text: # Licensed under a 3-clause BSD style license - see LICENSE.rst
  """
  Under the hood, there are 3 separate classes that perform different
  parts of the transformation:
  
     - `~astropy.wcs.Wcsprm`: Is a direct wrapper of the core WCS
       functionality in `wcslib`_.  (This includes TPV and TPD
       polynomial distortion, but not SIP distortion).
  
     - `~astropy.wcs.Sip`: Handles polynomial distortion as defined in the
       `SIP`_ convention.
  
     - `~astropy.wcs.DistortionLookupTable`: Handles `distortion paper`_
       lookup tables.
  
  Additionally, the class `WCS` aggregates all of these transformations
  together in a pipeline:
  
     - Detector to image plane correction (by a pair of
       `~astropy.wcs.DistortionLookupTable` objects).
  
     - `SIP`_ distortion correction (by an underlying `~astropy.wcs.Sip`
       object)
  
     - `distortion paper`_ table-lookup correction (by a pair of
       `~astropy.wcs.DistortionLookupTable` objects).
  
     - `wcslib`_ WCS transformation (by a `~astropy.wcs.Wcsprm` object)
  
  """
  
  # STDLIB
  import copy
  import uuid
  import io
  import itertools
  import os
  import re
  import textwrap
  import warnings
  import builtins
  
  # THIRD-PARTY
  import numpy as np
  
  # LOCAL
  from astropy import log
  from astropy.io import fits
  from . import docstrings
  from . import _wcs
  
  from astropy.utils.compat import possible_filename
  from astropy.utils.exceptions import AstropyWarning, AstropyUserWarning, AstropyDeprecationWarning
  
  
  # Mix-in class that provides the APE 14 API
  from .wcsapi.fitswcs import FITSWCSAPIMixin, SlicedFITSWCS
  
  __all__ = ['FITSFixedWarning', 'WCS', 'find_all_wcs',
             'DistortionLookupTable', 'Sip', 'Tabprm', 'Wcsprm',
             'WCSBase', 'validate', 'WcsError', 'SingularMatrixError',
             'InconsistentAxisTypesError', 'InvalidTransformError',
             'InvalidCoordinateError', 'NoSolutionError',
             'InvalidSubimageSpecificationError', 'NoConvergence',
             'NonseparableSubimageCoordinateSystemError',
             'NoWcsKeywordsFoundError', 'InvalidTabularParametersError']
  
  
  __doctest_skip__ = ['WCS.all_world2pix']
  
  
  if _wcs is not None:
      _parsed_version = _wcs.__version__.split('.')
      if int(_parsed_version[0]) == 5 and int(_parsed_version[1]) < 8:
          raise ImportError(
              "astropy.wcs is built with wcslib {0}, but only versions 5.8 and "
              "later on the 5.x series are known to work.  The version of wcslib "
              "that ships with astropy may be used.")
  
      if not _wcs._sanity_check():
          raise RuntimeError(
          "astropy.wcs did not pass its sanity check for your build "
          "on your platform.")
  
      WCSBase = _wcs._Wcs
      DistortionLookupTable = _wcs.DistortionLookupTable
      Sip = _wcs.Sip
      Wcsprm = _wcs.Wcsprm
      Tabprm = _wcs.Tabprm
      WcsError = _wcs.WcsError
      SingularMatrixError = _wcs.SingularMatrixError
      InconsistentAxisTypesError = _wcs.InconsistentAxisTypesError
      InvalidTransformError = _wcs.InvalidTransformError
      InvalidCoordinateError = _wcs.InvalidCoordinateError
      NoSolutionError = _wcs.NoSolutionError
      InvalidSubimageSpecificationError = _wcs.InvalidSubimageSpecificationError
      NonseparableSubimageCoordinateSystemError = _wcs.NonseparableSubimageCoordinateSystemError
      NoWcsKeywordsFoundError = _wcs.NoWcsKeywordsFoundError
      InvalidTabularParametersError = _wcs.InvalidTabularParametersError
  
      # Copy all the constants from the C extension into this module's namespace
      for key, val in _wcs.__dict__.items():
          if key.startswith(('WCSSUB', 'WCSHDR', 'WCSHDO')):
              locals()[key] = val
              __all__.append(key)
  
