Basic idea

In the following we work with a discrete closed curve \gamma: \mathbb{R}\rightarrow \mathbb{R}^2. What this program does is to take this curve and determine the surface of the minimal area that has this curve as boundary. Because we only do this in the 2-dimensional case, this is always a point. So what we expect is that the algorithms transform the curve fist into a circle and shrinks this circle to a point.

Discrete in this case means that we do not know the whole curve but therefore some points called vertices \lbrace v_1,...v_n\rbrace\in(\mathbb{R}^2)^n. If we would have infinite vertices they would fit the curve exactly. The more points we have the smoother the algorithms works. We define the length which is a important value

len(\gamma)=\sum\limits_{i=1}^n \parallel v_{i+1}-v_i\parallel.

Note that v_{n+1}=v_{1}. Now we take a variation \lbrace \eta_1,...,\eta_n\rbrace\in (\mathbb{R}^2)^n. Now we compute

\frac{d}{d\epsilon}|_{\epsilon=0} len(\gamma+\epsilon\eta)\\ =\frac{d}{d\epsilon} \sum\limits_{i=1}^n \parallel v_{i+1}+\epsilon\eta_{i+1}-v_i-\epsilon\eta_i\parallel\\ =-\sum\limits_{i=1}^n <-t_{i+1}+t_i,\eta_i>\\ \Rightarrow \Delta \epsilon|_i=-(t_i-t_{i+1}).

The definition of t is quite easy, it is the normalized difference between two points

t_i=\frac{v_i-v_{i-1}}{\parallel v_i-v_{i-1}\parallel}.

The main idea is now that we compute dx_i=\Delta \epsilon|_i=-(t_i-t_{i+1}) and afterwards v_i^{new}=v_i+step*dx where step is the stepsize which is topic of a different section, mainly for the higher order schemes, because the simplest scheme iterates with the maximal save stepsize.