text: # Licensed under a 3-clause BSD style license - see LICENSE.rst
  # -*- coding: utf-8 -*-
  
  """
  Implements projections--particularly sky projections defined in WCS Paper II
  [1]_.
  
  All angles are set and and displayed in degrees but internally computations are
  performed in radians. All functions expect inputs and outputs degrees.
  
  References
  ----------
  .. [1] Calabretta, M.R., Greisen, E.W., 2002, A&A, 395, 1077 (Paper II)
  """
  
  
  import abc
  
  import numpy as np
  
  from .core import Model
  from .parameters import Parameter, InputParameterError
  
  from .. import units as u
  
  from . import _projections
  from .utils import _to_radian, _to_orig_unit
  
  
  projcodes = [
      'AZP', 'SZP', 'TAN', 'STG', 'SIN', 'ARC', 'ZEA', 'AIR', 'CYP',
      'CEA', 'CAR', 'MER', 'SFL', 'PAR', 'MOL', 'AIT', 'COP', 'COE',
      'COD', 'COO', 'BON', 'PCO', 'TSC', 'CSC', 'QSC', 'HPX', 'XPH'
  ]
  
  
  __all__ = ['Projection', 'Pix2SkyProjection', 'Sky2PixProjection',
             'Zenithal', 'Cylindrical', 'PseudoCylindrical', 'Conic',
             'PseudoConic', 'QuadCube', 'HEALPix',
             'AffineTransformation2D',
             'projcodes',
  
             'Pix2Sky_ZenithalPerspective', 'Sky2Pix_ZenithalPerspective',
             'Pix2Sky_SlantZenithalPerspective', 'Sky2Pix_SlantZenithalPerspective',
             'Pix2Sky_Gnomonic', 'Sky2Pix_Gnomonic',
             'Pix2Sky_Stereographic', 'Sky2Pix_Stereographic',
             'Pix2Sky_SlantOrthographic', 'Sky2Pix_SlantOrthographic',
             'Pix2Sky_ZenithalEquidistant', 'Sky2Pix_ZenithalEquidistant',
             'Pix2Sky_ZenithalEqualArea', 'Sky2Pix_ZenithalEqualArea',
             'Pix2Sky_Airy', 'Sky2Pix_Airy',
             'Pix2Sky_CylindricalPerspective', 'Sky2Pix_CylindricalPerspective',
             'Pix2Sky_CylindricalEqualArea', 'Sky2Pix_CylindricalEqualArea',
             'Pix2Sky_PlateCarree', 'Sky2Pix_PlateCarree',
             'Pix2Sky_Mercator', 'Sky2Pix_Mercator',
             'Pix2Sky_SansonFlamsteed', 'Sky2Pix_SansonFlamsteed',
             'Pix2Sky_Parabolic', 'Sky2Pix_Parabolic',
             'Pix2Sky_Molleweide', 'Sky2Pix_Molleweide',
             'Pix2Sky_HammerAitoff', 'Sky2Pix_HammerAitoff',
             'Pix2Sky_ConicPerspective', 'Sky2Pix_ConicPerspective',
             'Pix2Sky_ConicEqualArea', 'Sky2Pix_ConicEqualArea',
             'Pix2Sky_ConicEquidistant', 'Sky2Pix_ConicEquidistant',
             'Pix2Sky_ConicOrthomorphic', 'Sky2Pix_ConicOrthomorphic',
             'Pix2Sky_BonneEqualArea', 'Sky2Pix_BonneEqualArea',
             'Pix2Sky_Polyconic', 'Sky2Pix_Polyconic',
             'Pix2Sky_TangentialSphericalCube', 'Sky2Pix_TangentialSphericalCube',
             'Pix2Sky_COBEQuadSphericalCube', 'Sky2Pix_COBEQuadSphericalCube',
             'Pix2Sky_QuadSphericalCube', 'Sky2Pix_QuadSphericalCube',
             'Pix2Sky_HEALPix', 'Sky2Pix_HEALPix',
             'Pix2Sky_HEALPixPolar', 'Sky2Pix_HEALPixPolar',
  
             # The following are short FITS WCS aliases
             'Pix2Sky_AZP', 'Sky2Pix_AZP',
             'Pix2Sky_SZP', 'Sky2Pix_SZP',
             'Pix2Sky_TAN', 'Sky2Pix_TAN',
             'Pix2Sky_STG', 'Sky2Pix_STG',
             'Pix2Sky_SIN', 'Sky2Pix_SIN',
             'Pix2Sky_ARC', 'Sky2Pix_ARC',
             'Pix2Sky_ZEA', 'Sky2Pix_ZEA',
             'Pix2Sky_AIR', 'Sky2Pix_AIR',
             'Pix2Sky_CYP', 'Sky2Pix_CYP',
             'Pix2Sky_CEA', 'Sky2Pix_CEA',
             'Pix2Sky_CAR', 'Sky2Pix_CAR',
             'Pix2Sky_MER', 'Sky2Pix_MER',
             'Pix2Sky_SFL', 'Sky2Pix_SFL',
             'Pix2Sky_PAR', 'Sky2Pix_PAR',
             'Pix2Sky_MOL', 'Sky2Pix_MOL',
             'Pix2Sky_AIT', 'Sky2Pix_AIT',
             'Pix2Sky_COP', 'Sky2Pix_COP',
             'Pix2Sky_COE', 'Sky2Pix_COE',
             'Pix2Sky_COD', 'Sky2Pix_COD',
             'Pix2Sky_COO', 'Sky2Pix_COO',
             'Pix2Sky_BON', 'Sky2Pix_BON',
             'Pix2Sky_PCO', 'Sky2Pix_PCO',
             'Pix2Sky_TSC', 'Sky2Pix_TSC',
             'Pix2Sky_CSC', 'Sky2Pix_CSC',
             'Pix2Sky_QSC', 'Sky2Pix_QSC',
             'Pix2Sky_HPX', 'Sky2Pix_HPX',
             'Pix2Sky_XPH', 'Sky2Pix_XPH'
  ]
  
  
  class Projection(Model):
      """Base class for all sky projections."""
  
      # Radius of the generating sphere.
      # This sets the circumference to 360 deg so that arc length is measured in deg.
      r0 = 180 * u.deg / np.pi
  
      _separable = False
  
      @property
      @abc.abstractmethod
      def inverse(self):
          """
          Inverse projection--all projection models must provide an inverse.
          """
  
  
  class Pix2SkyProjection(Projection):
      """Base class for all Pix2Sky projections."""
  
      inputs = ('x', 'y')
      outputs = ('phi', 'theta')
  
      _input_units_strict = True
      _input_units_allow_dimensionless = True
  
      @property
      def input_units(self):
          return {'x': u.deg, 'y': u.deg}
  
      @property
      def return_units(self):
          return {'phi': u.deg, 'theta': u.deg}
  
  
  class Sky2PixProjection(Projection):
      """Base class for all Sky2Pix projections."""
  
      inputs = ('phi', 'theta')
      outputs = ('x', 'y')
  
      _input_units_strict = True
      _input_units_allow_dimensionless = True
  
      @property
      def input_units(self):
          return {'phi': u.deg, 'theta': u.deg}
  
      @property
      def return_units(self):
          return {'x': u.deg, 'y': u.deg}
  
  
  class Zenithal(Projection):
      r"""Base class for all Zenithal projections.
  