      # Set coordinate extraction callback for WCS -TAB:
      def _load_tab_bintable(hdulist, extnam, extver, extlev, kind, ttype, row, ndim):
          arr = hdulist[(extnam, extver)].data[ttype][row - 1]
  
          if arr.ndim != ndim:
              if kind == 'c' and ndim == 2:
                  arr = arr.reshape((arr.size, 1))
              else:
                  raise ValueError("Bad TDIM")
  
          return np.ascontiguousarray(arr, dtype=np.double)
  
      _wcs.set_wtbarr_fitsio_callback(_load_tab_bintable)
  
  else:
      WCSBase = object
      Wcsprm = object
      DistortionLookupTable = object
      Sip = object
      Tabprm = object
      WcsError = None
      SingularMatrixError = None
      InconsistentAxisTypesError = None
      InvalidTransformError = None
      InvalidCoordinateError = None
      NoSolutionError = None
      InvalidSubimageSpecificationError = None
      NonseparableSubimageCoordinateSystemError = None
      NoWcsKeywordsFoundError = None
      InvalidTabularParametersError = None
  
  
  # Additional relax bit flags
  WCSHDO_SIP = 0x80000
  
  # Regular expression defining SIP keyword It matches keyword that starts with A
  # or B, optionally followed by P, followed by an underscore then a number in
  # range of 0-19, followed by an underscore and another number in range of 0-19.
  # Keyword optionally ends with a capital letter.
  SIP_KW = re.compile('''^[AB]P?_1?[0-9]_1?[0-9][A-Z]?$''')
  
  
  def _parse_keysel(keysel):
      keysel_flags = 0
      if keysel is not None:
          for element in keysel:
              if element.lower() == 'image':
                  keysel_flags |= _wcs.WCSHDR_IMGHEAD
              elif element.lower() == 'binary':
                  keysel_flags |= _wcs.WCSHDR_BIMGARR
              elif element.lower() == 'pixel':
                  keysel_flags |= _wcs.WCSHDR_PIXLIST
              else:
                  raise ValueError(
                      "keysel must be a list of 'image', 'binary' " +
                      "and/or 'pixel'")
      else:
          keysel_flags = -1
  
      return keysel_flags
  
  
  class NoConvergence(Exception):
      """
      An error class used to report non-convergence and/or divergence
      of numerical methods. It is used to report errors in the
      iterative solution used by
      the :py:meth:`~astropy.wcs.WCS.all_world2pix`.
  
      Attributes
      ----------
  
      best_solution : `numpy.ndarray`
          Best solution achieved by the numerical method.
  
      accuracy : `numpy.ndarray`
          Accuracy of the ``best_solution``.
  
      niter : `int`
          Number of iterations performed by the numerical method
          to compute ``best_solution``.
  
      divergent : None, `numpy.ndarray`
          Indices of the points in ``best_solution`` array
          for which the solution appears to be divergent. If the
          solution does not diverge, ``divergent`` will be set to `None`.
  
      slow_conv : None, `numpy.ndarray`
          Indices of the solutions in ``best_solution`` array
          for which the solution failed to converge within the
          specified maximum number of iterations. If there are no
          non-converging solutions (i.e., if the required accuracy
          has been achieved for all input data points)
          then ``slow_conv`` will be set to `None`.
  
      """
  
      def __init__(self, *args, best_solution=None, accuracy=None, niter=None,
                   divergent=None, slow_conv=None, **kwargs):
          super().__init__(*args)
  
          self.best_solution = best_solution
          self.accuracy = accuracy
          self.niter = niter
          self.divergent = divergent
          self.slow_conv = slow_conv
  
          if kwargs:
              warnings.warn("Function received unexpected arguments ({}) these "
                            "are ignored but will raise an Exception in the "
                            "future.".format(list(kwargs)),
                            AstropyDeprecationWarning)
  
  
  class FITSFixedWarning(AstropyWarning):
      """
      The warning raised when the contents of the FITS header have been
      modified to be standards compliant.
      """
      pass
  
  
  class WCS(FITSWCSAPIMixin, WCSBase):
      """WCS objects perform standard WCS transformations, and correct for
      `SIP`_ and `distortion paper`_ table-lookup transformations, based
      on the WCS keywords and supplementary data read from a FITS file.
  
