Calculating $\theta$ coordinate

Once $\vert\mathcal{J} \nabla \psi \vert$ is known, one can use Eq. (188) to calculate the value of $\theta$ of a point by performing the following line integral:

$$\theta_{i, j} = \int_0^{\mathbf{x}_{i, j}} \frac{R}{\vert\mathcal{J} \nabla \psi \vert} d l_p$$ (189)

where the line integration is along the contour $\Psi = \Psi_j$. It is here that the positive direction of $\theta$ can be selected. It is obvious that the sign of the Jacobian of the constructed coordinates may be different from the $\mathcal{J}$ appearing in Eq. (189), depending on the positive direction chosen for the poloidal coordinate. Denote the Jacobian of the constructed coordinates by $\mathcal{J}'$, then

$$\mathcal{J}' = \pm \mathcal{J}$$ (190)

This sign should be taken into account after the radial coordinate and the positive direction of the poloidal angle are chosen. In GTAW code, I choose the positive direction of $\theta$ to be in anticlockwise direction when observers look along the direction of $\hat{\ensuremath{\boldsymbol{\phi}}}$. To achieve this, the line integration in Eq. (189) should be along the anticlockwise direction. (I use the determination of the direction matrix, a well known method in graphic theory, to determine the direction from the given discrete points on a magnetic surface.)

In order to evaluate the integration in Eq. (189), we need to select a zero point for $\theta$ coordinate. The usual choice for $\theta = 0$ line is a horizontal ray on the midplane that starts from the magnetic axis and points to the low filed side of the device (this is my choice in the GTAW code).