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# Source code for astropy.modeling.functional_models

    # Licensed under a 3-clause BSD style license - see LICENSE.rst

    """Mathematical models."""

    # pylint: disable=line-too-long, too-many-lines, too-many-arguments, invalid-name
    import warnings

    import numpy as np

    from astropy import units as u
    from astropy.units import Quantity, UnitsError
    from astropy.utils.compat.optional_deps import HAS_SCIPY
    from astropy.utils.exceptions import AstropyDeprecationWarning

    from .core import Fittable1DModel, Fittable2DModel
    from .parameters import InputParameterError, Parameter
    from .utils import ellipse_extent

    __all__ = [
        "AiryDisk2D",
        "ArcCosine1D",
        "ArcSine1D",
        "ArcTangent1D",
        "Box1D",
        "Box2D",
        "Const1D",
        "Const2D",
        "Cosine1D",
        "Disk2D",
        "Ellipse2D",
        "Exponential1D",
        "Gaussian1D",
        "Gaussian2D",
        "GeneralSersic2D",
        "KingProjectedAnalytic1D",
        "Linear1D",
        "Logarithmic1D",
        "Lorentz1D",
        "Lorentz2D",
        "Moffat1D",
        "Moffat2D",
        "Multiply",
        "Planar2D",
        "RedshiftScaleFactor",
        "RickerWavelet1D",
        "RickerWavelet2D",
        "Ring2D",
        "Scale",
        "Sersic1D",
        "Sersic2D",
        "Shift",
        "Sine1D",
        "Tangent1D",
        "Trapezoid1D",
        "TrapezoidDisk2D",
        "Voigt1D",
    ]

    TWOPI = 2 * np.pi
    FLOAT_EPSILON = float(np.finfo(np.float32).tiny)

    # Note that we define this here rather than using the value defined in
    # astropy.stats to avoid importing astropy.stats every time astropy.modeling
    # is loaded.
    GAUSSIAN_SIGMA_TO_FWHM = 2.0 * np.sqrt(2.0 * np.log(2.0))

    class Gaussian1D(Fittable1DModel):
        """
        One dimensional Gaussian model.

        Parameters
        ----------
        amplitude : float or `~astropy.units.Quantity`.
            Amplitude (peak value) of the Gaussian - for a normalized profile
            (integrating to 1), set amplitude = 1 / (stddev * np.sqrt(2 * np.pi))
        mean : float or `~astropy.units.Quantity`.
            Mean of the Gaussian.
        stddev : float or `~astropy.units.Quantity`.
            Standard deviation of the Gaussian with FWHM = 2 * stddev * np.sqrt(2 * np.log(2)).

        Notes
        -----
        The ``x``, ``mean``, and ``stddev`` inputs must have compatible
        units or be unitless numbers.

        Model formula:

            .. math:: f(x) = A e^{- \\frac{\\left(x - x_{0}\\right)^{2}}{2 \\sigma^{2}}}

        Examples
        --------
        >>> from astropy.modeling import models
        >>> def tie_center(model):
        ...         mean = 50 * model.stddev
        ...         return mean
        >>> tied_parameters = {'mean': tie_center}

        Specify that 'mean' is a tied parameter in one of two ways:

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3,
        ...                             tied=tied_parameters)

        or

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3)
        >>> g1.mean.tied
        False
        >>> g1.mean.tied = tie_center
        >>> g1.mean.tied
        <function tie_center at 0x...>

        Fixed parameters:

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3,
        ...                        fixed={'stddev': True})
        >>> g1.stddev.fixed
        True

        or

        >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3)
        >>> g1.stddev.fixed
        False
        >>> g1.stddev.fixed = True
        >>> g1.stddev.fixed
        True

        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Gaussian1D

            plt.figure()
            s1 = Gaussian1D()
            r = np.arange(-5, 5, .01)

            for factor in range(1, 4):
                s1.amplitude = factor
                plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

            plt.axis([-5, 5, -1, 4])
            plt.show()

        See Also
        --------
        Gaussian2D, Box1D, Moffat1D, Lorentz1D
        """

        amplitude = Parameter(
            default=1, description="Amplitude (peak value) of the Gaussian"
        )
        mean = Parameter(default=0, description="Position of peak (Gaussian)")

        # Ensure stddev makes sense if its bounds are not explicitly set.
        # stddev must be non-zero and positive.
        stddev = Parameter(
            default=1,
            bounds=(FLOAT_EPSILON, None),
            description="Standard deviation of the Gaussian",
        )

        def bounding_box(self, factor=5.5):
            """
            Tuple defining the default ``bounding_box`` limits,
            ``(x_low, x_high)``.

            Parameters
            ----------
            factor : float
                The multiple of `stddev` used to define the limits.
                The default is 5.5, corresponding to a relative error < 1e-7.

            Examples
            --------
            >>> from astropy.modeling.models import Gaussian1D
            >>> model = Gaussian1D(mean=0, stddev=2)
            >>> model.bounding_box
            ModelBoundingBox(
                intervals={
                    x: Interval(lower=-11.0, upper=11.0)
                }
                model=Gaussian1D(inputs=('x',))
                order='C'
            )

            This range can be set directly (see: `Model.bounding_box
            <astropy.modeling.Model.bounding_box>`) or by using a different factor,
            like:

            >>> model.bounding_box = model.bounding_box(factor=2)
            >>> model.bounding_box
            ModelBoundingBox(
                intervals={
                    x: Interval(lower=-4.0, upper=4.0)
                }
                model=Gaussian1D(inputs=('x',))
                order='C'
            )
            """
            x0 = self.mean
            dx = factor * self.stddev

            return (x0 - dx, x0 + dx)

        @property
        def fwhm(self):
            """Gaussian full width at half maximum."""
            return self.stddev * GAUSSIAN_SIGMA_TO_FWHM

        @staticmethod
        def evaluate(x, amplitude, mean, stddev):
            """
            Gaussian1D model function.
            """
            return amplitude * np.exp(-0.5 * (x - mean) ** 2 / stddev**2)

        @staticmethod
        def fit_deriv(x, amplitude, mean, stddev):
            """
            Gaussian1D model function derivatives.
            """
            d_amplitude = np.exp(-0.5 / stddev**2 * (x - mean) ** 2)
            d_mean = amplitude * d_amplitude * (x - mean) / stddev**2
            d_stddev = amplitude * d_amplitude * (x - mean) ** 2 / stddev**3
            return [d_amplitude, d_mean, d_stddev]

        @property
        def input_units(self):
            if self.mean.input_unit is None:
                return None
            return {self.inputs[0]: self.mean.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "mean": inputs_unit[self.inputs[0]],
                "stddev": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Gaussian2D(Fittable2DModel):
        r"""
        Two dimensional Gaussian model.

        Parameters
        ----------
        amplitude : float or `~astropy.units.Quantity`.
            Amplitude (peak value) of the Gaussian.
        x_mean : float or `~astropy.units.Quantity`.
            Mean of the Gaussian in x.
        y_mean : float or `~astropy.units.Quantity`.
            Mean of the Gaussian in y.
        x_stddev : float or `~astropy.units.Quantity` or None.
            Standard deviation of the Gaussian in x before rotating by theta. Must
            be None if a covariance matrix (``cov_matrix``) is provided. If no
            ``cov_matrix`` is given, ``None`` means the default value (1).
        y_stddev : float or `~astropy.units.Quantity` or None.
            Standard deviation of the Gaussian in y before rotating by theta. Must
            be None if a covariance matrix (``cov_matrix``) is provided. If no
            ``cov_matrix`` is given, ``None`` means the default value (1).
        theta : float or `~astropy.units.Quantity`, optional.
            The rotation angle as an angular quantity
            (`~astropy.units.Quantity` or `~astropy.coordinates.Angle`)
            or a value in radians (as a float). The rotation angle
            increases counterclockwise. Must be `None` if a covariance matrix
            (``cov_matrix``) is provided. If no ``cov_matrix`` is given,
            `None` means the default value (0).
        cov_matrix : ndarray, optional
            A 2x2 covariance matrix. If specified, overrides the ``x_stddev``,
            ``y_stddev``, and ``theta`` defaults.

        Notes
        -----
        The ``x, y``, ``[x,y]_mean``, and ``[x,y]_stddev`` inputs must have
        compatible units or be unitless numbers.

        Model formula:

            .. math::

                f(x, y) = A e^{-a\left(x - x_{0}\right)^{2}  -b\left(x - x_{0}\right)
                \left(y - y_{0}\right)  -c\left(y - y_{0}\right)^{2}}

        Using the following definitions:

            .. math::
                a = \left(\frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} +
                \frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right)

                b = \left(\frac{\sin{\left (2 \theta \right )}}{2 \sigma_{x}^{2}} -
                \frac{\sin{\left (2 \theta \right )}}{2 \sigma_{y}^{2}}\right)

                c = \left(\frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} +
                \frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right)

        If using a ``cov_matrix``, the model is of the form:
            .. math::
                f(x, y) = A e^{-0.5 \left(
                        \vec{x} - \vec{x}_{0}\right)^{T} \Sigma^{-1} \left(\vec{x} - \vec{x}_{0}
                    \right)}

        where :math:`\vec{x} = [x, y]`, :math:`\vec{x}_{0} = [x_{0}, y_{0}]`,
        and :math:`\Sigma` is the covariance matrix:

            .. math::
                \Sigma = \left(\begin{array}{ccc}
                \sigma_x^2               & \rho \sigma_x \sigma_y \\
                \rho \sigma_x \sigma_y   & \sigma_y^2
                \end{array}\right)

        :math:`\rho` is the correlation between ``x`` and ``y``, which should
        be between -1 and +1.  Positive correlation corresponds to a
        ``theta`` in the range 0 to 90 degrees.  Negative correlation
        corresponds to a ``theta`` in the range of 0 to -90 degrees.

        See [1]_ for more details about the 2D Gaussian function.

        See Also
        --------
        Gaussian1D, Box2D, Moffat2D

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Gaussian_function
        """

        amplitude = Parameter(default=1, description="Amplitude of the Gaussian")
        x_mean = Parameter(
            default=0, description="Peak position (along x axis) of Gaussian"
        )
        y_mean = Parameter(
            default=0, description="Peak position (along y axis) of Gaussian"
        )
        x_stddev = Parameter(
            default=1, description="Standard deviation of the Gaussian (along x axis)"
        )
        y_stddev = Parameter(
            default=1, description="Standard deviation of the Gaussian (along y axis)"
        )
        theta = Parameter(
            default=0.0,
            description=(
                "Rotation angle either as a "
                "float (in radians) or a "
                "|Quantity| angle (optional)"
            ),
        )

        def __init__(
            self,
            amplitude=amplitude.default,
            x_mean=x_mean.default,
            y_mean=y_mean.default,
            x_stddev=None,
            y_stddev=None,
            theta=None,
            cov_matrix=None,
            **kwargs,
        ):
            if cov_matrix is None:
                if x_stddev is None:
                    x_stddev = self.__class__.x_stddev.default
                if y_stddev is None:
                    y_stddev = self.__class__.y_stddev.default
                if theta is None:
                    theta = self.__class__.theta.default
            else:
                if x_stddev is not None or y_stddev is not None or theta is not None:
                    raise InputParameterError(
                        "Cannot specify both cov_matrix and x/y_stddev/theta"
                    )
                # Compute principle coordinate system transformation
                cov_matrix = np.array(cov_matrix)

                if cov_matrix.shape != (2, 2):
                    raise ValueError("Covariance matrix must be 2x2")

                eig_vals, eig_vecs = np.linalg.eig(cov_matrix)
                x_stddev, y_stddev = np.sqrt(eig_vals)
                y_vec = eig_vecs[:, 0]
                theta = np.arctan2(y_vec[1], y_vec[0])

            # Ensure stddev makes sense if its bounds are not explicitly set.
            # stddev must be non-zero and positive.
            # TODO: Investigate why setting this in Parameter above causes
            #       convolution tests to hang.
            kwargs.setdefault("bounds", {})
            kwargs["bounds"].setdefault("x_stddev", (FLOAT_EPSILON, None))
            kwargs["bounds"].setdefault("y_stddev", (FLOAT_EPSILON, None))

            super().__init__(
                amplitude=amplitude,
                x_mean=x_mean,
                y_mean=y_mean,
                x_stddev=x_stddev,
                y_stddev=y_stddev,
                theta=theta,
                **kwargs,
            )

        @property
        def x_fwhm(self):
            """Gaussian full width at half maximum in X."""
            return self.x_stddev * GAUSSIAN_SIGMA_TO_FWHM

        @property
        def y_fwhm(self):
            """Gaussian full width at half maximum in Y."""
            return self.y_stddev * GAUSSIAN_SIGMA_TO_FWHM

        def bounding_box(self, factor=5.5):
            """
            Tuple defining the default ``bounding_box`` limits in each dimension,
            ``((y_low, y_high), (x_low, x_high))``.

            The default offset from the mean is 5.5-sigma, corresponding
            to a relative error < 1e-7. The limits are adjusted for rotation.

            Parameters
            ----------
            factor : float, optional
                The multiple of `x_stddev` and `y_stddev` used to define the limits.
                The default is 5.5.