      Zenithal (or azimuthal) projections map the sphere directly onto a
      plane.  All zenithal projections are specified by defining the
      radius as a function of native latitude, :math:`R_\theta`.
  
      The pixel-to-sky transformation is defined as:
  
      .. math::
          \phi &= \arg(-y, x) \\
          R_\theta &= \sqrt{x^2 + y^2}
  
      and the inverse (sky-to-pixel) is defined as:
  
      .. math::
          x &= R_\theta \sin \phi \\
          y &= R_\theta \cos \phi
      """
  
      _separable = False
  
  
  class Pix2Sky_ZenithalPerspective(Pix2SkyProjection, Zenithal):
      r"""
      Zenithal perspective projection - pixel to sky.
  
      Corresponds to the ``AZP`` projection in FITS WCS.
  
      .. math::
          \phi &= \arg(-y \cos \gamma, x) \\
          \theta &= \left\{\genfrac{}{}{0pt}{}{\psi - \omega}{\psi + \omega + 180^{\circ}}\right.
  
      where:
  
      .. math::
          \psi &= \arg(\rho, 1) \\
          \omega &= \sin^{-1}\left(\frac{\rho \mu}{\sqrt{\rho^2 + 1}}\right) \\
          \rho &= \frac{R}{\frac{180^{\circ}}{\pi}(\mu + 1) + y \sin \gamma} \\
          R &= \sqrt{x^2 + y^2 \cos^2 \gamma}
  
      Parameters
      --------------
      mu : float
          Distance from point of projection to center of sphere
          in spherical radii, μ.  Default is 0.
  
      gamma : float
          Look angle γ in degrees.  Default is 0°.
      """
  
      mu = Parameter(default=0.0)
      gamma = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
  
      def __init__(self, mu=mu.default, gamma=gamma.default, **kwargs):
          # units : mu - in spherical radii, gamma - in deg
          # TODO: Support quantity objects here and in similar contexts
          super().__init__(mu, gamma, **kwargs)
  
      @mu.validator
      def mu(self, value):
          if np.any(value == -1):
              raise InputParameterError(
                  "Zenithal perspective projection is not defined for mu = -1")
  
      @property
      def inverse(self):
          return Sky2Pix_ZenithalPerspective(self.mu.value, self.gamma.value)
  
      @classmethod
      def evaluate(cls, x, y, mu, gamma):
          return _projections.azpx2s(x, y, mu, _to_orig_unit(gamma))
  
  
  Pix2Sky_AZP = Pix2Sky_ZenithalPerspective
  
  
  class Sky2Pix_ZenithalPerspective(Sky2PixProjection, Zenithal):
      r"""
      Zenithal perspective projection - sky to pixel.
  
      Corresponds to the ``AZP`` projection in FITS WCS.
  
      .. math::
          x &= R \sin \phi \\
          y &= -R \sec \gamma \cos \theta
  
      where:
  
      .. math::
          R = \frac{180^{\circ}}{\pi} \frac{(\mu + 1) \cos \theta}{(\mu + \sin \theta) + \cos \theta \cos \phi \tan \gamma}
  
      Parameters
      ----------
      mu : float
          Distance from point of projection to center of sphere
          in spherical radii, μ. Default is 0.
  
      gamma : float
          Look angle γ in degrees. Default is 0°.
      """
  
      mu = Parameter(default=0.0)
      gamma = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
  
      @mu.validator
      def mu(self, value):
          if np.any(value == -1):
              raise InputParameterError(
                  "Zenithal perspective projection is not defined for mu = -1")
  
      @property
      def inverse(self):
          return Pix2Sky_AZP(self.mu.value, self.gamma.value)
  
      @classmethod
      def evaluate(cls, phi, theta, mu, gamma):
          return _projections.azps2x(
              phi, theta, mu, _to_orig_unit(gamma))
  
  
  Sky2Pix_AZP = Sky2Pix_ZenithalPerspective
  
  
  class Pix2Sky_SlantZenithalPerspective(Pix2SkyProjection, Zenithal):
      r"""
      Slant zenithal perspective projection - pixel to sky.
  
      Corresponds to the ``SZP`` projection in FITS WCS.
  
      Parameters
      --------------
      mu : float
          Distance from point of projection to center of sphere
          in spherical radii, μ.  Default is 0.
  
      phi0 : float
          The longitude φ₀ of the reference point, in degrees.  Default
          is 0°.
  
      theta0 : float
          The latitude θ₀ of the reference point, in degrees.  Default
          is 90°.
      """
  
      def _validate_mu(mu):
          if np.asarray(mu == -1).any():
              raise ValueError(
                  "Zenithal perspective projection is not defined for mu=-1")
          return mu
  
      mu = Parameter(default=0.0, setter=_validate_mu)
      phi0 = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
      theta0 = Parameter(default=90.0, getter=_to_orig_unit, setter=_to_radian)
  
      @property
      def inverse(self):
          return Sky2Pix_SlantZenithalPerspective(
              self.mu.value, self.phi0.value, self.theta0.value)
  
      @classmethod
      def evaluate(cls, x, y, mu, phi0, theta0):
          return _projections.szpx2s(
              x, y, mu, _to_orig_unit(phi0), _to_orig_unit(theta0))
  
  
  Pix2Sky_SZP = Pix2Sky_SlantZenithalPerspective
  
  
  class Sky2Pix_SlantZenithalPerspective(Sky2PixProjection, Zenithal):
      r"""
      Zenithal perspective projection - sky to pixel.
  
      Corresponds to the ``SZP`` projection in FITS WCS.
  
      Parameters
      ----------
      mu : float
          distance from point of projection to center of sphere
          in spherical radii, μ.  Default is 0.
  
      phi0 : float
          The longitude φ₀ of the reference point, in degrees.  Default
          is 0°.
  
      theta0 : float
          The latitude θ₀ of the reference point, in degrees.  Default
          is 90°.
      """
  
      def _validate_mu(mu):
          if np.asarray(mu == -1).any():
              raise ValueError("Zenithal perspective projection is not defined for mu=-1")
          return mu
  
      mu = Parameter(default=0.0, setter=_validate_mu)
      phi0 = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
      theta0 = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
  
      @property
      def inverse(self):
          return Pix2Sky_SlantZenithalPerspective(
              self.mu.value, self.phi0.value, self.theta0.value)
  
      @classmethod
      def evaluate(cls, phi, theta, mu, phi0, theta0):
          return _projections.szps2x(
              phi, theta, mu, _to_orig_unit(phi0), _to_orig_unit(theta0))
  
  
  Sky2Pix_SZP = Sky2Pix_SlantZenithalPerspective
  
  
  class Pix2Sky_Gnomonic(Pix2SkyProjection, Zenithal):
      r"""
      Gnomonic projection - pixel to sky.
  
