Mappings
This folder contains code describing tools for handling different coordinate systems.
The current coordinate transformations implemented are:
Analytically invertible coordinate transformations
Circular coordinate transformation
- Mapping (CircularToCartesian):
\[
\left\{
\begin{aligned}
& x(r,\theta) = r \cos(\theta), \\
& y(r,\theta) = r \sin(\theta).
\end{aligned}
\right.
\]
\[
J (r,\theta) =
\begin{bmatrix}
\cos(\theta) & -r\sin(\theta) \\
\sin(\theta) & r\cos(\theta)
\end{bmatrix}.
\]
with \(\det(J) = r\).
- Inverse mapping (CartesianToCircular):
\[
\left\{
\begin{aligned}
& r(x,y) = \sqrt{x^2 + y^2}, \\
& \theta(x,y) = \text{atan2}(y, x).
\end{aligned}
\right.
\]
Czarny coordinate transformation
- Mapping (CzarnyToCartesian): Taking \(\varepsilon \in ]0,1[\) and \(e\) as parameters.
\[
\left\{
\begin{aligned}
& x(r,\theta) = \frac{1}{\varepsilon} \left( 1 - \sqrt{1 + \varepsilon(\varepsilon + 2 r \cos(\theta))} \right), \\
& y(r,\theta) = \frac{e\xi r \sin(\theta)}{2 -\sqrt{1 + \varepsilon(\varepsilon + 2 r \cos(\theta))}}.
\end{aligned}
\right.
\]
with \(\xi = 1/\sqrt{1 - \varepsilon^2 /4}\).
\[
J (r,\theta)
%=
% \begin{bmatrix}
% -\frac{\cos(\theta)}{\sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
% &
% \frac{r\sin(\theta)}{\sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
% \\
% \frac{\cos(\theta)}{\sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
% \frac{e\varepsilon \xi r \sin(\theta)}{\left(2 - \sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}\right)^2}
% +
% \frac{e \xi \sin(\theta)}{2 - \sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
% &
% \frac{-r\sin(\theta)}{\sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
% \frac{e\varepsilon \xi r \sin(\theta)}{\left(2 - \sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}\right)^2}
% +
% \frac{e \xi r\cos(\theta)}{2 - \sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
% \\
% \end{bmatrix}
=
\frac{1}{\sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
\begin{bmatrix}
-1
&
0
\\
\frac{e\varepsilon \xi r \sin(\theta)}{\left(2 - \sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}\right)^2}
&
\frac{\sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}{2 - \sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}
\\
\end{bmatrix}
\begin{bmatrix}
\cos(\theta) & -r\sin(\theta) \\
\sin(\theta) & r\cos(\theta) \\
\end{bmatrix}.
\]
with \(\det(J) = \frac{-r}{\sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}\frac{e \xi}{2 - \sqrt{1 + \varepsilon(\varepsilon + 2r\cos(\theta))}}\).
- Inverse mapping (CartesianToCzarny): Taking \(\varepsilon \in ]0,1[\) and \(e\) as parameters.
\[
\left\{
\begin{aligned}
& r(x,y) = \sqrt{\frac{y^2 (1+\varepsilon x)^2}{e^2\xi^2+0.25(\varepsilon x^2-2x-\varepsilon)^2}}, \\
& \theta(x,y) = \text{atan2}(2y (1+\varepsilon x), e \xi (\varepsilon x^2 - 2x-\varepsilon)).
\end{aligned}
\right.
\]
Barycentric coordinate transformation
- Mapping (BarycentricToCartesian):
\[
(c_1, c_2, c_3) \rightarrow (x, y).
\]
Discrete coordinate transformation defined on B-splines
- Mapping (DiscreteToCartesian)
\[
\left\{
\begin{aligned}
& x(r,\theta) = \sum_k c_{x,k} B_k(r,\theta) , \\
& y(r,\theta) = \sum_k c_{y,k} B_k(r,\theta) .
\end{aligned}
\right.
\]
3D coordinate transformations
Cylindrical to Cartesian transformation
- Mapping (CylindricalToCartesian):
\[
\left\{
\begin{aligned}
& x(R,Z,\zeta) = R \cos(\zeta), \\
& y(R,Z,\zeta) = R \sin(\zeta), \\
& z(R,Z,\zeta) = Z
\end{aligned}
\right.
\]
\[
J (R, Z, \zeta) =
\begin{bmatrix}
\cos(\zeta) & 0 & -R\sin(\zeta) \\
\sin(\zeta) & 0 & R\cos(\zeta)
0 & 1 & 0 \\
\end{bmatrix}.
\]
with \(\det(J) = -R\).
- Inverse mapping (CartesianToCylindrical):
\[
\left\{
\begin{aligned}
& R(x,y,z) = \sqrt{x^2 + y^2}, \\
& \zeta(x,y,z) = \text{atan2}(y, x), \\
& Z(x,y,z) = z.
\end{aligned}
\right.
\]
Toroidal to Cylindrical transformation
The toroidal to cylindrical transformation combines a 2D transformation in the polar plane of the torus with an orthogonal transformation in the remaining dimension.
\[
\begin{aligned}
(\rho, \theta) = f(R, Z) \\
\phi = - \zeta
\end{aligned}
\]
where \(f\) is a 2D curvilinear to cartesian transformation.
\[
J (R, Z, \zeta) =
\begin{bmatrix}
J_f\.^{\rho}_{\;R} & J_f\.^{\rho}_{\;Z} & 0 \\
J_f\.^{\theta}_{\;R} & 0 & J_f\.^{\theta}_{\;Z} & 0
0 & 0 & -1 \\
\end{bmatrix}.
\]
Combined coordinate transformation which combines two of the coordinate transformations above
The tools are:
- InverseJacobianMatrix : this tool calculates the inverse Jacobian matrix on the specified coordinate system.
- InvJacobianOPoint : this tool calculates the inverse Jacobian matrix at the O-point on the specified coordinate system.
- MetricTensorEvaluator : this tool calculates the metric tensor associated with a coordinate transformation.
- VectorMapper : this tool helps when converting vectors stored in a
VectorFieldfrom one coordinate system to another. - other static analysis tools found in
mapping_tools.hpp