            Examples
            --------
            >>> from astropy.modeling.models import Gaussian2D
            >>> model = Gaussian2D(x_mean=0, y_mean=0, x_stddev=1, y_stddev=2)
            >>> model.bounding_box
            ModelBoundingBox(
                intervals={
                    x: Interval(lower=-5.5, upper=5.5)
                    y: Interval(lower=-11.0, upper=11.0)
                }
                model=Gaussian2D(inputs=('x', 'y'))
                order='C'
            )

            This range can be set directly (see: `Model.bounding_box
            <astropy.modeling.Model.bounding_box>`) or by using a different factor
            like:

            >>> model.bounding_box = model.bounding_box(factor=2)
            >>> model.bounding_box
            ModelBoundingBox(
                intervals={
                    x: Interval(lower=-2.0, upper=2.0)
                    y: Interval(lower=-4.0, upper=4.0)
                }
                model=Gaussian2D(inputs=('x', 'y'))
                order='C'
            )
            """
            a = factor * self.x_stddev
            b = factor * self.y_stddev
            dx, dy = ellipse_extent(a, b, self.theta)

            return (
                (self.y_mean - dy, self.y_mean + dy),
                (self.x_mean - dx, self.x_mean + dx),
            )

        @staticmethod
        def evaluate(x, y, amplitude, x_mean, y_mean, x_stddev, y_stddev, theta):
            """Two dimensional Gaussian function."""
            cost2 = np.cos(theta) ** 2
            sint2 = np.sin(theta) ** 2
            sin2t = np.sin(2.0 * theta)
            xstd2 = x_stddev**2
            ystd2 = y_stddev**2
            xdiff = x - x_mean
            ydiff = y - y_mean
            a = 0.5 * ((cost2 / xstd2) + (sint2 / ystd2))
            b = 0.5 * ((sin2t / xstd2) - (sin2t / ystd2))
            c = 0.5 * ((sint2 / xstd2) + (cost2 / ystd2))
            return amplitude * np.exp(
                -((a * xdiff**2) + (b * xdiff * ydiff) + (c * ydiff**2))
            )

        @staticmethod
        def fit_deriv(x, y, amplitude, x_mean, y_mean, x_stddev, y_stddev, theta):
            """Two dimensional Gaussian function derivative with respect to parameters."""
            cost = np.cos(theta)
            sint = np.sin(theta)
            cost2 = np.cos(theta) ** 2
            sint2 = np.sin(theta) ** 2
            cos2t = np.cos(2.0 * theta)
            sin2t = np.sin(2.0 * theta)
            xstd2 = x_stddev**2
            ystd2 = y_stddev**2
            xstd3 = x_stddev**3
            ystd3 = y_stddev**3
            xdiff = x - x_mean
            ydiff = y - y_mean
            xdiff2 = xdiff**2
            ydiff2 = ydiff**2
            a = 0.5 * ((cost2 / xstd2) + (sint2 / ystd2))
            b = 0.5 * ((sin2t / xstd2) - (sin2t / ystd2))
            c = 0.5 * ((sint2 / xstd2) + (cost2 / ystd2))
            g = amplitude * np.exp(-((a * xdiff2) + (b * xdiff * ydiff) + (c * ydiff2)))
            da_dtheta = sint * cost * ((1.0 / ystd2) - (1.0 / xstd2))
            da_dx_stddev = -cost2 / xstd3
            da_dy_stddev = -sint2 / ystd3
            db_dtheta = (cos2t / xstd2) - (cos2t / ystd2)
            db_dx_stddev = -sin2t / xstd3
            db_dy_stddev = sin2t / ystd3
            dc_dtheta = -da_dtheta
            dc_dx_stddev = -sint2 / xstd3
            dc_dy_stddev = -cost2 / ystd3
            dg_dA = g / amplitude
            dg_dx_mean = g * ((2.0 * a * xdiff) + (b * ydiff))
            dg_dy_mean = g * ((b * xdiff) + (2.0 * c * ydiff))
            dg_dx_stddev = g * (
                -(
                    da_dx_stddev * xdiff2
                    + db_dx_stddev * xdiff * ydiff
                    + dc_dx_stddev * ydiff2
                )
            )
            dg_dy_stddev = g * (
                -(
                    da_dy_stddev * xdiff2
                    + db_dy_stddev * xdiff * ydiff
                    + dc_dy_stddev * ydiff2
                )
            )
            dg_dtheta = g * (
                -(da_dtheta * xdiff2 + db_dtheta * xdiff * ydiff + dc_dtheta * ydiff2)
            )
            return [dg_dA, dg_dx_mean, dg_dy_mean, dg_dx_stddev, dg_dy_stddev, dg_dtheta]

        @property
        def input_units(self):
            x_unit = self.x_mean.input_unit
            y_unit = self.y_mean.input_unit

            if x_unit is None and y_unit is None:
                return None

            return {self.inputs[0]: x_unit, self.inputs[1]: y_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_mean": inputs_unit[self.inputs[0]],
                "y_mean": inputs_unit[self.inputs[0]],
                "x_stddev": inputs_unit[self.inputs[0]],
                "y_stddev": inputs_unit[self.inputs[0]],
                "theta": u.rad,
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Shift(Fittable1DModel):
        """
        Shift a coordinate.

        Parameters
        ----------
        offset : float
            Offset to add to a coordinate.
        """

        offset = Parameter(default=0, description="Offset to add to a model")
        linear = True

        _has_inverse_bounding_box = True

        @property
        def input_units(self):
            if self.offset.input_unit is None:
                return None
            return {self.inputs[0]: self.offset.input_unit}

        @property
        def inverse(self):
            """One dimensional inverse Shift model function."""
            inv = self.copy()
            inv.offset *= -1

            try:
                self.bounding_box  # noqa: B018
            except NotImplementedError:
                pass
            else:
                inv.bounding_box = tuple(
                    self.evaluate(x, self.offset) for x in self.bounding_box
                )

            return inv

        @staticmethod
        def evaluate(x, offset):
            """One dimensional Shift model function."""
            return x + offset

        @staticmethod
        def sum_of_implicit_terms(x):
            """Evaluate the implicit term (x) of one dimensional Shift model."""
            return x

        @staticmethod
        def fit_deriv(x, *params):
            """One dimensional Shift model derivative with respect to parameter."""
            d_offset = np.ones_like(x)
            return [d_offset]

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {"offset": outputs_unit[self.outputs[0]]}

    class Scale(Fittable1DModel):
        """
        Multiply a model by a dimensionless factor.

        Parameters
        ----------
        factor : float
            Factor by which to scale a coordinate.

        Notes
        -----
        If ``factor`` is a `~astropy.units.Quantity` then the units will be
        stripped before the scaling operation.

        """

        factor = Parameter(default=1, description="Factor by which to scale a model")
        linear = True
        fittable = True

        _input_units_strict = True
        _input_units_allow_dimensionless = True

        _has_inverse_bounding_box = True

        @property
        def input_units(self):
            if self.factor.input_unit is None:
                return None
            return {self.inputs[0]: self.factor.input_unit}

        @property
        def inverse(self):
            """One dimensional inverse Scale model function."""
            inv = self.copy()
            inv.factor = 1 / self.factor

            try:
                self.bounding_box  # noqa: B018
            except NotImplementedError:
                pass
            else:
                inv.bounding_box = tuple(
                    self.evaluate(x, self.factor) for x in self.bounding_box.bounding_box()
                )

            return inv

        @staticmethod
        def evaluate(x, factor):
            """One dimensional Scale model function."""
            if isinstance(factor, u.Quantity):
                factor = factor.value

            return factor * x

        @staticmethod
        def fit_deriv(x, *params):
            """One dimensional Scale model derivative with respect to parameter."""
            d_factor = x
            return [d_factor]

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {"factor": outputs_unit[self.outputs[0]]}

    class Multiply(Fittable1DModel):
        """
        Multiply a model by a quantity or number.

        Parameters
        ----------
        factor : float
            Factor by which to multiply a coordinate.
        """

        factor = Parameter(default=1, description="Factor by which to multiply a model")
        linear = True
        fittable = True

        _has_inverse_bounding_box = True

        @property
        def inverse(self):
            """One dimensional inverse multiply model function."""
            inv = self.copy()
            inv.factor = 1 / self.factor

            try:
                self.bounding_box  # noqa: B018
            except NotImplementedError:
                pass
            else:
                inv.bounding_box = tuple(
                    self.evaluate(x, self.factor) for x in self.bounding_box.bounding_box()
                )

            return inv

        @staticmethod
        def evaluate(x, factor):
            """One dimensional multiply model function."""
            return factor * x

        @staticmethod
        def fit_deriv(x, *params):
            """One dimensional multiply model derivative with respect to parameter."""
            d_factor = x
            return [d_factor]

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {"factor": outputs_unit[self.outputs[0]]}

    class RedshiftScaleFactor(Fittable1DModel):
        """
        One dimensional redshift scale factor model.

        Parameters
        ----------
        z : float
            Redshift value.

        Notes
        -----
        Model formula:

            .. math:: f(x) = x (1 + z)
        """

        z = Parameter(description="Redshift", default=0)

        _has_inverse_bounding_box = True

        @staticmethod
        def evaluate(x, z):
            """One dimensional RedshiftScaleFactor model function."""
            return (1 + z) * x

        @staticmethod
        def fit_deriv(x, z):
            """One dimensional RedshiftScaleFactor model derivative."""
            d_z = x
            return [d_z]

        @property
        def inverse(self):
            """Inverse RedshiftScaleFactor model."""
            inv = self.copy()
            inv.z = 1.0 / (1.0 + self.z) - 1.0

            try:
                self.bounding_box  # noqa: B018
            except NotImplementedError:
                pass
            else:
                inv.bounding_box = tuple(
                    self.evaluate(x, self.z) for x in self.bounding_box.bounding_box()
                )

            return inv

    class Sersic1D(Fittable1DModel):
        r"""
        One dimensional Sersic surface brightness profile.

        Parameters
        ----------
        amplitude : float
            Surface brightness at ``r_eff``.
        r_eff : float
            Effective (half-light) radius.
        n : float
            Sersic index controlling the shape of the profile. Particular
            values of ``n`` are equivalent to the following profiles:

                * n=4 : `de Vaucouleurs <https://en.wikipedia.org/wiki/De_Vaucouleurs%27s_law>`_ :math:`r^{1/4}` profile
                * n=1 : Exponential profile
                * n=0.5 : Gaussian profile

        See Also
        --------
        Gaussian1D, Moffat1D, Lorentz1D

        Notes
        -----
        The ``r`` and ``r_eff`` inputs must have compatible units or be
        unitless numbers.

        Model formula:

        .. math::

            I(r) = I_{e} \exp\left\{
                   -b_{n} \left[\left(\frac{r}{r_{e}}\right)^{(1/n)}
                   -1\right]\right\}

        where :math:`I_{e}` is the ``amplitude`` and :math:`r_{e}` is ``reff``.

        The constant :math:`b_{n}` is defined such that :math:`r_{e}`
        contains half the total luminosity. It can be solved for numerically
        from the following equation:

        .. math::

            \Gamma(2n) = 2\gamma (2n, b_{n})

        where :math:`\Gamma(a)` is the `gamma function
        <https://en.wikipedia.org/wiki/Gamma_function>`_ and
        :math:`\gamma(a, x)` is the `lower incomplete gamma function
        <https://en.wikipedia.org/wiki/Incomplete_gamma_function>`_.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            from astropy.modeling.models import Sersic1D
            import matplotlib.pyplot as plt

            plt.figure()
            plt.subplot(111, xscale='log', yscale='log')
            s1 = Sersic1D(amplitude=1, r_eff=5)
            r = np.arange(0, 100, 0.01)

            for n in range(1, 10):
                 s1.n = n
                 plt.plot(r, s1(r))

            plt.axis([1e-1, 30, 1e-2, 1e3])
            plt.xlabel('log Radius')
            plt.ylabel('log Surface Brightness')
            plt.text(0.25, 1.5, 'n=1')
            plt.text(0.25, 300, 'n=10')
            plt.xticks([])
            plt.yticks([])
            plt.show()

        References
        ----------
        .. [1] http://ned.ipac.caltech.edu/level5/March05/Graham/Graham2.html
        """

        amplitude = Parameter(default=1, description="Surface brightness at r_eff")
        r_eff = Parameter(default=1, description="Effective (half-light) radius")
        n = Parameter(default=4, description="Sersic index")
        _gammaincinv = None

        @classmethod
        def evaluate(cls, r, amplitude, r_eff, n):
            """One dimensional Sersic profile function."""
            if cls._gammaincinv is None:
                from scipy.special import gammaincinv

                cls._gammaincinv = gammaincinv

            return amplitude * np.exp(
                -cls._gammaincinv(2 * n, 0.5) * ((r / r_eff) ** (1 / n) - 1)
            )

        @property
        def input_units(self):
            if self.r_eff.input_unit is None:
                return None
            return {self.inputs[0]: self.r_eff.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "r_eff": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class _Trigonometric1D(Fittable1DModel):
        """
        Base class for one dimensional trigonometric and inverse trigonometric models.

        Parameters
        ----------
        amplitude : float
            Oscillation amplitude
        frequency : float
            Oscillation frequency
        phase : float
            Oscillation phase
        """

        amplitude = Parameter(default=1, description="Oscillation amplitude")
        frequency = Parameter(default=1, description="Oscillation frequency")
        phase = Parameter(default=0, description="Oscillation phase")

        @property
        def input_units(self):
            if self.frequency.input_unit is None:
                return None
            return {self.inputs[0]: 1.0 / self.frequency.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "frequency": inputs_unit[self.inputs[0]] ** -1,
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Sine1D(_Trigonometric1D):
        """
        One dimensional Sine model.

