly in a mathematical sense.
I also implemented its features as far as necessary to make sure I understand them fully.
The paper describes a discretization of a continuous model of elastic rods using a geometric approach. That means that it first looks at a continuous model and then uses geometric reasoning to find discrete analogs for continuous quantities and uses them to discretize the model:
The continuous rod is discretized as chain of vertices that are connected by straight lines. The curvature of the rod is discretized using the angle between two edges 2. It is is treated as an "integrated quantity", which means that it is distributed over half of the left and half of the right neighboring edge. This is pretty much the main difference to simulating a chain of straight rods.
The treatment of twisted rods is more intricate and the "proofs" are quite short in the original paper. It took me quite some time and additional resources to understand them.
While some of the arguments are straight-forward when looked at with intuition, it's incredibly hard to write them down in a way that uses correct notation and does not look like overkill.
The thesis can be found here. It is written in a very condensed way because it covers many topics and tries to treat them in a self-contained way (proofing almost everything along the way):
it deduces different ways to express rotations (rotation matrices, quaternions and the Cayley transformation) and hints at how they are connected;
it introduces continuous curves and parallel frames;
it sets up a continuous model of elastic rods; and then
it shows how discretized curves and frames can be defined and their similarities of their properties to the one's of the continuous concepts;
it presents the discretized rod model; and
it explains how these rods can be simulated using constraints that are enforced with an algorithm called "Fast Manifold Projection".
Last but not least, you can find the LaTeX code for the thesis here (Bazaar repository at launchpad).
actually it is defined as \(2 \left | \tan {\frac{\phi}{2}} \right |\), similar to the definition in the lecture "Discrete Differential Geometry of Curves and Surfaces" by Tim Hoffmann.↩