      See also: http://docs.astropy.org/en/stable/wcs/
  
      Parameters
      ----------
      header : astropy.io.fits header object, Primary HDU, Image HDU, string, dict-like, or None, optional
          If *header* is not provided or None, the object will be
          initialized to default values.
  
      fobj : An astropy.io.fits file (hdulist) object, optional
          It is needed when header keywords point to a `distortion
          paper`_ lookup table stored in a different extension.
  
      key : str, optional
          The name of a particular WCS transform to use.  This may be
          either ``' '`` or ``'A'``-``'Z'`` and corresponds to the
          ``\"a\"`` part of the ``CTYPEia`` cards.  *key* may only be
          provided if *header* is also provided.
  
      minerr : float, optional
          The minimum value a distortion correction must have in order
          to be applied. If the value of ``CQERRja`` is smaller than
          *minerr*, the corresponding distortion is not applied.
  
      relax : bool or int, optional
          Degree of permissiveness:
  
          - `True` (default): Admit all recognized informal extensions
            of the WCS standard.
  
          - `False`: Recognize only FITS keywords defined by the
            published WCS standard.
  
          - `int`: a bit field selecting specific extensions to accept.
            See :ref:`relaxread` for details.
  
      naxis : int or sequence, optional
          Extracts specific coordinate axes using
          :meth:`~astropy.wcs.Wcsprm.sub`.  If a header is provided, and
          *naxis* is not ``None``, *naxis* will be passed to
          :meth:`~astropy.wcs.Wcsprm.sub` in order to select specific
          axes from the header.  See :meth:`~astropy.wcs.Wcsprm.sub` for
          more details about this parameter.
  
      keysel : sequence of flags, optional
          A sequence of flags used to select the keyword types
          considered by wcslib.  When ``None``, only the standard image
          header keywords are considered (and the underlying wcspih() C
          function is called).  To use binary table image array or pixel
          list keywords, *keysel* must be set.
  
          Each element in the list should be one of the following
          strings:
  
          - 'image': Image header keywords
  
          - 'binary': Binary table image array keywords
  
          - 'pixel': Pixel list keywords
  
          Keywords such as ``EQUIna`` or ``RFRQna`` that are common to
          binary table image arrays and pixel lists (including
          ``WCSNna`` and ``TWCSna``) are selected by both 'binary' and
          'pixel'.
  
      colsel : sequence of int, optional
          A sequence of table column numbers used to restrict the WCS
          transformations considered to only those pertaining to the
          specified columns.  If `None`, there is no restriction.
  
      fix : bool, optional
          When `True` (default), call `~astropy.wcs.Wcsprm.fix` on
          the resulting object to fix any non-standard uses in the
          header.  `FITSFixedWarning` Warnings will be emitted if any
          changes were made.
  
      translate_units : str, optional
          Specify which potentially unsafe translations of non-standard
          unit strings to perform.  By default, performs none.  See
          `WCS.fix` for more information about this parameter.  Only
          effective when ``fix`` is `True`.
  
      Raises
      ------
      MemoryError
           Memory allocation failed.
  
      ValueError
           Invalid key.
  
      KeyError
           Key not found in FITS header.
  
      ValueError
           Lookup table distortion present in the header but *fobj* was
           not provided.
  
      Notes
      -----
  
      1. astropy.wcs supports arbitrary *n* dimensions for the core WCS
         (the transformations handled by WCSLIB).  However, the
         `distortion paper`_ lookup table and `SIP`_ distortions must be
         two dimensional.  Therefore, if you try to create a WCS object
         where the core WCS has a different number of dimensions than 2
         and that object also contains a `distortion paper`_ lookup
         table or `SIP`_ distortion, a `ValueError`
         exception will be raised.  To avoid this, consider using the
         *naxis* kwarg to select two dimensions from the core WCS.
  
      2. The number of coordinate axes in the transformation is not
         determined directly from the ``NAXIS`` keyword but instead from
         the highest of:
  
             - ``NAXIS`` keyword
  
             - ``WCSAXESa`` keyword
  
             - The highest axis number in any parameterized WCS keyword.
               The keyvalue, as well as the keyword, must be
               syntactically valid otherwise it will not be considered.
  