        Parameters
        ----------
        amplitude : float
            Oscillation amplitude
        frequency : float
            Oscillation frequency
        phase : float
            Oscillation phase

        See Also
        --------
        ArcSine1D, Cosine1D, Tangent1D, Const1D, Linear1D

        Notes
        -----
        Model formula:

            .. math:: f(x) = A \\sin(2 \\pi f x + 2 \\pi p)

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Sine1D

            plt.figure()
            s1 = Sine1D(amplitude=1, frequency=.25)
            r=np.arange(0, 10, .01)

            for amplitude in range(1,4):
                 s1.amplitude = amplitude
                 plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)

            plt.axis([0, 10, -5, 5])
            plt.show()
        """

        @staticmethod
        def evaluate(x, amplitude, frequency, phase):
            """One dimensional Sine model function."""
            # Note: If frequency and x are quantities, they should normally have
            # inverse units, so that argument ends up being dimensionless. However,
            # np.sin of a dimensionless quantity will crash, so we remove the
            # quantity-ness from argument in this case (another option would be to
            # multiply by * u.rad but this would be slower overall).
            argument = TWOPI * (frequency * x + phase)
            if isinstance(argument, Quantity):
                argument = argument.value
            return amplitude * np.sin(argument)

        @staticmethod
        def fit_deriv(x, amplitude, frequency, phase):
            """One dimensional Sine model derivative."""
            d_amplitude = np.sin(TWOPI * frequency * x + TWOPI * phase)
            d_frequency = (
                TWOPI * x * amplitude * np.cos(TWOPI * frequency * x + TWOPI * phase)
            )
            d_phase = TWOPI * amplitude * np.cos(TWOPI * frequency * x + TWOPI * phase)
            return [d_amplitude, d_frequency, d_phase]

        @property
        def inverse(self):
            """One dimensional inverse of Sine."""
            return ArcSine1D(
                amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
            )

    class Cosine1D(_Trigonometric1D):
        """
        One dimensional Cosine model.

        Parameters
        ----------
        amplitude : float
            Oscillation amplitude
        frequency : float
            Oscillation frequency
        phase : float
            Oscillation phase

        See Also
        --------
        ArcCosine1D, Sine1D, Tangent1D, Const1D, Linear1D

        Notes
        -----
        Model formula:

            .. math:: f(x) = A \\cos(2 \\pi f x + 2 \\pi p)

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Cosine1D

            plt.figure()
            s1 = Cosine1D(amplitude=1, frequency=.25)
            r=np.arange(0, 10, .01)

            for amplitude in range(1,4):
                 s1.amplitude = amplitude
                 plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)

            plt.axis([0, 10, -5, 5])
            plt.show()
        """

        @staticmethod
        def evaluate(x, amplitude, frequency, phase):
            """One dimensional Cosine model function."""
            # Note: If frequency and x are quantities, they should normally have
            # inverse units, so that argument ends up being dimensionless. However,
            # np.sin of a dimensionless quantity will crash, so we remove the
            # quantity-ness from argument in this case (another option would be to
            # multiply by * u.rad but this would be slower overall).
            argument = TWOPI * (frequency * x + phase)
            if isinstance(argument, Quantity):
                argument = argument.value
            return amplitude * np.cos(argument)

        @staticmethod
        def fit_deriv(x, amplitude, frequency, phase):
            """One dimensional Cosine model derivative."""
            d_amplitude = np.cos(TWOPI * frequency * x + TWOPI * phase)
            d_frequency = -(
                TWOPI * x * amplitude * np.sin(TWOPI * frequency * x + TWOPI * phase)
            )
            d_phase = -(TWOPI * amplitude * np.sin(TWOPI * frequency * x + TWOPI * phase))
            return [d_amplitude, d_frequency, d_phase]

        @property
        def inverse(self):
            """One dimensional inverse of Cosine."""
            return ArcCosine1D(
                amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
            )

    class Tangent1D(_Trigonometric1D):
        """
        One dimensional Tangent model.

        Parameters
        ----------
        amplitude : float
            Oscillation amplitude
        frequency : float
            Oscillation frequency
        phase : float
            Oscillation phase

        See Also
        --------
        Sine1D, Cosine1D, Const1D, Linear1D

        Notes
        -----
        Model formula:

            .. math:: f(x) = A \\tan(2 \\pi f x + 2 \\pi p)

        Note that the tangent function is undefined for inputs of the form
        pi/2 + n*pi for all integers n. Thus thus the default bounding box
        has been restricted to:

            .. math:: [(-1/4 - p)/f, (1/4 - p)/f]

        which is the smallest interval for the tangent function to be continuous
        on.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Tangent1D

            plt.figure()
            s1 = Tangent1D(amplitude=1, frequency=.25)
            r=np.arange(0, 10, .01)

            for amplitude in range(1,4):
                 s1.amplitude = amplitude
                 plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)

            plt.axis([0, 10, -5, 5])
            plt.show()
        """

        @staticmethod
        def evaluate(x, amplitude, frequency, phase):
            """One dimensional Tangent model function."""
            # Note: If frequency and x are quantities, they should normally have
            # inverse units, so that argument ends up being dimensionless. However,
            # np.sin of a dimensionless quantity will crash, so we remove the
            # quantity-ness from argument in this case (another option would be to
            # multiply by * u.rad but this would be slower overall).
            argument = TWOPI * (frequency * x + phase)
            if isinstance(argument, Quantity):
                argument = argument.value
            return amplitude * np.tan(argument)

        @staticmethod
        def fit_deriv(x, amplitude, frequency, phase):
            """One dimensional Tangent model derivative."""
            sec = 1 / (np.cos(TWOPI * frequency * x + TWOPI * phase)) ** 2

            d_amplitude = np.tan(TWOPI * frequency * x + TWOPI * phase)
            d_frequency = TWOPI * x * amplitude * sec
            d_phase = TWOPI * amplitude * sec
            return [d_amplitude, d_frequency, d_phase]

        @property
        def inverse(self):
            """One dimensional inverse of Tangent."""
            return ArcTangent1D(
                amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
            )

        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits,
            ``(x_low, x_high)``.
            """
            bbox = [
                (-1 / 4 - self.phase) / self.frequency,
                (1 / 4 - self.phase) / self.frequency,
            ]

            if self.frequency.unit is not None:
                bbox = bbox / self.frequency.unit

            return bbox

    class _InverseTrigonometric1D(_Trigonometric1D):
        """
        Base class for one dimensional inverse trigonometric models.
        """

        @property
        def input_units(self):
            if self.amplitude.input_unit is None:
                return None
            return {self.inputs[0]: self.amplitude.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "frequency": outputs_unit[self.outputs[0]] ** -1,
                "amplitude": inputs_unit[self.inputs[0]],
            }

    class ArcSine1D(_InverseTrigonometric1D):
        """
        One dimensional ArcSine model returning values between -pi/2 and pi/2
        only.

        Parameters
        ----------
        amplitude : float
            Oscillation amplitude for corresponding Sine
        frequency : float
            Oscillation frequency for corresponding Sine
        phase : float
            Oscillation phase for corresponding Sine

        See Also
        --------
        Sine1D, ArcCosine1D, ArcTangent1D

        Notes
        -----
        Model formula:

            .. math:: f(x) = ((arcsin(x / A) / 2pi) - p) / f

        The arcsin function being used for this model will only accept inputs
        in [-A, A]; otherwise, a runtime warning will be thrown and the result
        will be NaN. To avoid this, the bounding_box has been properly set to
        accommodate this; therefore, it is recommended that this model always
        be evaluated with the ``with_bounding_box=True`` option.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import ArcSine1D

            plt.figure()
            s1 = ArcSine1D(amplitude=1, frequency=.25)
            r=np.arange(-1, 1, .01)

            for amplitude in range(1,4):
                 s1.amplitude = amplitude
                 plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)

            plt.axis([-1, 1, -np.pi/2, np.pi/2])
            plt.show()
        """

        @staticmethod
        def evaluate(x, amplitude, frequency, phase):
            """One dimensional ArcSine model function."""
            # Note: If frequency and x are quantities, they should normally have
            # inverse units, so that argument ends up being dimensionless. However,
            # np.sin of a dimensionless quantity will crash, so we remove the
            # quantity-ness from argument in this case (another option would be to
            # multiply by * u.rad but this would be slower overall).

            argument = x / amplitude
            if isinstance(argument, Quantity):
                argument = argument.value
            arc_sine = np.arcsin(argument) / TWOPI

            return (arc_sine - phase) / frequency

        @staticmethod
        def fit_deriv(x, amplitude, frequency, phase):
            """One dimensional ArcSine model derivative."""
            d_amplitude = -x / (
                TWOPI * frequency * amplitude**2 * np.sqrt(1 - (x / amplitude) ** 2)
            )
            d_frequency = (phase - (np.arcsin(x / amplitude) / TWOPI)) / frequency**2
            d_phase = -1 / frequency * np.ones(x.shape)
            return [d_amplitude, d_frequency, d_phase]

        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits,
            ``(x_low, x_high)``.
            """
            return -1 * self.amplitude, 1 * self.amplitude

        @property
        def inverse(self):
            """One dimensional inverse of ArcSine."""
            return Sine1D(
                amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
            )

    class ArcCosine1D(_InverseTrigonometric1D):
        """
        One dimensional ArcCosine returning values between 0 and pi only.

        Parameters
        ----------
        amplitude : float
            Oscillation amplitude for corresponding Cosine
        frequency : float
            Oscillation frequency for corresponding Cosine
        phase : float
            Oscillation phase for corresponding Cosine

        See Also
        --------
        Cosine1D, ArcSine1D, ArcTangent1D

        Notes
        -----
        Model formula:

            .. math:: f(x) = ((arccos(x / A) / 2pi) - p) / f

        The arccos function being used for this model will only accept inputs
        in [-A, A]; otherwise, a runtime warning will be thrown and the result
        will be NaN. To avoid this, the bounding_box has been properly set to
        accommodate this; therefore, it is recommended that this model always
        be evaluated with the ``with_bounding_box=True`` option.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import ArcCosine1D

            plt.figure()
            s1 = ArcCosine1D(amplitude=1, frequency=.25)
            r=np.arange(-1, 1, .01)

            for amplitude in range(1,4):
                 s1.amplitude = amplitude
                 plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)

            plt.axis([-1, 1, 0, np.pi])
            plt.show()
        """

        @staticmethod
        def evaluate(x, amplitude, frequency, phase):
            """One dimensional ArcCosine model function."""
            # Note: If frequency and x are quantities, they should normally have
            # inverse units, so that argument ends up being dimensionless. However,
            # np.sin of a dimensionless quantity will crash, so we remove the
            # quantity-ness from argument in this case (another option would be to
            # multiply by * u.rad but this would be slower overall).

            argument = x / amplitude
            if isinstance(argument, Quantity):
                argument = argument.value
            arc_cos = np.arccos(argument) / TWOPI

            return (arc_cos - phase) / frequency

        @staticmethod
        def fit_deriv(x, amplitude, frequency, phase):
            """One dimensional ArcCosine model derivative."""
            d_amplitude = x / (
                TWOPI * frequency * amplitude**2 * np.sqrt(1 - (x / amplitude) ** 2)
            )
            d_frequency = (phase - (np.arccos(x / amplitude) / TWOPI)) / frequency**2
            d_phase = -1 / frequency * np.ones(x.shape)
            return [d_amplitude, d_frequency, d_phase]

        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits,
            ``(x_low, x_high)``.
            """
            return -1 * self.amplitude, 1 * self.amplitude

        @property
        def inverse(self):
            """One dimensional inverse of ArcCosine."""
            return Cosine1D(
                amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
            )

    class ArcTangent1D(_InverseTrigonometric1D):
        """
        One dimensional ArcTangent model returning values between -pi/2 and
        pi/2 only.

        Parameters
        ----------
        amplitude : float
            Oscillation amplitude for corresponding Tangent
        frequency : float
            Oscillation frequency for corresponding Tangent
        phase : float
            Oscillation phase for corresponding Tangent

        See Also
        --------
        Tangent1D, ArcSine1D, ArcCosine1D

        Notes
        -----
        Model formula:

            .. math:: f(x) = ((arctan(x / A) / 2pi) - p) / f

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import ArcTangent1D

            plt.figure()
            s1 = ArcTangent1D(amplitude=1, frequency=.25)
            r=np.arange(-10, 10, .01)

            for amplitude in range(1,4):
                 s1.amplitude = amplitude
                 plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2)

            plt.axis([-10, 10, -np.pi/2, np.pi/2])
            plt.show()
        """

        @staticmethod
        def evaluate(x, amplitude, frequency, phase):
            """One dimensional ArcTangent model function."""
            # Note: If frequency and x are quantities, they should normally have
            # inverse units, so that argument ends up being dimensionless. However,
            # np.sin of a dimensionless quantity will crash, so we remove the
            # quantity-ness from argument in this case (another option would be to
            # multiply by * u.rad but this would be slower overall).

            argument = x / amplitude
            if isinstance(argument, Quantity):
                argument = argument.value
            arc_cos = np.arctan(argument) / TWOPI

            return (arc_cos - phase) / frequency

        @staticmethod
        def fit_deriv(x, amplitude, frequency, phase):
            """One dimensional ArcTangent model derivative."""
            d_amplitude = -x / (
                TWOPI * frequency * amplitude**2 * (1 + (x / amplitude) ** 2)
            )
            d_frequency = (phase - (np.arctan(x / amplitude) / TWOPI)) / frequency**2
            d_phase = -1 / frequency * np.ones(x.shape)
            return [d_amplitude, d_frequency, d_phase]

        @property
        def inverse(self):
            """One dimensional inverse of ArcTangent."""
            return Tangent1D(
                amplitude=self.amplitude, frequency=self.frequency, phase=self.phase
            )

    class Linear1D(Fittable1DModel):
        """
        One dimensional Line model.