      Corresponds to the ``TAN`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          \theta = \tan^{-1}\left(\frac{180^{\circ}}{\pi R_\theta}\right)
      """
  
      @property
      def inverse(self):
          return Sky2Pix_Gnomonic()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.tanx2s(x, y)
  
  
  Pix2Sky_TAN = Pix2Sky_Gnomonic
  
  
  class Sky2Pix_Gnomonic(Sky2PixProjection, Zenithal):
      r"""
      Gnomonic Projection - sky to pixel.
  
      Corresponds to the ``TAN`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          R_\theta = \frac{180^{\circ}}{\pi}\cot \theta
      """
  
      @property
      def inverse(self):
          return Pix2Sky_Gnomonic()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.tans2x(phi, theta)
  
  
  Sky2Pix_TAN = Sky2Pix_Gnomonic
  
  
  class Pix2Sky_Stereographic(Pix2SkyProjection, Zenithal):
      r"""
      Stereographic Projection - pixel to sky.
  
      Corresponds to the ``STG`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          \theta = 90^{\circ} - 2 \tan^{-1}\left(\frac{\pi R_\theta}{360^{\circ}}\right)
      """
  
      @property
      def inverse(self):
          return Sky2Pix_Stereographic()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.stgx2s(x, y)
  
  
  Pix2Sky_STG = Pix2Sky_Stereographic
  
  
  class Sky2Pix_Stereographic(Sky2PixProjection, Zenithal):
      r"""
      Stereographic Projection - sky to pixel.
  
      Corresponds to the ``STG`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          R_\theta = \frac{180^{\circ}}{\pi}\frac{2 \cos \theta}{1 + \sin \theta}
      """
  
      @property
      def inverse(self):
          return Pix2Sky_Stereographic()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.stgs2x(phi, theta)
  
  
  Sky2Pix_STG = Sky2Pix_Stereographic
  
  
  class Pix2Sky_SlantOrthographic(Pix2SkyProjection, Zenithal):
      r"""
      Slant orthographic projection - pixel to sky.
  
      Corresponds to the ``SIN`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      The following transformation applies when :math:`\xi` and
      :math:`\eta` are both zero.
  
      .. math::
          \theta = \cos^{-1}\left(\frac{\pi}{180^{\circ}}R_\theta\right)
  
      The parameters :math:`\xi` and :math:`\eta` are defined from the
      reference point :math:`(\phi_c, \theta_c)` as:
  
      .. math::
          \xi &= \cot \theta_c \sin \phi_c \\
          \eta &= - \cot \theta_c \cos \phi_c
  
      Parameters
      ----------
      xi : float
          Obliqueness parameter, ξ.  Default is 0.0.
  
      eta : float
          Obliqueness parameter, η.  Default is 0.0.
      """
  
      xi = Parameter(default=0.0)
      eta = Parameter(default=0.0)
  
      @property
      def inverse(self):
          return Sky2Pix_SlantOrthographic(self.xi.value, self.eta.value)
  
      @classmethod
      def evaluate(cls, x, y, xi, eta):
          return _projections.sinx2s(x, y, xi, eta)
  
  
  Pix2Sky_SIN = Pix2Sky_SlantOrthographic
  
  
  class Sky2Pix_SlantOrthographic(Sky2PixProjection, Zenithal):
      r"""
      Slant orthographic projection - sky to pixel.
  
      Corresponds to the ``SIN`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      The following transformation applies when :math:`\xi` and
      :math:`\eta` are both zero.
  
      .. math::
          R_\theta = \frac{180^{\circ}}{\pi}\cos \theta
  
      But more specifically are:
  
      .. math::
          x &= \frac{180^\circ}{\pi}[\cos \theta \sin \phi + \xi(1 - \sin \theta)] \\
          y &= \frac{180^\circ}{\pi}[\cos \theta \cos \phi + \eta(1 - \sin \theta)]
      """
  
      xi = Parameter(default=0.0)
      eta = Parameter(default=0.0)
  
      @property
      def inverse(self):
          return Pix2Sky_SlantOrthographic(self.xi.value, self.eta.value)
  
      @classmethod
      def evaluate(cls, phi, theta, xi, eta):
          return _projections.sins2x(phi, theta, xi, eta)
  
  
  Sky2Pix_SIN = Sky2Pix_SlantOrthographic
  
  
  class Pix2Sky_ZenithalEquidistant(Pix2SkyProjection, Zenithal):
      r"""
      Zenithal equidistant projection - pixel to sky.
  
      Corresponds to the ``ARC`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          \theta = 90^\circ - R_\theta
      """
      @property
      def inverse(self):
          return Sky2Pix_ZenithalEquidistant()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.arcx2s(x, y)
  
  
  Pix2Sky_ARC = Pix2Sky_ZenithalEquidistant
  
  
  class Sky2Pix_ZenithalEquidistant(Sky2PixProjection, Zenithal):
      r"""
      Zenithal equidistant projection - sky to pixel.
  
      Corresponds to the ``ARC`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          R_\theta = 90^\circ - \theta
      """
      @property
      def inverse(self):
          return Pix2Sky_ZenithalEquidistant()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.arcs2x(phi, theta)
  
  
  Sky2Pix_ARC = Sky2Pix_ZenithalEquidistant
  
  
  class Pix2Sky_ZenithalEqualArea(Pix2SkyProjection, Zenithal):
      r"""
      Zenithal equidistant projection - pixel to sky.
  
      Corresponds to the ``ZEA`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          \theta = 90^\circ - 2 \sin^{-1} \left(\frac{\pi R_\theta}{360^\circ}\right)
      """
      @property
      def inverse(self):
          return Sky2Pix_ZenithalEqualArea()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.zeax2s(x, y)
  
  
  Pix2Sky_ZEA = Pix2Sky_ZenithalEqualArea
  
  
  class Sky2Pix_ZenithalEqualArea(Sky2PixProjection, Zenithal):
      r"""
      Zenithal equidistant projection - sky to pixel.
  
      Corresponds to the ``ZEA`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          R_\theta &= \frac{180^\circ}{\pi} \sqrt{2(1 - \sin\theta)} \\
                   &= \frac{360^\circ}{\pi} \sin\left(\frac{90^\circ - \theta}{2}\right)
      """
      @property
      def inverse(self):
          return Pix2Sky_ZenithalEqualArea()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.zeas2x(phi, theta)
  
  
  Sky2Pix_ZEA = Sky2Pix_ZenithalEqualArea
  
  
  class Pix2Sky_Airy(Pix2SkyProjection, Zenithal):
      r"""
      Airy projection - pixel to sky.
  
      Corresponds to the ``AIR`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      Parameters
      ----------
      theta_b : float
          The latitude :math:`\theta_b` at which to minimize the error,
          in degrees.  Default is 90°.
      """
      theta_b = Parameter(default=90.0)
  
      @property
      def inverse(self):
          return Sky2Pix_Airy(self.theta_b.value)
  
      @classmethod
      def evaluate(cls, x, y, theta_b):
          return _projections.airx2s(x, y, theta_b)
  
  
  Pix2Sky_AIR = Pix2Sky_Airy
  
  
  class Sky2Pix_Airy(Sky2PixProjection, Zenithal):
      r"""
      Airy - sky to pixel.
  