         If none of these keyword types is present, i.e. if the header
         only contains auxiliary WCS keywords for a particular
         coordinate representation, then no coordinate description is
         constructed for it.
  
         The number of axes, which is set as the ``naxis`` member, may
         differ for different coordinate representations of the same
         image.
  
      3. When the header includes duplicate keywords, in most cases the
         last encountered is used.
  
      4. `~astropy.wcs.Wcsprm.set` is called immediately after
         construction, so any invalid keywords or transformations will
         be raised by the constructor, not when subsequently calling a
         transformation method.
  
      """
  
      def __init__(self, header=None, fobj=None, key=' ', minerr=0.0,
                   relax=True, naxis=None, keysel=None, colsel=None,
                   fix=True, translate_units='', _do_set=True):
          close_fds = []
  
          if header is None:
              if naxis is None:
                  naxis = 2
              wcsprm = _wcs.Wcsprm(header=None, key=key,
                                   relax=relax, naxis=naxis)
              self.naxis = wcsprm.naxis
              # Set some reasonable defaults.
              det2im = (None, None)
              cpdis = (None, None)
              sip = None
          else:
              keysel_flags = _parse_keysel(keysel)
  
              if isinstance(header, (str, bytes)):
                  try:
                      is_path = (possible_filename(header) and
                                 os.path.exists(header))
                  except (OSError, ValueError):
                      is_path = False
  
                  if is_path:
                      if fobj is not None:
                          raise ValueError(
                              "Can not provide both a FITS filename to "
                              "argument 1 and a FITS file object to argument 2")
                      fobj = fits.open(header)
                      close_fds.append(fobj)
                      header = fobj[0].header
              elif isinstance(header, fits.hdu.image._ImageBaseHDU):
                  header = header.header
              elif not isinstance(header, fits.Header):
                  try:
                      # Accept any dict-like object
                      orig_header = header
                      header = fits.Header()
                      for dict_key in orig_header.keys():
                          header[dict_key] = orig_header[dict_key]
                  except TypeError:
                      raise TypeError(
                          "header must be a string, an astropy.io.fits.Header "
                          "object, or a dict-like object")
  
              if isinstance(header, fits.Header):
                  header_string = header.tostring().rstrip()
              else:
                  header_string = header
  
              # Importantly, header is a *copy* of the passed-in header
              # because we will be modifying it
              if isinstance(header_string, str):
                  header_bytes = header_string.encode('ascii')
                  header_string = header_string
              else:
                  header_bytes = header_string
                  header_string = header_string.decode('ascii')
  
              if not (fobj is None or isinstance(fobj, fits.HDUList)):
                  raise AssertionError("'fobj' must be either None or an "
                                       "astropy.io.fits.HDUList object.")
  
              try:
                  tmp_header = fits.Header.fromstring(header_string)
                  self._remove_sip_kw(tmp_header)
                  tmp_header_bytes = tmp_header.tostring().rstrip()
                  if isinstance(tmp_header_bytes, str):
                      tmp_header_bytes = tmp_header_bytes.encode('ascii')
                  tmp_wcsprm = _wcs.Wcsprm(header=tmp_header_bytes, key=key,
                                           relax=relax, keysel=keysel_flags,
                                           colsel=colsel, warnings=False,
                                           hdulist=fobj)
              except _wcs.NoWcsKeywordsFoundError:
                  est_naxis = 0
              else:
                  if naxis is not None:
                      try:
                          tmp_wcsprm.sub(naxis)
                      except ValueError:
                          pass
                      est_naxis = tmp_wcsprm.naxis
                  else:
                      est_naxis = 2
  
              header = fits.Header.fromstring(header_string)
  
              if est_naxis == 0:
                  est_naxis = 2
              self.naxis = est_naxis
  
              det2im = self._read_det2im_kw(header, fobj, err=minerr)
              cpdis = self._read_distortion_kw(
                  header, fobj, dist='CPDIS', err=minerr)
              sip = self._read_sip_kw(header, wcskey=key)
              self._remove_sip_kw(header)
  
              header_string = header.tostring()
              header_string = header_string.replace('END' + ' ' * 77, '')
  