        Parameters
        ----------
        slope : float
            Slope of the straight line

        intercept : float
            Intercept of the straight line

        See Also
        --------
        Const1D

        Notes
        -----
        Model formula:

            .. math:: f(x) = a x + b
        """

        slope = Parameter(default=1, description="Slope of the straight line")
        intercept = Parameter(default=0, description="Intercept of the straight line")
        linear = True

        @staticmethod
        def evaluate(x, slope, intercept):
            """One dimensional Line model function."""
            return slope * x + intercept

        @staticmethod
        def fit_deriv(x, *params):
            """One dimensional Line model derivative with respect to parameters."""
            d_slope = x
            d_intercept = np.ones_like(x)
            return [d_slope, d_intercept]

        @property
        def inverse(self):
            new_slope = self.slope**-1
            new_intercept = -self.intercept / self.slope
            return self.__class__(slope=new_slope, intercept=new_intercept)

        @property
        def input_units(self):
            if self.intercept.input_unit is None and self.slope.input_unit is None:
                return None
            return {self.inputs[0]: self.intercept.input_unit / self.slope.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "intercept": outputs_unit[self.outputs[0]],
                "slope": outputs_unit[self.outputs[0]] / inputs_unit[self.inputs[0]],
            }

    class Planar2D(Fittable2DModel):
        """
        Two dimensional Plane model.

        Parameters
        ----------
        slope_x : float
            Slope of the plane in X

        slope_y : float
            Slope of the plane in Y

        intercept : float
            Z-intercept of the plane

        Notes
        -----
        Model formula:

            .. math:: f(x, y) = a x + b y + c
        """

        slope_x = Parameter(default=1, description="Slope of the plane in X")
        slope_y = Parameter(default=1, description="Slope of the plane in Y")
        intercept = Parameter(default=0, description="Z-intercept of the plane")
        linear = True

        @staticmethod
        def evaluate(x, y, slope_x, slope_y, intercept):
            """Two dimensional Plane model function."""
            return slope_x * x + slope_y * y + intercept

        @staticmethod
        def fit_deriv(x, y, *params):
            """Two dimensional Plane model derivative with respect to parameters."""
            d_slope_x = x
            d_slope_y = y
            d_intercept = np.ones_like(x)
            return [d_slope_x, d_slope_y, d_intercept]

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "intercept": outputs_unit["z"],
                "slope_x": outputs_unit["z"] / inputs_unit["x"],
                "slope_y": outputs_unit["z"] / inputs_unit["y"],
            }

    class Lorentz1D(Fittable1DModel):
        """
        One dimensional Lorentzian model.

        Parameters
        ----------
        amplitude : float or `~astropy.units.Quantity`.
            Peak value. For a normalized profile (integrating to 1),
            set amplitude = 2 / (np.pi * fwhm).
        x_0 : float or `~astropy.units.Quantity`.
            Position of the peak.
        fwhm : float or `~astropy.units.Quantity`.
            Full width at half maximum (FWHM).

        See Also
        --------
        Lorentz2D, Gaussian1D, Box1D, RickerWavelet1D

        Notes
        -----
        The ``x``, ``x_0``, and ``fwhm`` inputs must have compatible units
        or be unitless numbers.

        Model formula:

        .. math::

            f(x) = \\frac{A \\gamma^{2}}{\\gamma^{2} + \\left(x - x_{0}\\right)^{2}}

        where :math:`\\gamma` is the half width at half maximum (HWHM),
        which is half the FWHM.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Lorentz1D

            plt.figure()
            s1 = Lorentz1D()
            r = np.arange(-5, 5, .01)

            for factor in range(1, 4):
                s1.amplitude = factor
                plt.plot(r, s1(r), lw=2, label=f'Amplitude={factor}')

            plt.axis([-5, 5, -1, 4])
            plt.legend()
            plt.show()
        """

        amplitude = Parameter(default=1, description="Peak value")
        x_0 = Parameter(default=0, description="Position of the peak")
        fwhm = Parameter(default=1, description="Full width at half maximum")

        @staticmethod
        def evaluate(x, amplitude, x_0, fwhm):
            """One dimensional Lorentzian model function."""
            return amplitude * ((fwhm / 2.0) ** 2) / ((x - x_0) ** 2 + (fwhm / 2.0) ** 2)

        @staticmethod
        def fit_deriv(x, amplitude, x_0, fwhm):
            """One dimensional Lorentzian model derivative with respect to parameters."""
            gamma = fwhm / 2.0
            denom = gamma**2 + (x - x_0) ** 2
            d_amplitude = gamma**2 / denom
            d_x_0 = amplitude * gamma**2 * 2 * (x - x_0) / denom**2
            d_fwhm = amplitude * gamma * (x - x_0) ** 2 / denom**2
            return [d_amplitude, d_x_0, d_fwhm]

        def bounding_box(self, factor=25):
            """Tuple defining the default ``bounding_box`` limits,
            ``(x_low, x_high)``.

            Parameters
            ----------
            factor : float
                The multiple of FWHM used to define the limits.
                Default is chosen to include most (99%) of the
                area under the curve, while still showing the
                central feature of interest.

            """
            x0 = self.x_0
            dx = factor * self.fwhm

            return (x0 - dx, x0 + dx)

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {self.inputs[0]: self.x_0.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "fwhm": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Lorentz2D(Fittable2DModel):
        """
        Two-dimensional Lorentzian model.

        Parameters
        ----------
        amplitude : float or `~astropy.units.Quantity`.
            Peak value.
        x_0 : float or `~astropy.units.Quantity`.
            Position of the peak in x.
        y_0 : float or `~astropy.units.Quantity`.
            Position of the peak in y.
        fwhm : float or `~astropy.units.Quantity`.
            Full width at half maximum (FWHM).

        See Also
        --------
        Lorentz1D, Gaussian2D, Moffat2D

        Notes
        -----
        The ``x``, ``y``, ``x_0``, ``y_0``, and ``fwhm`` inputs must have
        compatible units or as unitless numbers.

        Model formula:

        .. math::

            f(x, y) = \\frac{A \\gamma^{2}}{\\gamma^{2}
                      + \\left(x - x_{0}\\right)^2 + \\left(y - y_{0}\\right)^{2}}

        where :math:`\\gamma` is the half width at half maximum (HWHM), which
        is half the FWHM.

        The area under the `Lorentz2D` profile is infinite, therefore this
        model profile cannot be normalized to sum to 1.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Lorentz2D

            plt.figure()
            model = Lorentz2D(x_0=12, y_0=12, fwhm=3)
            yy, xx = np.mgrid[:25, :25]
            data = model(xx, yy)

            plt.imshow(data)
            plt.show()
        """

        amplitude = Parameter(default=1, description="Peak value")
        x_0 = Parameter(default=0, description="Position of the peak in x")
        y_0 = Parameter(default=0, description="Position of the peak in y")
        fwhm = Parameter(default=1, description="Full width at half maximum")

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, fwhm):
            """Two dimensional Lorentzian model function."""
            return amplitude * (
                ((fwhm / 2.0) ** 2) / ((x - x_0) ** 2 + (y - y_0) ** 2 + (fwhm / 2.0) ** 2)
            )

        @staticmethod
        def fit_deriv(x, y, amplitude, x_0, y_0, fwhm):
            """Two dimensional Lorentzian model derivative with respect to parameters."""
            gamma = fwhm / 2.0
            r2 = (x - x_0) ** 2 + (y - y_0) ** 2
            denom = gamma**2 + r2
            denom2 = denom**2
            d_amplitude = gamma**2 / denom
            d_x_0 = amplitude * gamma**2 * 2 * (x - x_0) / denom2
            d_y_0 = amplitude * gamma**2 * 2 * (y - y_0) / denom2
            d_fwhm = (amplitude * (fwhm / 2) * r2) / denom2
            return [d_amplitude, d_x_0, d_y_0, d_fwhm]

        def bounding_box(self, factor=25):
            """Tuple defining the default ``bounding_box`` limits,
            ``(x_low, x_high), (y_low, y_high)``.

            Parameters
            ----------
            factor : float
                The multiple of FWHM used to define the limits.
                Default is chosen to include most (99%) of the
                area under the curve, while still showing the
                central feature of interest.
            """
            x0 = self.x_0
            y0 = self.y_0
            dx = factor * self.fwhm
            return ((x0 - dx, x0 + dx), (y0 - dx, y0 + dx))

        @property
        def input_units(self):
            x_unit = self.x_0.input_unit
            y_unit = self.y_0.input_unit
            if x_unit is None and y_unit is None:
                return None

            return {self.inputs[0]: x_unit, self.inputs[1]: y_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "fwhm": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Voigt1D(Fittable1DModel):
        """
        One dimensional model for the Voigt profile.

        Parameters
        ----------
        x_0 : float or `~astropy.units.Quantity`
            Position of the peak
        amplitude_L : float or `~astropy.units.Quantity`.
            The Lorentzian amplitude (peak of the associated Lorentz function)
            - for a normalized profile (integrating to 1), set
            amplitude_L = 2 / (np.pi * fwhm_L)
        fwhm_L : float or `~astropy.units.Quantity`
            The Lorentzian full width at half maximum
        fwhm_G : float or `~astropy.units.Quantity`.
            The Gaussian full width at half maximum
        method : str, optional
            Algorithm for computing the complex error function; one of
            'Humlicek2' (default, fast and generally more accurate than ``rtol=3.e-5``) or
            'Scipy', alternatively 'wofz' (requires ``scipy``, almost as fast and
            reference in accuracy).

        See Also
        --------
        Gaussian1D, Lorentz1D

        Notes
        -----
        The ``x``, ``x_0``, and ``fwhm_*`` inputs must have compatible units
        or be unitless numbers.

        Voigt function is calculated as real part of the complex error function computed from either
        Humlicek's rational approximations (JQSRT 21:309, 1979; 27:437, 1982) following
        Schreier 2018 (MNRAS 479, 3068; and ``hum2zpf16m`` from his cpfX.py module); or
        `~scipy.special.wofz` (implementing 'Faddeeva.cc').

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            from astropy.modeling.models import Voigt1D
            import matplotlib.pyplot as plt

            plt.figure()
            x = np.arange(0, 10, 0.01)
            v1 = Voigt1D(x_0=5, amplitude_L=10, fwhm_L=0.5, fwhm_G=0.9)
            plt.plot(x, v1(x))
            plt.show()
        """

        x_0 = Parameter(default=0, description="Position of the peak")
        amplitude_L = Parameter(default=1, description="The Lorentzian amplitude")
        fwhm_L = Parameter(
            default=2 / np.pi, description="The Lorentzian full width at half maximum"
        )
        fwhm_G = Parameter(
            default=np.log(2), description="The Gaussian full width at half maximum"
        )

        sqrt_pi = np.sqrt(np.pi)
        sqrt_ln2 = np.sqrt(np.log(2))
        sqrt_ln2pi = np.sqrt(np.log(2) * np.pi)
        _last_z = np.zeros(1, dtype=complex)
        _last_w = np.zeros(1, dtype=float)
        _faddeeva = None

        def __init__(
            self,
            x_0=x_0.default,
            amplitude_L=amplitude_L.default,
            fwhm_L=fwhm_L.default,
            fwhm_G=fwhm_G.default,
            method=None,
            **kwargs,
        ):
            if str(method).lower() == "humlicek2" and HAS_SCIPY:
                warnings.warn(
                    f"{method} has been deprecated since Astropy 5.3 and will be removed in a future version.\n"
                    "It is recommended to always use the `~scipy.special.wofz` implementation "
                    "when `scipy` is installed.",
                    AstropyDeprecationWarning,
                )

            if method is None:
                if HAS_SCIPY:
                    method = "wofz"
                else:
                    method = "humlicek2"

            if str(method).lower() in ("wofz", "scipy"):
                from scipy.special import wofz

                self._faddeeva = wofz
            elif str(method).lower() == "humlicek2":
                self._faddeeva = self._hum2zpf16c
            else:
                raise ValueError(
                    f"Not a valid method for Voigt1D Faddeeva function: {method}."
                )
            self.method = self._faddeeva.__name__

            super().__init__(
                x_0=x_0, amplitude_L=amplitude_L, fwhm_L=fwhm_L, fwhm_G=fwhm_G, **kwargs
            )

        def _wrap_wofz(self, z):
            """Call complex error (Faddeeva) function w(z) implemented by algorithm `method`;
            cache results for consecutive calls from `evaluate`, `fit_deriv`.
            """
            if z.shape == self._last_z.shape and np.allclose(
                z, self._last_z, rtol=1.0e-14, atol=1.0e-15
            ):
                return self._last_w

            self._last_z = (
                z.to_value(u.dimensionless_unscaled) if isinstance(z, u.Quantity) else z
            )
            self._last_w = self._faddeeva(self._last_z)

            return self._last_w

        def evaluate(self, x, x_0, amplitude_L, fwhm_L, fwhm_G):
            """One dimensional Voigt function scaled to Lorentz peak amplitude."""
            z = np.atleast_1d(2 * (x - x_0) + 1j * fwhm_L) * self.sqrt_ln2 / fwhm_G
            # The normalised Voigt profile is w.real * self.sqrt_ln2 / (self.sqrt_pi * fwhm_G) * 2 ;
            # for the legacy definition we multiply with np.pi * fwhm_L / 2 * amplitude_L
            return self._wrap_wofz(z).real * self.sqrt_ln2pi / fwhm_G * fwhm_L * amplitude_L

        def fit_deriv(self, x, x_0, amplitude_L, fwhm_L, fwhm_G):
            """
            Derivative of the one dimensional Voigt function with respect to parameters.
            """
            s = self.sqrt_ln2 / fwhm_G
            z = np.atleast_1d(2 * (x - x_0) + 1j * fwhm_L) * s
            # V * constant from McLean implementation (== their Voigt function)
            w = self._wrap_wofz(z) * s * fwhm_L * amplitude_L * self.sqrt_pi