      Corresponds to the ``AIR`` projection in FITS WCS.
  
      See `Zenithal` for a definition of the full transformation.
  
      .. math::
          R_\theta = -2 \frac{180^\circ}{\pi}\left(\frac{\ln(\cos \xi)}{\tan \xi} + \frac{\ln(\cos \xi_b)}{\tan^2 \xi_b} \tan \xi \right)
  
      where:
  
      .. math::
          \xi &= \frac{90^\circ - \theta}{2} \\
          \xi_b &= \frac{90^\circ - \theta_b}{2}
  
      Parameters
      ----------
      theta_b : float
          The latitude :math:`\theta_b` at which to minimize the error,
          in degrees.  Default is 90°.
      """
      theta_b = Parameter(default=90.0)
  
      @property
      def inverse(self):
          return Pix2Sky_Airy(self.theta_b.value)
  
      @classmethod
      def evaluate(cls, phi, theta, theta_b):
          return _projections.airs2x(phi, theta, theta_b)
  
  
  Sky2Pix_AIR = Sky2Pix_Airy
  
  
  class Cylindrical(Projection):
      r"""Base class for Cylindrical projections.
  
      Cylindrical projections are so-named because the surface of
      projection is a cylinder.
      """
      _separable = True
  
  
  class Pix2Sky_CylindricalPerspective(Pix2SkyProjection, Cylindrical):
      r"""
      Cylindrical perspective - pixel to sky.
  
      Corresponds to the ``CYP`` projection in FITS WCS.
  
      .. math::
          \phi &= \frac{x}{\lambda} \\
          \theta &= \arg(1, \eta) + \sin{-1}\left(\frac{\eta \mu}{\sqrt{\eta^2 + 1}}\right)
  
      where:
  
      .. math::
          \eta = \frac{\pi}{180^{\circ}}\frac{y}{\mu + \lambda}
  
      Parameters
      ----------
      mu : float
          Distance from center of sphere in the direction opposite the
          projected surface, in spherical radii, μ. Default is 1.
  
      lam : float
          Radius of the cylinder in spherical radii, λ. Default is 1.
      """
  
      mu = Parameter(default=1.0)
      lam = Parameter(default=1.0)
  
      @mu.validator
      def mu(self, value):
          if np.any(value == -self.lam):
              raise InputParameterError(
                  "CYP projection is not defined for mu = -lambda")
  
      @lam.validator
      def lam(self, value):
          if np.any(value == -self.mu):
              raise InputParameterError(
                  "CYP projection is not defined for lambda = -mu")
  
      @property
      def inverse(self):
          return Sky2Pix_CylindricalPerspective(self.mu.value, self.lam.value)
  
      @classmethod
      def evaluate(cls, x, y, mu, lam):
          return _projections.cypx2s(x, y, mu, lam)
  
  
  Pix2Sky_CYP = Pix2Sky_CylindricalPerspective
  
  
  class Sky2Pix_CylindricalPerspective(Sky2PixProjection, Cylindrical):
      r"""
      Cylindrical Perspective - sky to pixel.
  
      Corresponds to the ``CYP`` projection in FITS WCS.
  
      .. math::
          x &= \lambda \phi \\
          y &= \frac{180^{\circ}}{\pi}\left(\frac{\mu + \lambda}{\mu + \cos \theta}\right)\sin \theta
  
      Parameters
      ----------
      mu : float
          Distance from center of sphere in the direction opposite the
          projected surface, in spherical radii, μ.  Default is 0.
  
      lam : float
          Radius of the cylinder in spherical radii, λ.  Default is 0.
      """
  
      mu = Parameter(default=1.0)
      lam = Parameter(default=1.0)
  
      @mu.validator
      def mu(self, value):
          if np.any(value == -self.lam):
              raise InputParameterError(
                  "CYP projection is not defined for mu = -lambda")
  
      @lam.validator
      def lam(self, value):
          if np.any(value == -self.mu):
              raise InputParameterError(
                  "CYP projection is not defined for lambda = -mu")
  
      @property
      def inverse(self):
          return Pix2Sky_CylindricalPerspective(self.mu, self.lam)
  
      @classmethod
      def evaluate(cls, phi, theta, mu, lam):
          return _projections.cyps2x(phi, theta, mu, lam)
  
  
  Sky2Pix_CYP = Sky2Pix_CylindricalPerspective
  
  
  class Pix2Sky_CylindricalEqualArea(Pix2SkyProjection, Cylindrical):
      r"""
      Cylindrical equal area projection - pixel to sky.
  
      Corresponds to the ``CEA`` projection in FITS WCS.
  
      .. math::
          \phi &= x \\
          \theta &= \sin^{-1}\left(\frac{\pi}{180^{\circ}}\lambda y\right)
  
      Parameters
      ----------
      lam : float
          Radius of the cylinder in spherical radii, λ.  Default is 0.
      """
  
      lam = Parameter(default=1)
  
      @property
      def inverse(self):
          return Sky2Pix_CylindricalEqualArea(self.lam)
  
      @classmethod
      def evaluate(cls, x, y, lam):
          return _projections.ceax2s(x, y, lam)
  
  
  Pix2Sky_CEA = Pix2Sky_CylindricalEqualArea
  
  
  class Sky2Pix_CylindricalEqualArea(Sky2PixProjection, Cylindrical):
      r"""
      Cylindrical equal area projection - sky to pixel.
  
      Corresponds to the ``CEA`` projection in FITS WCS.
  
      .. math::
          x &= \phi \\
          y &= \frac{180^{\circ}}{\pi}\frac{\sin \theta}{\lambda}
  
      Parameters
      ----------
      lam : float
          Radius of the cylinder in spherical radii, λ.  Default is 0.
      """
  
      lam = Parameter(default=1)
  
      @property
      def inverse(self):
          return Pix2Sky_CylindricalEqualArea(self.lam)
  
      @classmethod
      def evaluate(cls, phi, theta, lam):
          return _projections.ceas2x(phi, theta, lam)
  
  
  Sky2Pix_CEA = Sky2Pix_CylindricalEqualArea
  
  
  class Pix2Sky_PlateCarree(Pix2SkyProjection, Cylindrical):
      r"""
      Plate carrée projection - pixel to sky.
  
      Corresponds to the ``CAR`` projection in FITS WCS.
  
      .. math::
          \phi &= x \\
          \theta &= y
      """
  
      @property
      def inverse(self):
          return Sky2Pix_PlateCarree()
  
      @staticmethod
      def evaluate(x, y):
          # The intermediate variables are only used here for clarity
          phi = np.array(x, copy=True)
          theta = np.array(y, copy=True)
  
          return phi, theta
  
  
  Pix2Sky_CAR = Pix2Sky_PlateCarree
  
  
  class Sky2Pix_PlateCarree(Sky2PixProjection, Cylindrical):
      r"""
      Plate carrée projection - sky to pixel.
  