              if isinstance(header_string, str):
                  header_bytes = header_string.encode('ascii')
                  header_string = header_string
              else:
                  header_bytes = header_string
                  header_string = header_string.decode('ascii')
  
              try:
                  wcsprm = _wcs.Wcsprm(header=header_bytes, key=key,
                                       relax=relax, keysel=keysel_flags,
                                       colsel=colsel, hdulist=fobj)
              except _wcs.NoWcsKeywordsFoundError:
                  # The header may have SIP or distortions, but no core
                  # WCS.  That isn't an error -- we want a "default"
                  # (identity) core Wcs transformation in that case.
                  if colsel is None:
                      wcsprm = _wcs.Wcsprm(header=None, key=key,
                                           relax=relax, keysel=keysel_flags,
                                           colsel=colsel, hdulist=fobj)
                  else:
                      raise
  
              if naxis is not None:
                  wcsprm = wcsprm.sub(naxis)
              self.naxis = wcsprm.naxis
  
              if (wcsprm.naxis != 2 and
                  (det2im[0] or det2im[1] or cpdis[0] or cpdis[1] or sip)):
                  raise ValueError(
                      """
  FITS WCS distortion paper lookup tables and SIP distortions only work
  in 2 dimensions.  However, WCSLIB has detected {} dimensions in the
  core WCS keywords.  To use core WCS in conjunction with FITS WCS
  distortion paper lookup tables or SIP distortion, you must select or
  reduce these to 2 dimensions using the naxis kwarg.
  """.format(wcsprm.naxis))
  
              header_naxis = header.get('NAXIS', None)
              if header_naxis is not None and header_naxis < wcsprm.naxis:
                  warnings.warn(
                      "The WCS transformation has more axes ({:d}) than the "
                      "image it is associated with ({:d})".format(
                          wcsprm.naxis, header_naxis), FITSFixedWarning)
  
          self._get_naxis(header)
          WCSBase.__init__(self, sip, cpdis, wcsprm, det2im)
  
          if fix:
              if header is None:
                  with warnings.catch_warnings():
                      warnings.simplefilter('ignore', FITSFixedWarning)
                      self.fix(translate_units=translate_units)
              else:
                  self.fix(translate_units=translate_units)
  
          if _do_set:
              self.wcs.set()
  
          for fd in close_fds:
              fd.close()
  
          self._pixel_bounds = None
  
      def __copy__(self):
          new_copy = self.__class__()
          WCSBase.__init__(new_copy, self.sip,
                           (self.cpdis1, self.cpdis2),
                           self.wcs,
                           (self.det2im1, self.det2im2))
          new_copy.__dict__.update(self.__dict__)
          return new_copy
  
      def __deepcopy__(self, memo):
          from copy import deepcopy
  
          new_copy = self.__class__()
          new_copy.naxis = deepcopy(self.naxis, memo)
          WCSBase.__init__(new_copy, deepcopy(self.sip, memo),
                           (deepcopy(self.cpdis1, memo),
                            deepcopy(self.cpdis2, memo)),
                           deepcopy(self.wcs, memo),
                           (deepcopy(self.det2im1, memo),
                            deepcopy(self.det2im2, memo)))
          for key, val in self.__dict__.items():
              new_copy.__dict__[key] = deepcopy(val, memo)
          return new_copy
  
      def copy(self):
          """
          Return a shallow copy of the object.
  
          Convenience method so user doesn't have to import the
          :mod:`copy` stdlib module.
  
          .. warning::
              Use `deepcopy` instead of `copy` unless you know why you need a
              shallow copy.
          """
          return copy.copy(self)
  
      def deepcopy(self):
          """
          Return a deep copy of the object.
  