            # Schreier (2018) Eq. 6 == (dvdx + 1j * dvdy) / (sqrt(pi) * fwhm_L * amplitude_L)
            dwdz = -2 * z * w + 2j * s * fwhm_L * amplitude_L

            return [
                -dwdz.real * 2 * s,
                w.real / amplitude_L,
                w.real / fwhm_L - dwdz.imag * s,
                (-w.real - s * (2 * (x - x_0) * dwdz.real - fwhm_L * dwdz.imag)) / fwhm_G,
            ]

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {self.inputs[0]: self.x_0.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "fwhm_L": inputs_unit[self.inputs[0]],
                "fwhm_G": inputs_unit[self.inputs[0]],
                "amplitude_L": outputs_unit[self.outputs[0]],
            }

        @staticmethod
        def _hum2zpf16c(z, s=10.0):
            """Complex error function w(z = x + iy) combining Humlicek's rational approximations.

            |x| + y > 10:  Humlicek (JQSRT, 1982) rational approximation for region II;
            else:          Humlicek (JQSRT, 1979) rational approximation with n=16 and delta=y0=1.35

            Version using a mask and np.place;
            single complex argument version of Franz Schreier's cpfX.hum2zpf16m.
            Originally licensed under a 3-clause BSD style license - see
            https://atmos.eoc.dlr.de/tools/lbl4IR/cpfX.py
            """
            # Optimized (single fraction) Humlicek region I rational approximation for n=16, delta=1.35

            # fmt: off
            AA = np.array(
                [
                    +46236.3358828121,   -147726.58393079657j,
                    -206562.80451354137,  281369.1590631087j,
                    +183092.74968253175, -184787.96830696272j,
                    -66155.39578477248,   57778.05827983565j,
                    +11682.770904216826, -9442.402767960672j,
                    -1052.8438624933142,  814.0996198624186j,
                    +45.94499030751872,  -34.59751573708725j,
                    -0.7616559377907136,  0.5641895835476449j,
                ]
            )  # 1j/sqrt(pi) to the 12. digit

            bb = np.array(
                [
                    +7918.06640624997,
                    -126689.0625,
                    +295607.8125,
                    -236486.25,
                    +84459.375,
                    -15015.0,
                    +1365.0,
                    -60.0,
                    +1.0,
                ]
            )
            # fmt: on

            sqrt_piinv = 1.0 / np.sqrt(np.pi)

            zz = z * z
            w = 1j * (z * (zz * sqrt_piinv - 1.410474)) / (0.75 + zz * (zz - 3.0))

            if np.any(z.imag < s):
                mask = abs(z.real) + z.imag < s  # returns true for interior points

                # returns small complex array covering only the interior region
                Z = z[np.where(mask)] + 1.35j
                ZZ = Z * Z

                # fmt: off
                # Recursive algorithms for the polynomials in Z with coefficients AA, bb
                # numer = 0.0
                # for A in AA[::-1]:
                #     numer = numer * Z + A
                # Explicitly unrolled above loop for speed
                numer = (((((((((((((((AA[15]*Z + AA[14])*Z + AA[13])*Z + AA[12])*Z + AA[11])*Z +
                                   AA[10])*Z + AA[9])*Z + AA[8])*Z + AA[7])*Z + AA[6])*Z +
                              AA[5])*Z + AA[4])*Z+AA[3])*Z + AA[2])*Z + AA[1])*Z + AA[0])

                # denom = 0.0
                # for b in bb[::-1]:
                #     denom = denom * ZZ + b
                # Explicitly unrolled above loop for speed
                denom = (((((((ZZ + bb[7])*ZZ + bb[6])*ZZ + bb[5])*ZZ+bb[4])*ZZ + bb[3])*ZZ +
                          bb[2])*ZZ + bb[1])*ZZ + bb[0]
                # fmt: on

                np.place(w, mask, numer / denom)

            return w

    class Const1D(Fittable1DModel):
        """
        One dimensional Constant model.

        Parameters
        ----------
        amplitude : float
            Value of the constant function

        See Also
        --------
        Const2D

        Notes
        -----
        Model formula:

            .. math:: f(x) = A

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Const1D

            plt.figure()
            s1 = Const1D()
            r = np.arange(-5, 5, .01)

            for factor in range(1, 4):
                s1.amplitude = factor
                plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

            plt.axis([-5, 5, -1, 4])
            plt.show()
        """

        amplitude = Parameter(
            default=1, description="Value of the constant function", mag=True
        )
        linear = True

        @staticmethod
        def evaluate(x, amplitude):
            """One dimensional Constant model function."""
            if amplitude.size == 1:
                # This is slightly faster than using ones_like and multiplying
                x = np.empty_like(amplitude, shape=x.shape, dtype=x.dtype)
                x.fill(amplitude.item())
            else:
                # This case is less likely but could occur if the amplitude
                # parameter is given an array-like value
                x = amplitude * np.ones_like(x, subok=False)

            if isinstance(amplitude, Quantity):
                return Quantity(x, unit=amplitude.unit, copy=False, subok=True)
            return x

        @staticmethod
        def fit_deriv(x, amplitude):
            """One dimensional Constant model derivative with respect to parameters."""
            d_amplitude = np.ones_like(x)
            return [d_amplitude]

        @property
        def input_units(self):
            return None

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {"amplitude": outputs_unit[self.outputs[0]]}

    class Const2D(Fittable2DModel):
        """
        Two dimensional Constant model.

        Parameters
        ----------
        amplitude : float
            Value of the constant function

        See Also
        --------
        Const1D

        Notes
        -----
        Model formula:

            .. math:: f(x, y) = A
        """

        amplitude = Parameter(
            default=1, description="Value of the constant function", mag=True
        )
        linear = True

        @staticmethod
        def evaluate(x, y, amplitude):
            """Two dimensional Constant model function."""
            if amplitude.size == 1:
                # This is slightly faster than using ones_like and multiplying
                x = np.empty_like(amplitude, shape=x.shape, dtype=x.dtype)
                x.fill(amplitude.item())
            else:
                # This case is less likely but could occur if the amplitude
                # parameter is given an array-like value
                x = amplitude * np.ones_like(x, subok=False)

            if isinstance(amplitude, Quantity):
                return Quantity(x, unit=amplitude.unit, copy=False, subok=True)
            return x

        @property
        def input_units(self):
            return None

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {"amplitude": outputs_unit[self.outputs[0]]}

    class Ellipse2D(Fittable2DModel):
        """
        A 2D Ellipse model.

        Parameters
        ----------
        amplitude : float
            Value of the ellipse.

        x_0 : float
            x position of the center of the disk.

        y_0 : float
            y position of the center of the disk.

        a : float
            The length of the semimajor axis.

        b : float
            The length of the semiminor axis.

        theta : float or `~astropy.units.Quantity`, optional
            The rotation angle as an angular quantity
            (`~astropy.units.Quantity` or `~astropy.coordinates.Angle`)
            or a value in radians (as a float). The rotation angle
            increases counterclockwise from the positive x axis.

        See Also
        --------
        Disk2D, Box2D

        Notes
        -----
        Model formula:

        .. math::

            f(x, y) = \\left \\{
                        \\begin{array}{ll}
                          \\mathrm{amplitude} & : \\left[\\frac{(x - x_0) \\cos
                            \\theta + (y - y_0) \\sin \\theta}{a}\\right]^2 +
                            \\left[\\frac{-(x - x_0) \\sin \\theta + (y - y_0)
                            \\cos \\theta}{b}\\right]^2  \\leq 1 \\\\
                          0 & : \\mathrm{otherwise}
                        \\end{array}
                      \\right.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            from astropy.modeling.models import Ellipse2D
            from astropy.coordinates import Angle
            import matplotlib.pyplot as plt
            import matplotlib.patches as mpatches
            x0, y0 = 25, 25
            a, b = 20, 10
            theta = Angle(30, 'deg')
            e = Ellipse2D(amplitude=100., x_0=x0, y_0=y0, a=a, b=b,
                          theta=theta.radian)
            y, x = np.mgrid[0:50, 0:50]
            fig, ax = plt.subplots(1, 1)
            ax.imshow(e(x, y), origin='lower', interpolation='none', cmap='Greys_r')
            e2 = mpatches.Ellipse((x0, y0), 2*a, 2*b, angle=theta.degree, edgecolor='red',
                                  facecolor='none')
            ax.add_patch(e2)
            plt.show()
        """

        amplitude = Parameter(default=1, description="Value of the ellipse", mag=True)
        x_0 = Parameter(default=0, description="X position of the center of the disk.")
        y_0 = Parameter(default=0, description="Y position of the center of the disk.")
        a = Parameter(default=1, description="The length of the semimajor axis")
        b = Parameter(default=1, description="The length of the semiminor axis")
        theta = Parameter(
            default=0.0,
            description=(
                "Rotation angle either as a float (in radians) or a |Quantity| angle"
            ),
        )

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, a, b, theta):
            """Two dimensional Ellipse model function."""
            xx = x - x_0
            yy = y - y_0
            cost = np.cos(theta)
            sint = np.sin(theta)
            numerator1 = (xx * cost) + (yy * sint)
            numerator2 = -(xx * sint) + (yy * cost)
            in_ellipse = ((numerator1 / a) ** 2 + (numerator2 / b) ** 2) <= 1.0
            result = np.select([in_ellipse], [amplitude])

            if isinstance(amplitude, Quantity):
                return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
            return result

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits.

            ``((y_low, y_high), (x_low, x_high))``
            """
            a = self.a
            b = self.b
            theta = self.theta
            dx, dy = ellipse_extent(a, b, theta)

            return ((self.y_0 - dy, self.y_0 + dy), (self.x_0 - dx, self.x_0 + dx))

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {
                self.inputs[0]: self.x_0.input_unit,
                self.inputs[1]: self.y_0.input_unit,
            }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "a": inputs_unit[self.inputs[0]],
                "b": inputs_unit[self.inputs[0]],
                "theta": u.rad,
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Disk2D(Fittable2DModel):
        """
        Two dimensional radial symmetric Disk model.

        Parameters
        ----------
        amplitude : float
            Value of the disk function
        x_0 : float
            x position center of the disk
        y_0 : float
            y position center of the disk
        R_0 : float
            Radius of the disk

        See Also
        --------
        Box2D, TrapezoidDisk2D

        Notes
        -----
        Model formula:

            .. math::

                f(r) = \\left \\{
                         \\begin{array}{ll}
                           A & : r \\leq R_0 \\\\
                           0 & : r > R_0
                         \\end{array}
                       \\right.
        """

        amplitude = Parameter(default=1, description="Value of disk function", mag=True)
        x_0 = Parameter(default=0, description="X position of center of the disk")
        y_0 = Parameter(default=0, description="Y position of center of the disk")
        R_0 = Parameter(default=1, description="Radius of the disk")

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, R_0):
            """Two dimensional Disk model function."""
            rr = (x - x_0) ** 2 + (y - y_0) ** 2
            result = np.select([rr <= R_0**2], [amplitude])

            if isinstance(amplitude, Quantity):
                return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
            return result

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits.

            ``((y_low, y_high), (x_low, x_high))``
            """
            return (
                (self.y_0 - self.R_0, self.y_0 + self.R_0),
                (self.x_0 - self.R_0, self.x_0 + self.R_0),
            )

        @property
        def input_units(self):
            x_unit = self.x_0.input_unit
            y_unit = self.y_0.input_unit

            if x_unit is None and y_unit is None:
                return None
            return {self.inputs[0]: x_unit, self.inputs[1]: y_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "R_0": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Ring2D(Fittable2DModel):
        """
        Two dimensional radial symmetric Ring model.

        Parameters
        ----------
        amplitude : float
            Value of the disk function
        x_0 : float
            x position center of the disk
        y_0 : float
            y position center of the disk
        r_in : float
            Inner radius of the ring
        width : float
            Width of the ring.
        r_out : float
            Outer Radius of the ring. Can be specified instead of width.

        See Also
        --------
        Disk2D, TrapezoidDisk2D

        Notes
        -----
        Model formula:

            .. math::

                f(r) = \\left \\{
                         \\begin{array}{ll}
                           A & : r_{in} \\leq r \\leq r_{out} \\\\
                           0 & : \\text{else}
                         \\end{array}
                       \\right.

