# VMEC PEST1 Coordinates

The coordinates output by the VMEC code are not straight field line coordinates. In order to convert to straight field line coordinates (PEST1) the VMEC theta coordinate must be transformed to the straight field line coordinate. In general the VMEC coordinates are written
$R_{m,n}\left(s\right)=\frac{2*N}{\left(2\pi\right)^2}\int^{2\pi/L}_0 dv \int^{2\pi}_0 d\theta_* R\left(s,\theta_*,v\right) \cos\left(m\theta_*-nv\right)$
where
$\theta_*= u + \lambda\left(s,u,v\right)$
In VMEC, the lambda quantity is in terms of normalized flux. Using these definitions, the following identity can be written
$R_{m,n}\left(s\right)=\frac{2*N}{\left(2\pi\right)^2}\int^{2\pi/L}_0 dv \int^{2\pi}_0 du \left(1+\frac{\partial \lambda}{\partial u}\right) R\left(s,u,v\right) \cos\left(m\left(u+\lambda\right)-nv\right) - g_{00}\frac{2*N}{\left(2\pi\right)^2}\int^{2\pi/L}_0 dv \int^{2\pi}_0 du \left(1+\frac{\partial \lambda}{\partial u}\right) R\left(s,u,v\right)$
As a result the quantities in VMEC coordinates must first be transformed to suv space, then Fourier decomposed to the PEST1 coordinates.