          Convenience method so user doesn't have to import the
          :mod:`copy` stdlib module.
          """
          return copy.deepcopy(self)
  
      def sub(self, axes=None):
  
          copy = self.deepcopy()
  
          # We need to know which axes have been dropped, but there is no easy
          # way to do this with the .sub function, so instead we assign UUIDs to
          # the CNAME parameters in copy.wcs. We can later access the original
          # CNAME properties from self.wcs.
          cname_uuid = [str(uuid.uuid4()) for i in range(copy.wcs.naxis)]
          copy.wcs.cname = cname_uuid
  
          # Subset the WCS
          copy.wcs = copy.wcs.sub(axes)
          copy.naxis = copy.wcs.naxis
  
          # Construct a list of dimensions from the original WCS in the order
          # in which they appear in the final WCS.
          keep = [cname_uuid.index(cname) if cname in cname_uuid else None for cname in copy.wcs.cname]
  
          # Restore the original CNAMEs
          copy.wcs.cname = ['' if i is None else self.wcs.cname[i] for i in keep]
  
          # Subset pixel_shape and pixel_bounds
          if self.pixel_shape:
              copy.pixel_shape = tuple([None if i is None else self.pixel_shape[i] for i in keep])
          if self.pixel_bounds:
              copy.pixel_bounds = [None if i is None else self.pixel_bounds[i] for i in keep]
  
          return copy
  
      if _wcs is not None:
          sub.__doc__ = _wcs.Wcsprm.sub.__doc__
  
      def _fix_scamp(self):
          """
          Remove SCAMP's PVi_m distortion parameters if SIP distortion parameters
          are also present. Some projects (e.g., Palomar Transient Factory)
          convert SCAMP's distortion parameters (which abuse the PVi_m cards) to
          SIP. However, wcslib gets confused by the presence of both SCAMP and
          SIP distortion parameters.
  
          See https://github.com/astropy/astropy/issues/299.
          """
          # Nothing to be done if no WCS attached
          if self.wcs is None:
              return
  
          # Nothing to be done if no PV parameters attached
          pv = self.wcs.get_pv()
          if not pv:
              return
  
          # Nothing to be done if axes don't use SIP distortion parameters
          if self.sip is None:
              return
  
          # Nothing to be done if any radial terms are present...
          # Loop over list to find any radial terms.
          # Certain values of the `j' index are used for storing
          # radial terms; refer to Equation (1) in
          # <http://web.ipac.caltech.edu/staff/shupe/reprints/SIP_to_PV_SPIE2012.pdf>.
          pv = np.asarray(pv)
          # Loop over distinct values of `i' index
          for i in set(pv[:, 0]):
              # Get all values of `j' index for this value of `i' index
              js = set(pv[:, 1][pv[:, 0] == i])
              # Find max value of `j' index
              max_j = max(js)
              for j in (3, 11, 23, 39):
                  if j < max_j and j in js:
                      return
  
          self.wcs.set_pv([])
          warnings.warn("Removed redundant SCAMP distortion parameters " +
              "because SIP parameters are also present", FITSFixedWarning)
  
      def fix(self, translate_units='', naxis=None):
          """
          Perform the fix operations from wcslib, and warn about any
          changes it has made.
  
          Parameters
          ----------
          translate_units : str, optional
              Specify which potentially unsafe translations of
              non-standard unit strings to perform.  By default,
              performs none.
  
              Although ``"S"`` is commonly used to represent seconds,
              its translation to ``"s"`` is potentially unsafe since the
              standard recognizes ``"S"`` formally as Siemens, however
              rarely that may be used.  The same applies to ``"H"`` for
              hours (Henry), and ``"D"`` for days (Debye).
  
              This string controls what to do in such cases, and is
              case-insensitive.
  
              - If the string contains ``"s"``, translate ``"S"`` to
                ``"s"``.
  
              - If the string contains ``"h"``, translate ``"H"`` to
                ``"h"``.
  
              - If the string contains ``"d"``, translate ``"D"`` to
                ``"d"``.
  
              Thus ``''`` doesn't do any unsafe translations, whereas
              ``'shd'`` does all of them.
  
          naxis : int array[naxis], optional
              Image axis lengths.  If this array is set to zero or
              ``None``, then `~astropy.wcs.Wcsprm.cylfix` will not be
              invoked.
          """
          if self.wcs is not None:
              self._fix_scamp()
              fixes = self.wcs.fix(translate_units, naxis)
              for key, val in fixes.items():
                  if val != "No change":
                      warnings.warn(
                          ("'{0}' made the change '{1}'.").
                          format(key, val),
                          FITSFixedWarning)
  
      def calc_footprint(self, header=None, undistort=True, axes=None, center=True):
          """
          Calculates the footprint of the image on the sky.
  