        Where :math:`r_{out} = r_{in} + r_{width}`.
        """

        amplitude = Parameter(default=1, description="Value of the disk function", mag=True)
        x_0 = Parameter(default=0, description="X position of center of disc")
        y_0 = Parameter(default=0, description="Y position of center of disc")
        r_in = Parameter(default=1, description="Inner radius of the ring")
        width = Parameter(default=1, description="Width of the ring")

        def __init__(
            self,
            amplitude=amplitude.default,
            x_0=x_0.default,
            y_0=y_0.default,
            r_in=None,
            width=None,
            r_out=None,
            **kwargs,
        ):
            if (r_in is None) and (r_out is None) and (width is None):
                r_in = self.r_in.default
                width = self.width.default
            elif (r_in is not None) and (r_out is None) and (width is None):
                width = self.width.default
            elif (r_in is None) and (r_out is not None) and (width is None):
                r_in = self.r_in.default
                width = r_out - r_in
            elif (r_in is None) and (r_out is None) and (width is not None):
                r_in = self.r_in.default
            elif (r_in is not None) and (r_out is not None) and (width is None):
                width = r_out - r_in
            elif (r_in is None) and (r_out is not None) and (width is not None):
                r_in = r_out - width
            elif (r_in is not None) and (r_out is not None) and (width is not None):
                if np.any(width != (r_out - r_in)):
                    raise InputParameterError("Width must be r_out - r_in")

            if np.any(r_in < 0) or np.any(width < 0):
                raise InputParameterError(f"{r_in=} and {width=} must both be >=0")

            super().__init__(
                amplitude=amplitude, x_0=x_0, y_0=y_0, r_in=r_in, width=width, **kwargs
            )

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, r_in, width):
            """Two dimensional Ring model function."""
            rr = (x - x_0) ** 2 + (y - y_0) ** 2
            r_range = np.logical_and(rr >= r_in**2, rr <= (r_in + width) ** 2)
            result = np.select([r_range], [amplitude])

            if isinstance(amplitude, Quantity):
                return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
            return result

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box``.

            ``((y_low, y_high), (x_low, x_high))``
            """
            dr = self.r_in + self.width

            return ((self.y_0 - dr, self.y_0 + dr), (self.x_0 - dr, self.x_0 + dr))

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {
                self.inputs[0]: self.x_0.input_unit,
                self.inputs[1]: self.y_0.input_unit,
            }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "r_in": inputs_unit[self.inputs[0]],
                "width": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Box1D(Fittable1DModel):
        """
        One dimensional Box model.

        Parameters
        ----------
        amplitude : float
            Amplitude A
        x_0 : float
            Position of the center of the box function
        width : float
            Width of the box

        See Also
        --------
        Box2D, TrapezoidDisk2D

        Notes
        -----
        Model formula:

          .. math::

                f(x) = \\left \\{
                         \\begin{array}{ll}
                           A & : x_0 - w/2 \\leq x \\leq x_0 + w/2 \\\\
                           0 & : \\text{else}
                         \\end{array}
                       \\right.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Box1D

            plt.figure()
            s1 = Box1D()
            r = np.arange(-5, 5, .01)

            for factor in range(1, 4):
                s1.amplitude = factor
                s1.width = factor
                plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

            plt.axis([-5, 5, -1, 4])
            plt.show()
        """

        amplitude = Parameter(default=1, description="Amplitude A", mag=True)
        x_0 = Parameter(default=0, description="Position of center of box function")
        width = Parameter(default=1, description="Width of the box")

        @staticmethod
        def evaluate(x, amplitude, x_0, width):
            """One dimensional Box model function."""
            inside = np.logical_and(x >= x_0 - width / 2.0, x <= x_0 + width / 2.0)
            return np.select([inside], [amplitude], 0)

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits.

            ``(x_low, x_high))``
            """
            dx = self.width / 2

            return (self.x_0 - dx, self.x_0 + dx)

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {self.inputs[0]: self.x_0.input_unit}

        @property
        def return_units(self):
            if self.amplitude.unit is None:
                return None
            return {self.outputs[0]: self.amplitude.unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "width": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Box2D(Fittable2DModel):
        """
        Two dimensional Box model.

        Parameters
        ----------
        amplitude : float
            Amplitude
        x_0 : float
            x position of the center of the box function
        x_width : float
            Width in x direction of the box
        y_0 : float
            y position of the center of the box function
        y_width : float
            Width in y direction of the box

        See Also
        --------
        Box1D, Gaussian2D, Moffat2D

        Notes
        -----
        Model formula:

          .. math::

                f(x, y) = \\left \\{
                         \\begin{array}{ll}
                A : & x_0 - w_x/2 \\leq x \\leq x_0 + w_x/2 \\text{ and} \\\\
                    & y_0 - w_y/2 \\leq y \\leq y_0 + w_y/2 \\\\
                0 : & \\text{else}
                         \\end{array}
                       \\right.

        """

        amplitude = Parameter(default=1, description="Amplitude", mag=True)
        x_0 = Parameter(
            default=0, description="X position of the center of the box function"
        )
        y_0 = Parameter(
            default=0, description="Y position of the center of the box function"
        )
        x_width = Parameter(default=1, description="Width in x direction of the box")
        y_width = Parameter(default=1, description="Width in y direction of the box")

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, x_width, y_width):
            """Two dimensional Box model function."""
            x_range = np.logical_and(x >= x_0 - x_width / 2.0, x <= x_0 + x_width / 2.0)
            y_range = np.logical_and(y >= y_0 - y_width / 2.0, y <= y_0 + y_width / 2.0)

            result = np.select([np.logical_and(x_range, y_range)], [amplitude], 0)

            if isinstance(amplitude, Quantity):
                return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
            return result

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box``.

            ``((y_low, y_high), (x_low, x_high))``
            """
            dx = self.x_width / 2
            dy = self.y_width / 2

            return ((self.y_0 - dy, self.y_0 + dy), (self.x_0 - dx, self.x_0 + dx))

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {
                self.inputs[0]: self.x_0.input_unit,
                self.inputs[1]: self.y_0.input_unit,
            }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[1]],
                "x_width": inputs_unit[self.inputs[0]],
                "y_width": inputs_unit[self.inputs[1]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Trapezoid1D(Fittable1DModel):
        """
        One dimensional Trapezoid model.

        Parameters
        ----------
        amplitude : float
            Amplitude of the trapezoid
        x_0 : float
            Center position of the trapezoid
        width : float
            Width of the constant part of the trapezoid.
        slope : float
            Slope of the tails of the trapezoid

        See Also
        --------
        Box1D, Gaussian1D, Moffat1D

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Trapezoid1D

            plt.figure()
            s1 = Trapezoid1D()
            r = np.arange(-5, 5, .01)

            for factor in range(1, 4):
                s1.amplitude = factor
                s1.width = factor
                plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

            plt.axis([-5, 5, -1, 4])
            plt.show()
        """

        amplitude = Parameter(default=1, description="Amplitude of the trapezoid")
        x_0 = Parameter(default=0, description="Center position of the trapezoid")
        width = Parameter(default=1, description="Width of constant part of the trapezoid")
        slope = Parameter(default=1, description="Slope of the tails of trapezoid")

        @staticmethod
        def evaluate(x, amplitude, x_0, width, slope):
            """One dimensional Trapezoid model function."""
            # Compute the four points where the trapezoid changes slope
            # x1 <= x2 <= x3 <= x4
            x2 = x_0 - width / 2.0
            x3 = x_0 + width / 2.0
            x1 = x2 - amplitude / slope
            x4 = x3 + amplitude / slope

            # Compute model values in pieces between the change points
            range_a = np.logical_and(x >= x1, x < x2)
            range_b = np.logical_and(x >= x2, x < x3)
            range_c = np.logical_and(x >= x3, x < x4)
            val_a = slope * (x - x1)
            val_b = amplitude
            val_c = slope * (x4 - x)
            result = np.select([range_a, range_b, range_c], [val_a, val_b, val_c])

            if isinstance(amplitude, Quantity):
                return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
            return result

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits.

            ``(x_low, x_high))``
            """
            dx = self.width / 2 + self.amplitude / self.slope

            return (self.x_0 - dx, self.x_0 + dx)

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {self.inputs[0]: self.x_0.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "width": inputs_unit[self.inputs[0]],
                "slope": outputs_unit[self.outputs[0]] / inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class TrapezoidDisk2D(Fittable2DModel):
        """
        Two dimensional circular Trapezoid model.

        Parameters
        ----------
        amplitude : float
            Amplitude of the trapezoid
        x_0 : float
            x position of the center of the trapezoid
        y_0 : float
            y position of the center of the trapezoid
        R_0 : float
            Radius of the constant part of the trapezoid.
        slope : float
            Slope of the tails of the trapezoid in x direction.

        See Also
        --------
        Disk2D, Box2D
        """

        amplitude = Parameter(default=1, description="Amplitude of the trapezoid")
        x_0 = Parameter(default=0, description="X position of the center of the trapezoid")
        y_0 = Parameter(default=0, description="Y position of the center of the trapezoid")
        R_0 = Parameter(default=1, description="Radius of constant part of trapezoid")
        slope = Parameter(
            default=1, description="Slope of tails of trapezoid in x direction"
        )

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, R_0, slope):
            """Two dimensional Trapezoid Disk model function."""
            r = np.sqrt((x - x_0) ** 2 + (y - y_0) ** 2)
            range_1 = r <= R_0
            range_2 = np.logical_and(r > R_0, r <= R_0 + amplitude / slope)
            val_1 = amplitude
            val_2 = amplitude + slope * (R_0 - r)
            result = np.select([range_1, range_2], [val_1, val_2])

            if isinstance(amplitude, Quantity):
                return Quantity(result, unit=amplitude.unit, copy=False, subok=True)
            return result

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box``.

            ``((y_low, y_high), (x_low, x_high))``
            """
            dr = self.R_0 + self.amplitude / self.slope

            return ((self.y_0 - dr, self.y_0 + dr), (self.x_0 - dr, self.x_0 + dr))

        @property
        def input_units(self):
            x_unit = self.x_0.input_unit
            y_unit = self.y_0.input_unit

            if x_unit is None and y_unit is None:
                return None

            return {self.inputs[0]: x_unit, self.inputs[1]: y_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit["x"] != inputs_unit["y"]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "R_0": inputs_unit[self.inputs[0]],
                "slope": outputs_unit[self.outputs[0]] / inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class RickerWavelet1D(Fittable1DModel):
        """
        One dimensional Ricker Wavelet model (sometimes known as a "Mexican Hat"
        model).

        .. note::

            See https://github.com/astropy/astropy/pull/9445 for discussions
            related to renaming of this model.

        Parameters
        ----------
        amplitude : float
            Amplitude
        x_0 : float
            Position of the peak
        sigma : float
            Width of the Ricker wavelet

        See Also
        --------
        RickerWavelet2D, Box1D, Gaussian1D, Trapezoid1D

        Notes
        -----
        Model formula:

        .. math::

            f(x) = {A \\left(1 - \\frac{\\left(x - x_{0}\\right)^{2}}{\\sigma^{2}}\\right)
            e^{- \\frac{\\left(x - x_{0}\\right)^{2}}{2 \\sigma^{2}}}}

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import RickerWavelet1D

            plt.figure()
            s1 = RickerWavelet1D()
            r = np.arange(-5, 5, .01)

            for factor in range(1, 4):
                s1.amplitude = factor
                s1.width = factor
                plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

            plt.axis([-5, 5, -2, 4])
            plt.show()
        """

        amplitude = Parameter(default=1, description="Amplitude (peak) value")
        x_0 = Parameter(default=0, description="Position of the peak")
        sigma = Parameter(default=1, description="Width of the Ricker wavelet")

        @staticmethod
        def evaluate(x, amplitude, x_0, sigma):
            """One dimensional Ricker Wavelet model function."""
            xx_ww = (x - x_0) ** 2 / (2 * sigma**2)
            return amplitude * (1 - 2 * xx_ww) * np.exp(-xx_ww)

        def bounding_box(self, factor=10.0):
            """Tuple defining the default ``bounding_box`` limits,
            ``(x_low, x_high)``.

            Parameters
            ----------
            factor : float
                The multiple of sigma used to define the limits.

            """
            x0 = self.x_0
            dx = factor * self.sigma

            return (x0 - dx, x0 + dx)

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {self.inputs[0]: self.x_0.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "sigma": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class RickerWavelet2D(Fittable2DModel):
        """
        Two dimensional Ricker Wavelet model (sometimes known as a "Mexican Hat"
        model).

        .. note::

            See https://github.com/astropy/astropy/pull/9445 for discussions
            related to renaming of this model.

