Oblate Homoeoidal Coordinates: Effect of Eccentricity

Visualize how the radial coordinate lines (R-contours) in the OblateHomoeoidalCoordinateSystem change as a function of eccentricity.

import matplotlib.pyplot as plt

In this example, we’ll showcase a more complex coordinate system: OblateHomoeoidalCoordinateSystem.

The oblate homoeoidal coordinate system is a fully curvilinear coordinate system (non-diagonal metric) which inherits the angular coordinates of spherical coordinates but instead relies on a modified effective radius composed of con-centric ellipsoids.

In this example, we’ll show off how to make these coordinate systems and then display some of its properties.

import numpy as np

from pymetric import OblateHomoeoidalCoordinateSystem

To visualize, we’ll use an x-z grid and then convert it into the relevant coordinate systems with different eccentricities. We can then plot contours for the effective radius \(\xi\).

# Create the cartesian grid.
x = np.linspace(-1.5, 1.5, 200)
z = np.linspace(-1.5, 1.5, 200)
X, Z = np.meshgrid(x, z)

The class requires an eccentricity parameter when initialized, so we’ll iterate through each of them, create a coordinate system class, and then perform the conversions.

eccentricities = [0.0, 0.1, 0.3, 0.6, 0.9, 0.99]

fig, axes = plt.subplots(
    int(np.ceil(len(eccentricities) / 3)), 3, sharex=True, sharey=True
)

for i, ecc in enumerate(eccentricities):
    ax = axes.ravel()[i]
    csys = OblateHomoeoidalCoordinateSystem(ecc=ecc)

    # Convert Cartesian (x, z) to native coordinates (λ, μ, φ)
    Lambda, Mu, Phi = csys.from_cartesian(X, 0, Z)

    # Plot λ contours (these correspond to elliptical shells)
    contour = ax.contour(X, Z, Lambda, levels=15, cmap="viridis")
    ax.set_title(f"$\\varepsilon = {ecc}$")
    ax.set_aspect("equal")
    ax.set_xlabel("x")
    if i == 0:
        ax.set_ylabel("z")

fig.suptitle(r"Oblate Homoeoidal Coordinate Contours ($\lambda$-lines)", fontsize=14)
plt.tight_layout()
plt.show()