r"""
.. _shell2-spatial-model:

Shell2 spatial model
====================

This is a spatial model parametrizing a projected radiating shell.

The shell spatial model is defined by the following equations:

.. math::
    \phi(lon, lat) = \frac{3}{2 \pi (r_{out}^3 - r_{in}^3)} \cdot
            \begin{cases}
                \sqrt{r_{out}^2 - \theta^2} - \sqrt{r_{in}^2 - \theta^2} &
                             \text{for } \theta \lt r_{in} \\
                \sqrt{r_{out}^2 - \theta^2} &
                             \text{for } r_{in} \leq \theta \lt r_{out} \\
                0 & \text{for } \theta > r_{out}
            \end{cases}

where :math:`\theta` is the sky separation, :math:`r_{\text{out}}` is the outer radius
and  :math:`r_{\text{in}}` is the inner radius.

For Shell2SpatialModel, the radius parameter  r_0 correspond to :math:`r_{\text{out}}`.
The relative width parameter, eta, is given as \eta = :math:`(r_{\text{out}} - r_{\text{in}})/r_{\text{out}}`
so we have :math:`r_{\text{in}} = (1-\eta) r_{\text{out}}`.

Note that the normalization is a small angle approximation,
although that approximation is still very good even for 10 deg radius shells.


"""

# %%
# Example plot
# ------------
# Here is an example plot of the shell model for the parametrization using outer radius and relative width.
# In this case the relative width, eta, acts as a shape parameter.

import matplotlib.pyplot as plt
from gammapy.modeling.models import (
    Models,
    PowerLawSpectralModel,
    Shell2SpatialModel,
    SkyModel,
)

tags = [
    r"Disk-like, $\eta \rightarrow 0$",
    r"Shell, $\eta=0.25$",
    r"Peaked, $\eta\rightarrow 1$",
]
eta_range = [0.001, 0.25, 1]
fig, axes = plt.subplots(1, 3, figsize=(9, 6))
for ax, eta, tag in zip(axes, eta_range, tags):
    model = Shell2SpatialModel(
        lon_0="10 deg",
        lat_0="20 deg",
        r_0="2 deg",
        eta=eta,
        frame="galactic",
    )
    model.plot(ax=ax)
    ax.set_title(tag)
    ax.set_xticks([])
    ax.set_yticks([])
plt.tight_layout()

# %%
# YAML representation
# -------------------
# Here is an example YAML file using the model:

pwl = PowerLawSpectralModel()
shell2 = Shell2SpatialModel()

model = SkyModel(spectral_model=pwl, spatial_model=shell2, name="pwl-shell2-model")

models = Models([model])

print(models.to_yaml())