      Corresponds to the ``CAR`` projection in FITS WCS.
  
      .. math::
          x &= \phi \\
          y &= \theta
      """
  
      @property
      def inverse(self):
          return Pix2Sky_PlateCarree()
  
      @staticmethod
      def evaluate(phi, theta):
          # The intermediate variables are only used here for clarity
          x = np.array(phi, copy=True)
          y = np.array(theta, copy=True)
  
          return x, y
  
  
  Sky2Pix_CAR = Sky2Pix_PlateCarree
  
  
  class Pix2Sky_Mercator(Pix2SkyProjection, Cylindrical):
      r"""
      Mercator - pixel to sky.
  
      Corresponds to the ``MER`` projection in FITS WCS.
  
      .. math::
          \phi &= x \\
          \theta &= 2 \tan^{-1}\left(e^{y \pi / 180^{\circ}}\right)-90^{\circ}
      """
  
      @property
      def inverse(self):
          return Sky2Pix_Mercator()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.merx2s(x, y)
  
  
  Pix2Sky_MER = Pix2Sky_Mercator
  
  
  class Sky2Pix_Mercator(Sky2PixProjection, Cylindrical):
      r"""
      Mercator - sky to pixel.
  
      Corresponds to the ``MER`` projection in FITS WCS.
  
      .. math::
          x &= \phi \\
          y &= \frac{180^{\circ}}{\pi}\ln \tan \left(\frac{90^{\circ} + \theta}{2}\right)
      """
  
      @property
      def inverse(self):
          return Pix2Sky_Mercator()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.mers2x(phi, theta)
  
  
  Sky2Pix_MER = Sky2Pix_Mercator
  
  
  class PseudoCylindrical(Projection):
      r"""Base class for pseudocylindrical projections.
  
      Pseudocylindrical projections are like cylindrical projections
      except the parallels of latitude are projected at diminishing
      lengths toward the polar regions in order to reduce lateral
      distortion there.  Consequently, the meridians are curved.
      """
  
      _separable = True
  
  
  class Pix2Sky_SansonFlamsteed(Pix2SkyProjection, PseudoCylindrical):
      r"""
      Sanson-Flamsteed projection - pixel to sky.
  
      Corresponds to the ``SFL`` projection in FITS WCS.
  
      .. math::
          \phi &= \frac{x}{\cos y} \\
          \theta &= y
      """
  
      @property
      def inverse(self):
          return Sky2Pix_SansonFlamsteed()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.sflx2s(x, y)
  
  
  Pix2Sky_SFL = Pix2Sky_SansonFlamsteed
  
  
  class Sky2Pix_SansonFlamsteed(Sky2PixProjection, PseudoCylindrical):
      r"""
      Sanson-Flamsteed projection - sky to pixel.
  
      Corresponds to the ``SFL`` projection in FITS WCS.
  
      .. math::
          x &= \phi \cos \theta \\
          y &= \theta
      """
  
      @property
      def inverse(self):
          return Pix2Sky_SansonFlamsteed()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.sfls2x(phi, theta)
  
  
  Sky2Pix_SFL = Sky2Pix_SansonFlamsteed
  
  
  class Pix2Sky_Parabolic(Pix2SkyProjection, PseudoCylindrical):
      r"""
      Parabolic projection - pixel to sky.
  
      Corresponds to the ``PAR`` projection in FITS WCS.
  
      .. math::
          \phi &= \frac{180^\circ}{\pi} \frac{x}{1 - 4(y / 180^\circ)^2} \\
          \theta &= 3 \sin^{-1}\left(\frac{y}{180^\circ}\right)
      """
  
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_Parabolic()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.parx2s(x, y)
  
  
  Pix2Sky_PAR = Pix2Sky_Parabolic
  
  
  class Sky2Pix_Parabolic(Sky2PixProjection, PseudoCylindrical):
      r"""
      Parabolic projection - sky to pixel.
  
      Corresponds to the ``PAR`` projection in FITS WCS.
  
      .. math::
          x &= \phi \left(2\cos\frac{2\theta}{3} - 1\right) \\
          y &= 180^\circ \sin \frac{\theta}{3}
      """
  
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_Parabolic()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.pars2x(phi, theta)
  
  
  Sky2Pix_PAR = Sky2Pix_Parabolic
  
  
  class Pix2Sky_Molleweide(Pix2SkyProjection, PseudoCylindrical):
      r"""
      Molleweide's projection - pixel to sky.
  
      Corresponds to the ``MOL`` projection in FITS WCS.
  
      .. math::
          \phi &= \frac{\pi x}{2 \sqrt{2 - \left(\frac{\pi}{180^\circ}y\right)^2}} \\
          \theta &= \sin^{-1}\left(\frac{1}{90^\circ}\sin^{-1}\left(\frac{\pi}{180^\circ}\frac{y}{\sqrt{2}}\right) + \frac{y}{180^\circ}\sqrt{2 - \left(\frac{\pi}{180^\circ}y\right)^2}\right)
      """
  
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_Molleweide()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.molx2s(x, y)
  
  
  Pix2Sky_MOL = Pix2Sky_Molleweide
  
  
  class Sky2Pix_Molleweide(Sky2PixProjection, PseudoCylindrical):
      r"""
      Molleweide's projection - sky to pixel.
  
      Corresponds to the ``MOL`` projection in FITS WCS.
  
      .. math::
          x &= \frac{2 \sqrt{2}}{\pi} \phi \cos \gamma \\
          y &= \sqrt{2} \frac{180^\circ}{\pi} \sin \gamma
  
      where :math:`\gamma` is defined as the solution of the
      transcendental equation:
  
      .. math::
  
          \sin \theta = \frac{\gamma}{90^\circ} + \frac{\sin 2 \gamma}{\pi}
      """
  
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_Molleweide()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.mols2x(phi, theta)
  
  
  Sky2Pix_MOL = Sky2Pix_Molleweide
  
  
  class Pix2Sky_HammerAitoff(Pix2SkyProjection, PseudoCylindrical):
      r"""
      Hammer-Aitoff projection - pixel to sky.
  
      Corresponds to the ``AIT`` projection in FITS WCS.
  
      .. math::
          \phi &= 2 \arg \left(2Z^2 - 1, \frac{\pi}{180^\circ} \frac{Z}{2}x\right) \\
          \theta &= \sin^{-1}\left(\frac{\pi}{180^\circ}yZ\right)
      """
  
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_HammerAitoff()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.aitx2s(x, y)
  
  
  Pix2Sky_AIT = Pix2Sky_HammerAitoff
  
  
  class Sky2Pix_HammerAitoff(Sky2PixProjection, PseudoCylindrical):
      r"""
      Hammer-Aitoff projection - sky to pixel.
  
      Corresponds to the ``AIT`` projection in FITS WCS.
  