        Parameters
        ----------
        amplitude : float
            Amplitude
        x_0 : float
            x position of the peak
        y_0 : float
            y position of the peak
        sigma : float
            Width of the Ricker wavelet

        See Also
        --------
        RickerWavelet1D, Gaussian2D

        Notes
        -----
        Model formula:

        .. math::

            f(x, y) = A \\left(1 - \\frac{\\left(x - x_{0}\\right)^{2}
            + \\left(y - y_{0}\\right)^{2}}{\\sigma^{2}}\\right)
            e^{\\frac{- \\left(x - x_{0}\\right)^{2}
            - \\left(y - y_{0}\\right)^{2}}{2 \\sigma^{2}}}
        """

        amplitude = Parameter(default=1, description="Amplitude (peak) value")
        x_0 = Parameter(default=0, description="X position of the peak")
        y_0 = Parameter(default=0, description="Y position of the peak")
        sigma = Parameter(default=1, description="Width of the Ricker wavelet")

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, sigma):
            """Two dimensional Ricker Wavelet model function."""
            rr_ww = ((x - x_0) ** 2 + (y - y_0) ** 2) / (2 * sigma**2)
            return amplitude * (1 - rr_ww) * np.exp(-rr_ww)

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {
                self.inputs[0]: self.x_0.input_unit,
                self.inputs[1]: self.y_0.input_unit,
            }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "sigma": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class AiryDisk2D(Fittable2DModel):
        """
        Two dimensional Airy disk model.

        Parameters
        ----------
        amplitude : float
            Amplitude of the Airy function.
        x_0 : float
            x position of the maximum of the Airy function.
        y_0 : float
            y position of the maximum of the Airy function.
        radius : float
            The radius of the Airy disk (radius of the first zero).

        See Also
        --------
        Box2D, TrapezoidDisk2D, Gaussian2D

        Notes
        -----
        Model formula:

            .. math:: f(r) = A \\left[
                    \\frac{2 J_1(\\frac{\\pi r}{R/R_z})}{\\frac{\\pi r}{R/R_z}}
                \\right]^2

        Where :math:`J_1` is the first order Bessel function of the first
        kind, :math:`r` is radial distance from the maximum of the Airy
        function (:math:`r = \\sqrt{(x - x_0)^2 + (y - y_0)^2}`), :math:`R`
        is the input ``radius`` parameter, and :math:`R_z =
        1.2196698912665045`).

        For an optical system, the radius of the first zero represents the
        limiting angular resolution and is approximately 1.22 * lambda / D,
        where lambda is the wavelength of the light and D is the diameter of
        the aperture.

        See [1]_ for more details about the Airy disk.

        References
        ----------
        .. [1] https://en.wikipedia.org/wiki/Airy_disk
        """

        amplitude = Parameter(
            default=1, description="Amplitude (peak value) of the Airy function"
        )
        x_0 = Parameter(default=0, description="X position of the peak")
        y_0 = Parameter(default=0, description="Y position of the peak")
        radius = Parameter(
            default=1,
            description="The radius of the Airy disk (radius of first zero crossing)",
        )
        _rz = None
        _j1 = None

        @classmethod
        def evaluate(cls, x, y, amplitude, x_0, y_0, radius):
            """Two dimensional Airy model function."""
            if cls._rz is None:
                from scipy.special import j1, jn_zeros

                cls._rz = jn_zeros(1, 1)[0] / np.pi
                cls._j1 = j1

            r = np.sqrt((x - x_0) ** 2 + (y - y_0) ** 2) / (radius / cls._rz)

            if isinstance(r, Quantity):
                # scipy function cannot handle Quantity, so turn into array.
                r = r.to_value(u.dimensionless_unscaled)

            # Since r can be zero, we have to take care to treat that case
            # separately so as not to raise a numpy warning
            z = np.ones(r.shape)
            rt = np.pi * r[r > 0]
            z[r > 0] = (2.0 * cls._j1(rt) / rt) ** 2

            if isinstance(amplitude, Quantity):
                # make z quantity too, otherwise in-place multiplication fails.
                z = Quantity(z, u.dimensionless_unscaled, copy=False, subok=True)

            z *= amplitude
            return z

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {
                self.inputs[0]: self.x_0.input_unit,
                self.inputs[1]: self.y_0.input_unit,
            }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "radius": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Moffat1D(Fittable1DModel):
        """
        One dimensional Moffat model.

        Parameters
        ----------
        amplitude : float
            Amplitude of the model.
        x_0 : float
            x position of the maximum of the Moffat model.
        gamma : float
            Core width of the Moffat model.
        alpha : float
            Power index of the Moffat model.

        See Also
        --------
        Gaussian1D, Box1D

        Notes
        -----
        Model formula:

        .. math::

            f(x) = A \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha}

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            import matplotlib.pyplot as plt

            from astropy.modeling.models import Moffat1D

            plt.figure()
            s1 = Moffat1D()
            r = np.arange(-5, 5, .01)

            for factor in range(1, 4):
                s1.amplitude = factor
                s1.width = factor
                plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

            plt.axis([-5, 5, -1, 4])
            plt.show()
        """

        amplitude = Parameter(default=1, description="Amplitude of the model")
        x_0 = Parameter(default=0, description="X position of maximum of Moffat model")
        gamma = Parameter(default=1, description="Core width of Moffat model")
        alpha = Parameter(default=1, description="Power index of the Moffat model")

        @property
        def fwhm(self):
            """
            Moffat full width at half maximum.
            Derivation of the formula is available in
            `this notebook by Yoonsoo Bach
            <https://nbviewer.jupyter.org/github/ysbach/AO_2017/blob/master/04_Ground_Based_Concept.ipynb#1.2.-Moffat>`_.
            """
            return 2.0 * np.abs(self.gamma) * np.sqrt(2.0 ** (1.0 / self.alpha) - 1.0)

        @staticmethod
        def evaluate(x, amplitude, x_0, gamma, alpha):
            """One dimensional Moffat model function."""
            return amplitude * (1 + ((x - x_0) / gamma) ** 2) ** (-alpha)

        @staticmethod
        def fit_deriv(x, amplitude, x_0, gamma, alpha):
            """One dimensional Moffat model derivative with respect to parameters."""
            fac = 1 + (x - x_0) ** 2 / gamma**2
            d_A = fac ** (-alpha)
            d_x_0 = 2 * amplitude * alpha * (x - x_0) * d_A / (fac * gamma**2)
            d_gamma = 2 * amplitude * alpha * (x - x_0) ** 2 * d_A / (fac * gamma**3)
            d_alpha = -amplitude * d_A * np.log(fac)
            return [d_A, d_x_0, d_gamma, d_alpha]

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            return {self.inputs[0]: self.x_0.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "gamma": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Moffat2D(Fittable2DModel):
        """
        Two dimensional Moffat model.

        Parameters
        ----------
        amplitude : float
            Amplitude of the model.
        x_0 : float
            x position of the maximum of the Moffat model.
        y_0 : float
            y position of the maximum of the Moffat model.
        gamma : float
            Core width of the Moffat model.
        alpha : float
            Power index of the Moffat model.

        See Also
        --------
        Gaussian2D, Box2D

        Notes
        -----
        Model formula:

        .. math::

            f(x, y) = A \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} +
            \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha}

        Note that if :math:`\\alpha` is 1, the `Moffat2D` profile is a
        `Lorentz2D` profile. In that case, the integral of the profile is
        infinite.
        """

        amplitude = Parameter(default=1, description="Amplitude (peak value) of the model")
        x_0 = Parameter(
            default=0, description="X position of the maximum of the Moffat model"
        )
        y_0 = Parameter(
            default=0, description="Y position of the maximum of the Moffat model"
        )
        gamma = Parameter(default=1, description="Core width of the Moffat model")
        alpha = Parameter(default=1, description="Power index of the Moffat model")

        @property
        def fwhm(self):
            """
            Moffat full width at half maximum.
            Derivation of the formula is available in
            `this notebook by Yoonsoo Bach
            <https://nbviewer.jupyter.org/github/ysbach/AO_2017/blob/master/04_Ground_Based_Concept.ipynb#1.2.-Moffat>`_.
            """
            return 2.0 * np.abs(self.gamma) * np.sqrt(2.0 ** (1.0 / self.alpha) - 1.0)

        @staticmethod
        def evaluate(x, y, amplitude, x_0, y_0, gamma, alpha):
            """Two dimensional Moffat model function."""
            rr_gg = ((x - x_0) ** 2 + (y - y_0) ** 2) / gamma**2
            return amplitude * (1 + rr_gg) ** (-alpha)

        @staticmethod
        def fit_deriv(x, y, amplitude, x_0, y_0, gamma, alpha):
            """Two dimensional Moffat model derivative with respect to parameters."""
            rr_gg = ((x - x_0) ** 2 + (y - y_0) ** 2) / gamma**2
            d_A = (1 + rr_gg) ** (-alpha)
            d_x_0 = 2 * amplitude * alpha * d_A * (x - x_0) / (gamma**2 * (1 + rr_gg))
            d_y_0 = 2 * amplitude * alpha * d_A * (y - y_0) / (gamma**2 * (1 + rr_gg))
            d_alpha = -amplitude * d_A * np.log(1 + rr_gg)
            d_gamma = 2 * amplitude * alpha * d_A * rr_gg / (gamma * (1 + rr_gg))
            return [d_A, d_x_0, d_y_0, d_gamma, d_alpha]

        @property
        def input_units(self):
            if self.x_0.input_unit is None:
                return None
            else:
                return {
                    self.inputs[0]: self.x_0.input_unit,
                    self.inputs[1]: self.y_0.input_unit,
                }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "gamma": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Sersic2D(Fittable2DModel):
        r"""
        Two dimensional Sersic surface brightness profile.

        Parameters
        ----------
        amplitude : float
            Surface brightness at ``r_eff``.
        r_eff : float
            Effective (half-light) radius.
        n : float
            Sersic index controlling the shape of the profile. Particular
            values of ``n`` are equivalent to the following profiles:

                * n=4 : `de Vaucouleurs <https://en.wikipedia.org/wiki/De_Vaucouleurs%27s_law>`_ :math:`r^{1/4}` profile
                * n=1 : Exponential profile
                * n=0.5 : Gaussian profile
        x_0 : float, optional
            x position of the center.
        y_0 : float, optional
            y position of the center.
        ellip : float, optional
            Ellipticity of the isophote, defined as 1.0 minus the ratio of
            the lengths of the semimajor and semiminor axes:

            .. math:: ellip = 1 - \frac{b}{a}
        theta : float or `~astropy.units.Quantity`, optional
            The rotation angle as an angular quantity
            (`~astropy.units.Quantity` or `~astropy.coordinates.Angle`)
            or a value in radians (as a float). The rotation angle
            increases counterclockwise from the positive x axis.

        See Also
        --------
        GeneralSersic2D, Gaussian2D, Moffat2D

        Notes
        -----
        The ``x``, ``y``, ``x_0``, ``y_0``, and ``r_eff`` inputs must have
        compatible units or be unitless numbers.

        Model formula:

        .. math::

            I(x, y) = I_{e} \exp\left\{
                      -b_{n} \left[\left(\frac{r(x, y)}{r_{e}}\right)^{(1/n)}
                      -1\right]\right\}

        where :math:`I_{e}` is the ``amplitude``, :math:`r_{e}` is ``reff``,
        and :math:`r(x, y)` is a rotated ellipse defined as:

        .. math::

            r(x, y)^2 = A^2 + \left(\frac{B}{1 - ellip}\right)^2

        .. math::

            A = (x - x_0) \cos(\theta) + (y - y_0) \sin(\theta)

        .. math::

            B = -(x - x_0) \sin(\theta) + (y - y_0) \cos(\theta)

        The constant :math:`b_{n}` is defined such that :math:`r_{e}`
        contains half the total luminosity. It can be solved for numerically
        from the following equation:

        .. math::

            \Gamma(2n) = 2\gamma (2n, b_{n})

        where :math:`\Gamma(a)` is the `gamma function
        <https://en.wikipedia.org/wiki/Gamma_function>`_ and
        :math:`\gamma(a, x)` is the `lower incomplete gamma function
        <https://en.wikipedia.org/wiki/Incomplete_gamma_function>`_.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            from astropy.modeling.models import Sersic2D
            import matplotlib.pyplot as plt

            x, y = np.meshgrid(np.arange(100), np.arange(100))

            mod = Sersic2D(amplitude=1, r_eff=25, n=4, x_0=50, y_0=50,
                           ellip=0.5, theta=-1)
            img = mod(x, y)
            log_img = np.log10(img)

            fig, ax = plt.subplots()
            im = ax.imshow(log_img, origin='lower', interpolation='nearest',
                           vmin=-1, vmax=2)
            cbar = fig.colorbar(im, ax=ax)
            cbar.set_label('Log Brightness', rotation=270, labelpad=25)
            cbar.set_ticks([-1, 0, 1, 2])
            plt.xlabel('x')
            plt.ylabel('y')
            plt.show()

        References
        ----------
        .. [1] http://ned.ipac.caltech.edu/level5/March05/Graham/Graham2.html
        """

        amplitude = Parameter(default=1, description="Surface brightness at r_eff")
        r_eff = Parameter(default=1, description="Effective (half-light) radius")
        n = Parameter(default=4, description="Sersic Index")
        x_0 = Parameter(default=0, description="X position of the center")
        y_0 = Parameter(default=0, description="Y position of the center")
        ellip = Parameter(default=0, description="Ellipticity")
        theta = Parameter(
            default=0.0,
            description=(
                "Rotation angle either as a float (in radians) or a |Quantity| angle"
            ),
        )

        @classmethod
        def evaluate(cls, x, y, amplitude, r_eff, n, x_0, y_0, ellip, theta, c=0):
            """Two dimensional Sersic profile function."""
            from scipy.special import gammaincinv

            bn = gammaincinv(2.0 * n, 0.5)
            cos_theta = np.cos(theta)
            sin_theta = np.sin(theta)
            x_maj = np.abs((x - x_0) * cos_theta + (y - y_0) * sin_theta)
            x_min = np.abs(-(x - x_0) * sin_theta + (y - y_0) * cos_theta)

            b = (1 - ellip) * r_eff
            expon = 2.0 + c
            inv_expon = 1.0 / expon
            z = ((x_maj / r_eff) ** expon + (x_min / b) ** expon) ** inv_expon
            return amplitude * np.exp(-bn * (z ** (1 / n) - 1.0))

        @property
        def input_units(self):
            if self.x_0.input_unit is not None:
                return {
                    self.inputs[0]: self.x_0.input_unit,
                    self.inputs[1]: self.y_0.input_unit,
                }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            # Note that here we need to make sure that x and y are in the same
            # units otherwise this can lead to issues since rotation is not well
            # defined.
            if inputs_unit[self.inputs[0]] != inputs_unit[self.inputs[1]]:
                raise UnitsError("Units of 'x' and 'y' inputs should match")
            return {
                "x_0": inputs_unit[self.inputs[0]],
                "y_0": inputs_unit[self.inputs[0]],
                "r_eff": inputs_unit[self.inputs[0]],
                "theta": u.rad,
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class GeneralSersic2D(Sersic2D):
        r"""
        Generalized two dimensional Sersic surface brightness profile that
        allows for "boxy" or "disky" (kite-like) isophote shapes.