          A footprint is defined as the positions of the corners of the
          image on the sky after all available distortions have been
          applied.
  
          Parameters
          ----------
          header : `~astropy.io.fits.Header` object, optional
              Used to get ``NAXIS1`` and ``NAXIS2``
              header and axes are mutually exclusive, alternative ways
              to provide the same information.
  
          undistort : bool, optional
              If `True`, take SIP and distortion lookup table into
              account
  
          axes : length 2 sequence ints, optional
              If provided, use the given sequence as the shape of the
              image.  Otherwise, use the ``NAXIS1`` and ``NAXIS2``
              keywords from the header that was used to create this
              `WCS` object.
  
          center : bool, optional
              If `True` use the center of the pixel, otherwise use the corner.
  
          Returns
          -------
          coord : (4, 2) array of (*x*, *y*) coordinates.
              The order is clockwise starting with the bottom left corner.
          """
          if axes is not None:
              naxis1, naxis2 = axes
          else:
              if header is None:
                  try:
                      # classes that inherit from WCS and define naxis1/2
                      # do not require a header parameter
                      naxis1, naxis2 = self.pixel_shape
                  except (AttributeError, TypeError):
                      warnings.warn("Need a valid header in order to calculate footprint\n", AstropyUserWarning)
                      return None
              else:
                  naxis1 = header.get('NAXIS1', None)
                  naxis2 = header.get('NAXIS2', None)
  
          if naxis1 is None or naxis2 is None:
              raise ValueError(
                      "Image size could not be determined.")
  
          if center:
              corners = np.array([[1, 1],
                                  [1, naxis2],
                                  [naxis1, naxis2],
                                  [naxis1, 1]], dtype=np.float64)
          else:
              corners = np.array([[0.5, 0.5],
                                  [0.5, naxis2 + 0.5],
                                  [naxis1 + 0.5, naxis2 + 0.5],
                                  [naxis1 + 0.5, 0.5]], dtype=np.float64)
  
          if undistort:
              return self.all_pix2world(corners, 1)
          else:
              return self.wcs_pix2world(corners, 1)
  
      def _read_det2im_kw(self, header, fobj, err=0.0):
          """
          Create a `distortion paper`_ type lookup table for detector to
          image plane correction.
          """
          if fobj is None:
              return (None, None)
  
          if not isinstance(fobj, fits.HDUList):
              return (None, None)
  
          try:
              axiscorr = header['AXISCORR']
              d2imdis = self._read_d2im_old_format(header, fobj, axiscorr)
              return d2imdis
          except KeyError:
              pass
  
          dist = 'D2IMDIS'
          d_kw = 'D2IM'
          err_kw = 'D2IMERR'
          tables = {}
          for i in range(1, self.naxis + 1):
              d_error = header.get(err_kw + str(i), 0.0)
              if d_error < err:
                  tables[i] = None
                  continue
              distortion = dist + str(i)
              if distortion in header:
                  dis = header[distortion].lower()
                  if dis == 'lookup':
                      del header[distortion]
                      assert isinstance(fobj, fits.HDUList), ('An astropy.io.fits.HDUList'
                                  'is required for Lookup table distortion.')
                      dp = (d_kw + str(i)).strip()
                      dp_extver_key = dp + '.EXTVER'
                      if dp_extver_key in header:
                          d_extver = header[dp_extver_key]
                          del header[dp_extver_key]
                      else:
                          d_extver = 1
                      dp_axis_key = dp + f'.AXIS.{i:d}'
                      if i == header[dp_axis_key]:
                          d_data = fobj['D2IMARR', d_extver].data
                      else:
                          d_data = (fobj['D2IMARR', d_extver].data).transpose()
                      del header[dp_axis_key]
                      d_header = fobj['D2IMARR', d_extver].header
                      d_crpix = (d_header.get('CRPIX1', 0.0), d_header.get('CRPIX2', 0.0))
                      d_crval = (d_header.get('CRVAL1', 0.0), d_header.get('CRVAL2', 0.0))
                      d_cdelt = (d_header.get('CDELT1', 1.0), d_header.get('CDELT2', 1.0))
                      d_lookup = DistortionLookupTable(d_data, d_crpix,
                                                       d_crval, d_cdelt)
                      tables[i] = d_lookup
                  else:
                      warnings.warn('Polynomial distortion is not implemented.\n', AstropyUserWarning)
                  for key in set(header):
                      if key.startswith(dp + '.'):
                          del header[key]
              else:
                  tables[i] = None
          if not tables:
              return (None, None)
          else:
              return (tables.get(1), tables.get(2))
  