      .. math::
          x &= 2 \gamma \cos \theta \sin \frac{\phi}{2} \\
          y &= \gamma \sin \theta
  
      where:
  
      .. math::
          \gamma = \frac{180^\circ}{\pi} \sqrt{\frac{2}{1 + \cos \theta \cos(\phi / 2)}}
      """
  
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_HammerAitoff()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.aits2x(phi, theta)
  
  
  Sky2Pix_AIT = Sky2Pix_HammerAitoff
  
  
  class Conic(Projection):
      r"""Base class for conic projections.
  
      In conic projections, the sphere is thought to be projected onto
      the surface of a cone which is then opened out.
  
      In a general sense, the pixel-to-sky transformation is defined as:
  
      .. math::
  
          \phi &= \arg\left(\frac{Y_0 - y}{R_\theta}, \frac{x}{R_\theta}\right) / C \\
          R_\theta &= \mathrm{sign} \theta_a \sqrt{x^2 + (Y_0 - y)^2}
  
      and the inverse (sky-to-pixel) is defined as:
  
      .. math::
          x &= R_\theta \sin (C \phi) \\
          y &= R_\theta \cos (C \phi) + Y_0
  
      where :math:`C` is the "constant of the cone":
  
      .. math::
          C = \frac{180^\circ \cos \theta}{\pi R_\theta}
      """
      sigma = Parameter(default=90.0, getter=_to_orig_unit, setter=_to_radian)
      delta = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
  
      _separable = False
  
  
  class Pix2Sky_ConicPerspective(Pix2SkyProjection, Conic):
      r"""
      Colles' conic perspective projection - pixel to sky.
  
      Corresponds to the ``COP`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
          C &= \sin \theta_a \\
          R_\theta &= \frac{180^\circ}{\pi} \cos \eta [ \cot \theta_a - \tan(\theta - \theta_a)] \\
          Y_0 &= \frac{180^\circ}{\pi} \cos \eta \cot \theta_a
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Sky2Pix_ConicPerspective(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, x, y, sigma, delta):
          return _projections.copx2s(x, y, _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Pix2Sky_COP = Pix2Sky_ConicPerspective
  
  
  class Sky2Pix_ConicPerspective(Sky2PixProjection, Conic):
      r"""
      Colles' conic perspective projection - sky to pixel.
  
      Corresponds to the ``COP`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
          C &= \sin \theta_a \\
          R_\theta &= \frac{180^\circ}{\pi} \cos \eta [ \cot \theta_a - \tan(\theta - \theta_a)] \\
          Y_0 &= \frac{180^\circ}{\pi} \cos \eta \cot \theta_a
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Pix2Sky_ConicPerspective(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, phi, theta, sigma, delta):
          return _projections.cops2x(phi, theta,
                                     _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Sky2Pix_COP = Sky2Pix_ConicPerspective
  
  
  class Pix2Sky_ConicEqualArea(Pix2SkyProjection, Conic):
      r"""
      Alber's conic equal area projection - pixel to sky.
  
      Corresponds to the ``COE`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
          C &= \gamma / 2 \\
          R_\theta &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin \theta} \\
          Y_0 &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin((\theta_1 + \theta_2)/2)}
  
      where:
  
      .. math::
          \gamma = \sin \theta_1 + \sin \theta_2
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Sky2Pix_ConicEqualArea(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, x, y, sigma, delta):
          return _projections.coex2s(x, y, _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Pix2Sky_COE = Pix2Sky_ConicEqualArea
  
  
  class Sky2Pix_ConicEqualArea(Sky2PixProjection, Conic):
      r"""
      Alber's conic equal area projection - sky to pixel.
  
      Corresponds to the ``COE`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
          C &= \gamma / 2 \\
          R_\theta &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin \theta} \\
          Y_0 &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin((\theta_1 + \theta_2)/2)}
  
      where:
  
      .. math::
          \gamma = \sin \theta_1 + \sin \theta_2
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Pix2Sky_ConicEqualArea(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, phi, theta, sigma, delta):
          return _projections.coes2x(phi, theta,
                                     _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Sky2Pix_COE = Sky2Pix_ConicEqualArea
  
  
  class Pix2Sky_ConicEquidistant(Pix2SkyProjection, Conic):
      r"""
      Conic equidistant projection - pixel to sky.
  
      Corresponds to the ``COD`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
  
          C &= \frac{180^\circ}{\pi} \frac{\sin\theta_a\sin\eta}{\eta} \\
          R_\theta &= \theta_a - \theta + \eta\cot\eta\cot\theta_a \\
          Y_0 = \eta\cot\eta\cot\theta_a
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Sky2Pix_ConicEquidistant(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, x, y, sigma, delta):
          return _projections.codx2s(x, y, _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Pix2Sky_COD = Pix2Sky_ConicEquidistant
  
  
  class Sky2Pix_ConicEquidistant(Sky2PixProjection, Conic):
      r"""
      Conic equidistant projection - sky to pixel.
  
      Corresponds to the ``COD`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
  
          C &= \frac{180^\circ}{\pi} \frac{\sin\theta_a\sin\eta}{\eta} \\
          R_\theta &= \theta_a - \theta + \eta\cot\eta\cot\theta_a \\
          Y_0 = \eta\cot\eta\cot\theta_a
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Pix2Sky_ConicEquidistant(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, phi, theta, sigma, delta):
          return _projections.cods2x(phi, theta,
                                     _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Sky2Pix_COD = Sky2Pix_ConicEquidistant
  
  
  class Pix2Sky_ConicOrthomorphic(Pix2SkyProjection, Conic):
      r"""
      Conic orthomorphic projection - pixel to sky.
  
      Corresponds to the ``COO`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
  
          C &= \frac{\ln \left( \frac{\cos\theta_2}{\cos\theta_1} \right)}
                    {\ln \left[ \frac{\tan\left(\frac{90^\circ-\theta_2}{2}\right)}
                                     {\tan\left(\frac{90^\circ-\theta_1}{2}\right)} \right] } \\
          R_\theta &= \psi \left[ \tan \left( \frac{90^\circ - \theta}{2} \right) \right]^C \\
          Y_0 &= \psi \left[ \tan \left( \frac{90^\circ - \theta_a}{2} \right) \right]^C
  
      where:
  
      .. math::
  
          \psi = \frac{180^\circ}{\pi} \frac{\cos \theta}
                 {C\left[\tan\left(\frac{90^\circ-\theta}{2}\right)\right]^C}
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Sky2Pix_ConicOrthomorphic(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, x, y, sigma, delta):
          return _projections.coox2s(x, y, _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Pix2Sky_COO = Pix2Sky_ConicOrthomorphic
  
  
  class Sky2Pix_ConicOrthomorphic(Sky2PixProjection, Conic):
      r"""
      Conic orthomorphic projection - sky to pixel.
  
      Corresponds to the ``COO`` projection in FITS WCS.
  
      See `Conic` for a description of the entire equation.
  
