Geometry

Defining curves

The effective use of the Boundary Wall Method (BWM) requires equal-length segments. and the need to interpolate uneven curves. While one might have an equispaced \(x-y\) domain, the arc length \(\mathrm{d}s\) need not be homogeneous.

The general steps for using this toolbox with a custom array of points are the following:

1. Define \((x_i,y_i)\)
   • (Re)discretize using

For closed curves, we implement the discretizeCurve function in order to interpolate the whole arc length into \(N\) equal segments. For open curves, one must use discretizeResonatorCurve (it can be used for closed curves as well, but be mindful of using \(N\) vs. \(N+1\) points). Interpolating with Interpolations.jl will work as well.

Once one has a relatively accurate arc length discretization, one must calculate midpoints using the calcMidpoints function. Note that the total number of midpoints must strictly match the dimensions of the \(\mathbf{T}\) matrix.

Finally, one might want to obtain the distance matrix representing \([\mathbf{r}]_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}\). The following example

using BoundaryWall, WGLMakie # hide
θ     = LinRange(-pi, pi/2, 300)
x,y   = cos.(θ),sin.(θ)

x,y   = divideResonatorCurve(x,y, 300)
xm,ym = calcMidpoints(x, y)
ds    = calcArcLength(x,y)
rij   = calcDistances(xm, ym)
lines(ds)

A good way to check if one has done the discretization correctly is to check for the symmetry of the scattering \(\mathbf{T}\) matrix, or to graph diff(ds) and check for homogeneous behaviour.

Builtin methods

The package provides some useful functions such as createEllipse and createCircle in order to create analogues to rods.