        Parameters
        ----------
        amplitude : float
            Surface brightness at ``r_eff``.
        r_eff : float
            Effective (half-light) radius.
        n : float
            Sersic index controlling the shape of the profile. Particular
            values of ``n`` are equivalent to the following profiles:

                * n=4 : `de Vaucouleurs <https://en.wikipedia.org/wiki/De_Vaucouleurs%27s_law>`_ :math:`r^{1/4}` profile
                * n=1 : Exponential profile
                * n=0.5 : Gaussian profile
        x_0 : float, optional
            x position of the center.
        y_0 : float, optional
            y position of the center.
        ellip : float, optional
            Ellipticity of the isophote, defined as 1.0 minus the ratio of
            the lengths of the semimajor and semiminor axes:

            .. math:: ellip = 1 - \frac{b}{a}
        theta : float or `~astropy.units.Quantity`, optional
            The rotation angle as an angular quantity
            (`~astropy.units.Quantity` or `~astropy.coordinates.Angle`)
            or a value in radians (as a float). The rotation angle
            increases counterclockwise from the positive x axis.
        c : float, optional
            Parameter controlling the shape of the generalized ellipses.
            Negative values correspond to disky (kite-like) isophotes and
            positive values correspond to boxy isophotes. Setting ``c=0``
            provides perfectly elliptical isophotes (the same model as
            `Sersic2D`).

        See Also
        --------
        Sersic2D, Gaussian2D, Moffat2D

        Notes
        -----
        Model formula:

        .. math::

            I(x, y) = I_{e} \exp\left\{
                      -b_{n} \left[\left(\frac{r(x, y)}{r_{e}}\right)^{(1/n)}
                      -1\right]\right\}

        where :math:`I_{e}` is the ``amplitude``, :math:`r_{e}`
        is ``reff``, and :math:`r(x, y)` is a rotated
        "generalized" ellipse (see `Athanassoula et al. 1990
        <https://ui.adsabs.harvard.edu/abs/1990MNRAS.245..130A/abstract>`_)
        defined as:

        .. math::

            r(x, y)^2 = |A|^{c + 2}
                        + \left(\frac{|B|}{1 - ellip}\right)^{c + 2}

        .. math::

            A = (x - x_0) \cos(\theta) + (y - y_0) \sin(\theta)

        .. math::

            B = -(x - x_0) \sin(\theta) + (y - y_0) \cos(\theta)

        The constant :math:`b_{n}` is defined such that :math:`r_{e}`
        contains half the total luminosity. It can be solved for numerically
        from the following equation:

        .. math::

            \Gamma(2n) = 2\gamma (2n, b_{n})

        where :math:`\Gamma(a)` is the `gamma function
        <https://en.wikipedia.org/wiki/Gamma_function>`_ and
        :math:`\gamma(a, x)` is the `lower incomplete gamma function
        <https://en.wikipedia.org/wiki/Incomplete_gamma_function>`_.

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            from astropy.modeling.models import GeneralSersic2D
            import matplotlib.pyplot as plt

            x, y = np.meshgrid(np.arange(100), np.arange(100))

            mod = GeneralSersic2D(amplitude=1, r_eff=25, n=4, x_0=50, y_0=50,
                                  c=-1.0, ellip=0.5, theta=-1)
            img = mod(x, y)
            log_img = np.log10(img)

            fig, ax = plt.subplots()
            im = ax.imshow(log_img, origin='lower', interpolation='nearest',
                           vmin=-1, vmax=2)
            cbar = fig.colorbar(im, ax=ax)
            cbar.set_label('Log Brightness', rotation=270, labelpad=25)
            cbar.set_ticks([-1, 0, 1, 2])
            plt.title('Disky isophote with c=-1.0')
            plt.xlabel('x')
            plt.ylabel('y')
            plt.show()

        .. plot::
            :include-source:

            import numpy as np
            from astropy.modeling.models import GeneralSersic2D
            import matplotlib.pyplot as plt

            x, y = np.meshgrid(np.arange(100), np.arange(100))

            mod = GeneralSersic2D(amplitude=1, r_eff=25, n=4, x_0=50, y_0=50,
                                  c=1.0, ellip=0.5, theta=-1)
            img = mod(x, y)
            log_img = np.log10(img)

            fig, ax = plt.subplots()
            im = ax.imshow(log_img, origin='lower', interpolation='nearest',
                           vmin=-1, vmax=2)
            cbar = fig.colorbar(im, ax=ax)
            cbar.set_label('Log Brightness', rotation=270, labelpad=25)
            cbar.set_ticks([-1, 0, 1, 2])
            plt.title('Boxy isophote with c=1.0')
            plt.xlabel('x')
            plt.ylabel('y')
            plt.show()

        References
        ----------
        .. [1] http://ned.ipac.caltech.edu/level5/March05/Graham/Graham2.html
        .. [2] https://ui.adsabs.harvard.edu/abs/1990MNRAS.245..130A/abstract
        """

        amplitude = Parameter(default=1, description="Surface brightness at r_eff")
        r_eff = Parameter(default=1, description="Effective (half-light) radius")
        n = Parameter(default=4, description="Sersic Index")
        x_0 = Parameter(default=0, description="X position of the center")
        y_0 = Parameter(default=0, description="Y position of the center")
        ellip = Parameter(default=0, description="Ellipticity")
        theta = Parameter(
            default=0.0,
            description=(
                "Rotation angle either as a float (in radians) or a |Quantity| angle"
            ),
        )
        c = Parameter(default=0, description="Isophote shape parameter")

    class KingProjectedAnalytic1D(Fittable1DModel):
        """
        Projected (surface density) analytic King Model.

        Parameters
        ----------
        amplitude : float
            Amplitude or scaling factor.
        r_core : float
            Core radius (f(r_c) ~ 0.5 f_0)
        r_tide : float
            Tidal radius.

        Notes
        -----
        This model approximates a King model with an analytic function. The derivation of this
        equation can be found in King '62 (equation 14). This is just an approximation of the
        full model and the parameters derived from this model should be taken with caution.
        It usually works for models with a concentration (c = log10(r_t/r_c) parameter < 2.

        Model formula:

        .. math::

            f(x) = A r_c^2  \\left(\\frac{1}{\\sqrt{(x^2 + r_c^2)}} -
            \\frac{1}{\\sqrt{(r_t^2 + r_c^2)}}\\right)^2

        Examples
        --------
        .. plot::
            :include-source:

            import numpy as np
            from astropy.modeling.models import KingProjectedAnalytic1D
            import matplotlib.pyplot as plt

            plt.figure()
            rt_list = [1, 2, 5, 10, 20]
            for rt in rt_list:
                r = np.linspace(0.1, rt, 100)

                mod = KingProjectedAnalytic1D(amplitude = 1, r_core = 1., r_tide = rt)
                sig = mod(r)

                plt.loglog(r, sig/sig[0], label=f"c ~ {mod.concentration:0.2f}")

            plt.xlabel("r")
            plt.ylabel(r"$\\sigma/\\sigma_0$")
            plt.legend()
            plt.show()

        References
        ----------
        .. [1] https://ui.adsabs.harvard.edu/abs/1962AJ.....67..471K
        """

        amplitude = Parameter(
            default=1,
            bounds=(FLOAT_EPSILON, None),
            description="Amplitude or scaling factor",
        )
        r_core = Parameter(
            default=1, bounds=(FLOAT_EPSILON, None), description="Core Radius"
        )
        r_tide = Parameter(
            default=2, bounds=(FLOAT_EPSILON, None), description="Tidal Radius"
        )

        @property
        def concentration(self):
            """Concentration parameter of the king model."""
            return np.log10(np.abs(self.r_tide / self.r_core))

        @staticmethod
        def _core_func(x, r_core, r_tide, power=1):
            return (
                1.0 / np.sqrt(x**2 + r_core**2) ** power
                - 1.0 / np.sqrt(r_tide**2 + r_core**2) ** power
            )

        @staticmethod
        def _filter(x, r_tide, result):
            """Set invalid r values to 0"""
            bounds = (x >= r_tide) | (x < 0)
            result[bounds] = result[bounds] * 0.0

        def evaluate(self, x, amplitude, r_core, r_tide):
            """
            Analytic King model function.
            """
            result = amplitude * r_core**2 * self._core_func(x, r_core, r_tide) ** 2
            self._filter(x, r_tide, result)

            return result

        def fit_deriv(self, x, amplitude, r_core, r_tide):
            """
            Analytic King model function derivatives.
            """
            d_amplitude = r_core**2 * self._core_func(x, r_core, r_tide) ** 2
            self._filter(x, r_tide, d_amplitude)

            d_r_core = (
                -2.0
                * amplitude
                * r_core**3
                * self._core_func(x, r_core, r_tide, power=3)
                * self._core_func(x, r_core, r_tide)
                + 2 * amplitude * r_core * self._core_func(x, r_core, r_tide) ** 2
            )
            self._filter(x, r_tide, d_r_core)

            d_r_tide = (
                2 * amplitude * r_core**2 * r_tide * self._core_func(x, r_core, r_tide)
            ) / (r_core**2 + r_tide**2) ** (3 / 2)
            self._filter(x, r_tide, d_r_tide)

            return [d_amplitude, d_r_core, d_r_tide]

        @property
        def bounding_box(self):
            """
            Tuple defining the default ``bounding_box`` limits.

            The model is not defined for r > r_tide.

            ``(r_low, r_high)``
            """
            return (0 * self.r_tide, 1 * self.r_tide)

        @property
        def input_units(self):
            if self.r_core.input_unit is None:
                return None
            return {self.inputs[0]: self.r_core.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "r_core": inputs_unit[self.inputs[0]],
                "r_tide": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Logarithmic1D(Fittable1DModel):
        """
        One dimensional logarithmic model.

        Parameters
        ----------
        amplitude : float, optional
        tau : float, optional

        See Also
        --------
        Exponential1D, Gaussian1D
        """

        amplitude = Parameter(default=1)
        tau = Parameter(default=1)

        @staticmethod
        def evaluate(x, amplitude, tau):
            return amplitude * np.log(x / tau)

        @staticmethod
        def fit_deriv(x, amplitude, tau):
            d_amplitude = np.log(x / tau)
            d_tau = np.zeros(x.shape) - (amplitude / tau)
            return [d_amplitude, d_tau]

        @property
        def inverse(self):
            new_amplitude = self.tau
            new_tau = self.amplitude
            return Exponential1D(amplitude=new_amplitude, tau=new_tau)

        def _tau_validator(self, val):
            if np.all(val == 0):
                raise ValueError("0 is not an allowed value for tau")

        tau._validator = _tau_validator

        @property
        def input_units(self):
            if self.tau.input_unit is None:
                return None
            return {self.inputs[0]: self.tau.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "tau": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }

    class Exponential1D(Fittable1DModel):
        """
        One dimensional exponential model.

        Parameters
        ----------
        amplitude : float, optional
        tau : float, optional

        See Also
        --------
        Logarithmic1D, Gaussian1D
        """

        amplitude = Parameter(default=1)
        tau = Parameter(default=1)

        @staticmethod
        def evaluate(x, amplitude, tau):
            return amplitude * np.exp(x / tau)

        @staticmethod
        def fit_deriv(x, amplitude, tau):
            """Derivative with respect to parameters."""
            d_amplitude = np.exp(x / tau)
            d_tau = -amplitude * (x / tau**2) * np.exp(x / tau)
            return [d_amplitude, d_tau]

        @property
        def inverse(self):
            new_amplitude = self.tau
            new_tau = self.amplitude
            return Logarithmic1D(amplitude=new_amplitude, tau=new_tau)

        def _tau_validator(self, val):
            """tau cannot be 0."""
            if np.all(val == 0):
                raise ValueError("0 is not an allowed value for tau")

        tau._validator = _tau_validator

        @property
        def input_units(self):
            if self.tau.input_unit is None:
                return None
            return {self.inputs[0]: self.tau.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            return {
                "tau": inputs_unit[self.inputs[0]],
                "amplitude": outputs_unit[self.outputs[0]],
            }