      The projection formulae are:
  
      .. math::
  
          C &= \frac{\ln \left( \frac{\cos\theta_2}{\cos\theta_1} \right)}
                    {\ln \left[ \frac{\tan\left(\frac{90^\circ-\theta_2}{2}\right)}
                                     {\tan\left(\frac{90^\circ-\theta_1}{2}\right)} \right] } \\
          R_\theta &= \psi \left[ \tan \left( \frac{90^\circ - \theta}{2} \right) \right]^C \\
          Y_0 &= \psi \left[ \tan \left( \frac{90^\circ - \theta_a}{2} \right) \right]^C
  
      where:
  
      .. math::
  
          \psi = \frac{180^\circ}{\pi} \frac{\cos \theta}
                 {C\left[\tan\left(\frac{90^\circ-\theta}{2}\right)\right]^C}
  
      Parameters
      ----------
      sigma : float
          :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 90.
  
      delta : float
          :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and
          :math:`\theta_2` are the latitudes of the standard parallels,
          in degrees.  Default is 0.
      """
      @property
      def inverse(self):
          return Pix2Sky_ConicOrthomorphic(self.sigma.value, self.delta.value)
  
      @classmethod
      def evaluate(cls, phi, theta, sigma, delta):
          return _projections.coos2x(phi, theta,
                                     _to_orig_unit(sigma), _to_orig_unit(delta))
  
  
  Sky2Pix_COO = Sky2Pix_ConicOrthomorphic
  
  
  class PseudoConic(Projection):
      r"""Base class for pseudoconic projections.
  
      Pseudoconics are a subclass of conics with concentric parallels.
      """
  
  
  class Pix2Sky_BonneEqualArea(Pix2SkyProjection, PseudoConic):
      r"""
      Bonne's equal area pseudoconic projection - pixel to sky.
  
      Corresponds to the ``BON`` projection in FITS WCS.
  
      .. math::
  
          \phi &= \frac{\pi}{180^\circ} A_\phi R_\theta / \cos \theta \\
          \theta &= Y_0 - R_\theta
  
      where:
  
      .. math::
  
          R_\theta &= \mathrm{sign} \theta_1 \sqrt{x^2 + (Y_0 - y)^2} \\
          A_\phi &= \arg\left(\frac{Y_0 - y}{R_\theta}, \frac{x}{R_\theta}\right)
  
      Parameters
      ----------
      theta1 : float
          Bonne conformal latitude, in degrees.
      """
      theta1 = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
      _separable = True
  
      @property
      def inverse(self):
          return Sky2Pix_BonneEqualArea(self.theta1.value)
  
      @classmethod
      def evaluate(cls, x, y, theta1):
          return _projections.bonx2s(x, y, _to_orig_unit(theta1))
  
  
  Pix2Sky_BON = Pix2Sky_BonneEqualArea
  
  
  class Sky2Pix_BonneEqualArea(Sky2PixProjection, PseudoConic):
      r"""
      Bonne's equal area pseudoconic projection - sky to pixel.
  
      Corresponds to the ``BON`` projection in FITS WCS.
  
      .. math::
          x &= R_\theta \sin A_\phi \\
          y &= -R_\theta \cos A_\phi + Y_0
  
      where:
  
      .. math::
          A_\phi &= \frac{180^\circ}{\pi R_\theta} \phi \cos \theta \\
          R_\theta &= Y_0 - \theta \\
          Y_0 &= \frac{180^\circ}{\pi} \cot \theta_1 + \theta_1
  
      Parameters
      ----------
      theta1 : float
          Bonne conformal latitude, in degrees.
      """
      theta1 = Parameter(default=0.0, getter=_to_orig_unit, setter=_to_radian)
      _separable = True
  
      @property
      def inverse(self):
          return Pix2Sky_BonneEqualArea(self.theta1.value)
  
      @classmethod
      def evaluate(cls, phi, theta, theta1):
          return _projections.bons2x(phi, theta,
                                     _to_orig_unit(theta1))
  
  
  Sky2Pix_BON = Sky2Pix_BonneEqualArea
  
  
  class Pix2Sky_Polyconic(Pix2SkyProjection, PseudoConic):
      r"""
      Polyconic projection - pixel to sky.
  
      Corresponds to the ``PCO`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_Polyconic()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.pcox2s(x, y)
  
  
  Pix2Sky_PCO = Pix2Sky_Polyconic
  
  
  class Sky2Pix_Polyconic(Sky2PixProjection, PseudoConic):
      r"""
      Polyconic projection - sky to pixel.
  
      Corresponds to the ``PCO`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_Polyconic()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.pcos2x(phi, theta)
  
  
  Sky2Pix_PCO = Sky2Pix_Polyconic
  
  
  class QuadCube(Projection):
      r"""Base class for quad cube projections.
  
      Quadrilateralized spherical cube (quad-cube) projections belong to
      the class of polyhedral projections in which the sphere is
      projected onto the surface of an enclosing polyhedron.
  
      The six faces of the quad-cube projections are numbered and laid
      out as::
  
                0
          4 3 2 1 4 3 2
                5
  
      """
  
  
  class Pix2Sky_TangentialSphericalCube(Pix2SkyProjection, QuadCube):
      r"""
      Tangential spherical cube projection - pixel to sky.
  
      Corresponds to the ``TSC`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_TangentialSphericalCube()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.tscx2s(x, y)
  
  
  Pix2Sky_TSC = Pix2Sky_TangentialSphericalCube
  
  
  class Sky2Pix_TangentialSphericalCube(Sky2PixProjection, QuadCube):
      r"""
      Tangential spherical cube projection - sky to pixel.
  
      Corresponds to the ``PCO`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_TangentialSphericalCube()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.tscs2x(phi, theta)
  
  
  Sky2Pix_TSC = Sky2Pix_TangentialSphericalCube
  
  
  class Pix2Sky_COBEQuadSphericalCube(Pix2SkyProjection, QuadCube):
      r"""
      COBE quadrilateralized spherical cube projection - pixel to sky.
  
      Corresponds to the ``CSC`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_COBEQuadSphericalCube()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.cscx2s(x, y)
  
  
  Pix2Sky_CSC = Pix2Sky_COBEQuadSphericalCube
  
  
  class Sky2Pix_COBEQuadSphericalCube(Sky2PixProjection, QuadCube):
      r"""
      COBE quadrilateralized spherical cube projection - sky to pixel.
  
      Corresponds to the ``CSC`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_COBEQuadSphericalCube()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.cscs2x(phi, theta)
  
  
  Sky2Pix_CSC = Sky2Pix_COBEQuadSphericalCube
  
  
  class Pix2Sky_QuadSphericalCube(Pix2SkyProjection, QuadCube):
      r"""
      Quadrilateralized spherical cube projection - pixel to sky.
  
      Corresponds to the ``QSC`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_QuadSphericalCube()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.qscx2s(x, y)
  
  
  Pix2Sky_QSC = Pix2Sky_QuadSphericalCube
  
  
  class Sky2Pix_QuadSphericalCube(Sky2PixProjection, QuadCube):
      r"""
      Quadrilateralized spherical cube projection - sky to pixel.
  
