Editing Writhe (section)

== Numerically approximating the Gauss integral for writhe of a curve in space ==
Since writhe for a curve in space is defined as a [[double integral]], we can approximate its value numerically by first representing our curve as a finite chain of <math>N</math> line segments. A procedure that was first derived by Michael Levitt<ref name=levitt_computationofwrithe_1986 /> for the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowski<ref name=klenin_computationofwrithe_2000 /> is to compute

:<math>
\operatorname{Wr}=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\Omega_{ij}}{4\pi}=2\sum_{i=2}^{N}\sum_{j<i}\frac{\Omega_{ij}}{4\pi}</math>,

where <math>\Omega_{ij}/{4\pi}</math> is the exact evaluation of the double integral over line segments <math>i</math> and <math>j</math>; note that <math>\Omega_{ij}=\Omega_{ji}</math> and <math>\Omega_{i,i+1}=\Omega_{ii}=0</math>.<ref name=klenin_computationofwrithe_2000 />

To evaluate <math>\Omega_{ij}/{4\pi}</math> for given segments numbered <math>i</math> and <math>j</math>, number the endpoints of the two segments 1, 2, 3, and 4. Let <math>r_{pq}</math> be the vector that begins at endpoint <math>p</math> and ends at endpoint <math>q</math>. Define the following quantities:<ref name=klenin_computationofwrithe_2000 />

:<math>
n_{1}=\frac{r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|},\; n_{2}=\frac{r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|},\; n_{3}=\frac{r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|},\; n_{4}=\frac{r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}
</math>

Then we calculate<ref name=klenin_computationofwrithe_2000 />

:<math>
\Omega^{*}=\arcsin\left(n_{1}\cdot n_{2}\right)+\arcsin\left(n_{2}\cdot n_{3}\right)+\arcsin\left(n_{3}\cdot n_{4}\right)+\arcsin\left(n_{4}\cdot n_{1}\right).
</math>

Finally, we compensate for the possible sign difference and divide by <math>4\pi</math> to obtain<ref name=klenin_computationofwrithe_2000 />

:<math>
\frac{\Omega}{4\pi}=\frac{\Omega^{*}}{4\pi}\text{sign}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).
</math>

In addition, other methods to calculate writhe can be fully described mathematically and algorithmically, some of them outperform method above (which has quadratic computational complexity, by having linear complexity).<ref name=klenin_computationofwrithe_2000 />

[[File:Simulation of an elastic rod relieving torsional stress by forming coils.ogv|thumb|A simulation of an elastic rod relieving torsional stress by forming coils]]