      Corresponds to the ``QSC`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_QuadSphericalCube()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.qscs2x(phi, theta)
  
  
  Sky2Pix_QSC = Sky2Pix_QuadSphericalCube
  
  
  class HEALPix(Projection):
      r"""Base class for HEALPix projections.
      """
  
  
  class Pix2Sky_HEALPix(Pix2SkyProjection, HEALPix):
      r"""
      HEALPix - pixel to sky.
  
      Corresponds to the ``HPX`` projection in FITS WCS.
  
      Parameters
      ----------
      H : float
          The number of facets in longitude direction.
  
      X : float
          The number of facets in latitude direction.
      """
      _separable = True
  
      H = Parameter(default=4.0)
      X = Parameter(default=3.0)
  
      @property
      def inverse(self):
          return Sky2Pix_HEALPix(self.H.value, self.X.value)
  
      @classmethod
      def evaluate(cls, x, y, H, X):
          return _projections.hpxx2s(x, y, H, X)
  
  
  Pix2Sky_HPX = Pix2Sky_HEALPix
  
  
  class Sky2Pix_HEALPix(Sky2PixProjection, HEALPix):
      r"""
      HEALPix projection - sky to pixel.
  
      Corresponds to the ``HPX`` projection in FITS WCS.
  
      Parameters
      ----------
      H : float
          The number of facets in longitude direction.
  
      X : float
          The number of facets in latitude direction.
      """
      _separable = True
  
      H = Parameter(default=4.0)
      X = Parameter(default=3.0)
  
      @property
      def inverse(self):
          return Pix2Sky_HEALPix(self.H.value, self.X.value)
  
      @classmethod
      def evaluate(cls, phi, theta, H, X):
          return _projections.hpxs2x(phi, theta, H, X)
  
  
  Sky2Pix_HPX = Sky2Pix_HEALPix
  
  
  class Pix2Sky_HEALPixPolar(Pix2SkyProjection, HEALPix):
      r"""
      HEALPix polar, aka "butterfly" projection - pixel to sky.
  
      Corresponds to the ``XPH`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Sky2Pix_HEALPix()
  
      @classmethod
      def evaluate(cls, x, y):
          return _projections.xphx2s(x, y)
  
  
  Pix2Sky_XPH = Pix2Sky_HEALPixPolar
  
  
  class Sky2Pix_HEALPixPolar(Sky2PixProjection, HEALPix):
      r"""
      HEALPix polar, aka "butterfly" projection - pixel to sky.
  
      Corresponds to the ``XPH`` projection in FITS WCS.
      """
      _separable = False
  
      @property
      def inverse(self):
          return Pix2Sky_HEALPix()
  
      @classmethod
      def evaluate(cls, phi, theta):
          return _projections.hpxs2x(phi, theta)
  
  
  Sky2Pix_XPH = Sky2Pix_HEALPixPolar
  
  
  class AffineTransformation2D(Model):
      """
      Perform an affine transformation in 2 dimensions.
  
      Parameters
      ----------
      matrix : array
          A 2x2 matrix specifying the linear transformation to apply to the
          inputs
  
      translation : array
          A 2D vector (given as either a 2x1 or 1x2 array) specifying a
          translation to apply to the inputs
      """
  
      inputs = ('x', 'y')
      outputs = ('x', 'y')
  
      standard_broadcasting = False
  
      _separable = False
  
      matrix = Parameter(default=[[1.0, 0.0], [0.0, 1.0]])
      translation = Parameter(default=[0.0, 0.0])
  
      @matrix.validator
      def matrix(self, value):
          """Validates that the input matrix is a 2x2 2D array."""
  
          if np.shape(value) != (2, 2):
              raise InputParameterError(
                  "Expected transformation matrix to be a 2x2 array")
  
      @translation.validator
      def translation(self, value):
          """
          Validates that the translation vector is a 2D vector.  This allows
          either a "row" vector or a "column" vector where in the latter case the
          resultant Numpy array has ``ndim=2`` but the shape is ``(1, 2)``.
          """
  
          if not ((np.ndim(value) == 1 and np.shape(value) == (2,)) or
                  (np.ndim(value) == 2 and np.shape(value) == (1, 2))):
              raise InputParameterError(
                  "Expected translation vector to be a 2 element row or column "
                  "vector array")
  
      @property
      def inverse(self):
          """
          Inverse transformation.
  
          Raises `~astropy.modeling.InputParameterError` if the transformation cannot be inverted.
          """
  
          det = np.linalg.det(self.matrix.value)
  
          if det == 0:
              raise InputParameterError(
                  "Transformation matrix is singular; {0} model does not "
                  "have an inverse".format(self.__class__.__name__))
  
          matrix = np.linalg.inv(self.matrix.value)
          if self.matrix.unit is not None:
              matrix = matrix * self.matrix.unit
          # If matrix has unit then translation has unit, so no need to assign it.
          translation = -np.dot(matrix, self.translation.value)
          return self.__class__(matrix=matrix, translation=translation)
  
      @classmethod
      def evaluate(cls, x, y, matrix, translation):
          """
          Apply the transformation to a set of 2D Cartesian coordinates given as
          two lists--one for the x coordinates and one for a y coordinates--or a
          single coordinate pair.
  
          Parameters
          ----------
          x, y : array, float
                x and y coordinates
          """
          if x.shape != y.shape:
              raise ValueError("Expected input arrays to have the same shape")
  
          shape = x.shape or (1,)
          inarr = np.vstack([x.flatten(), y.flatten(), np.ones(x.size)])
  
          if inarr.shape[0] != 3 or inarr.ndim != 2:
              raise ValueError("Incompatible input shapes")
  
          augmented_matrix = cls._create_augmented_matrix(matrix, translation)
          result = np.dot(augmented_matrix, inarr)
          x, y = result[0], result[1]
          x.shape = y.shape = shape
  
          return x, y
  
      @staticmethod
      def _create_augmented_matrix(matrix, translation):
          unit = None
          if any([hasattr(translation, 'unit'), hasattr(matrix, 'unit')]):
              if not all([hasattr(translation, 'unit'), hasattr(matrix, 'unit')]):
                  raise ValueError("To use AffineTransformation with quantities, "
                                   "both matrix and unit need to be quantities.")
              unit = translation.unit
              # matrix should have the same units as translation
              if not (matrix.unit / translation.unit) == u.dimensionless_unscaled:
                  raise ValueError("matrix and translation must have the same units.")
  
          augmented_matrix = np.empty((3, 3), dtype=float)
          augmented_matrix[0:2, 0:2] = matrix
          augmented_matrix[0:2, 2:].flat = translation
          augmented_matrix[2] = [0, 0, 1]
          if unit is not None:
              return augmented_matrix * unit
          else:
              return augmented_matrix
  
      @property
      def input_units(self):
          if self.translation.unit is None and self.matrix.unit is None:
              return None
          elif self.translation.unit is not None:
              return {'x': self.translation.unit,
                      'y': self.translation.unit
                      }
          else:
              return {'x': self.matrix.unit,
                      'y': self.matrix.unit
                      }
  