      def _read_d2im_old_format(self, header, fobj, axiscorr):
          warnings.warn("The use of ``AXISCORR`` for D2IM correction has been deprecated."
                        "`~astropy.wcs` will read in files with ``AXISCORR`` but ``to_fits()`` will write "
                        "out files without it.",
                        AstropyDeprecationWarning)
          cpdis = [None, None]
          crpix = [0., 0.]
          crval = [0., 0.]
          cdelt = [1., 1.]
          try:
              d2im_data = fobj[('D2IMARR', 1)].data
          except KeyError:
              return (None, None)
          except AttributeError:
              return (None, None)
  
          d2im_data = np.array([d2im_data])
          d2im_hdr = fobj[('D2IMARR', 1)].header
          naxis = d2im_hdr['NAXIS']
  
          for i in range(1, naxis + 1):
              crpix[i - 1] = d2im_hdr.get('CRPIX' + str(i), 0.0)
              crval[i - 1] = d2im_hdr.get('CRVAL' + str(i), 0.0)
              cdelt[i - 1] = d2im_hdr.get('CDELT' + str(i), 1.0)
  
          cpdis = DistortionLookupTable(d2im_data, crpix, crval, cdelt)
  
          if axiscorr == 1:
              return (cpdis, None)
          elif axiscorr == 2:
              return (None, cpdis)
          else:
              warnings.warn("Expected AXISCORR to be 1 or 2", AstropyUserWarning)
              return (None, None)
  
      def _write_det2im(self, hdulist):
          """
          Writes a `distortion paper`_ type lookup table to the given
          `astropy.io.fits.HDUList`.
          """
  
          if self.det2im1 is None and self.det2im2 is None:
              return
          dist = 'D2IMDIS'
          d_kw = 'D2IM'
          err_kw = 'D2IMERR'
  
          def write_d2i(num, det2im):
              if det2im is None:
                  return
              f'{dist}{num:d}',
              hdulist[0].header[f'{dist}{num:d}'] = (
                  'LOOKUP', 'Detector to image correction type')
              hdulist[0].header[f'{d_kw}{num:d}.EXTVER'] = (
                  num, 'Version number of WCSDVARR extension')
              hdulist[0].header[f'{d_kw}{num:d}.NAXES'] = (
                  len(det2im.data.shape), 'Number of independent variables in d2im function')
              for i in range(det2im.data.ndim):
                  hdulist[0].header['{}{:d}.AXIS.{:d}'.format(d_kw, num, i + 1)] = (
                      i + 1, 'Axis number of the jth independent variable in a d2im function')
  
              image = fits.ImageHDU(det2im.data, name='D2IMARR')
              header = image.header
  
              header['CRPIX1'] = (det2im.crpix[0],
                                       'Coordinate system reference pixel')
              header['CRPIX2'] = (det2im.crpix[1],
                                       'Coordinate system reference pixel')
              header['CRVAL1'] = (det2im.crval[0],
                                       'Coordinate system value at reference pixel')
              header['CRVAL2'] = (det2im.crval[1],
                                       'Coordinate system value at reference pixel')
              header['CDELT1'] = (det2im.cdelt[0],
                                       'Coordinate increment along axis')
              header['CDELT2'] = (det2im.cdelt[1],
                                       'Coordinate increment along axis')
              image.ver = int(hdulist[0].header[f'{d_kw}{num:d}.EXTVER'])
              hdulist.append(image)
          write_d2i(1, self.det2im1)
          write_d2i(2, self.det2im2)
  
      def _read_distortion_kw(self, header, fobj, dist='CPDIS', err=0.0):
          """
          Reads `distortion paper`_ table-lookup keywords and data, and
          returns a 2-tuple of `~astropy.wcs.DistortionLookupTable`
          objects.
  
          If no `distortion paper`_ keywords are found, ``(None, None)``
          is returned.
          """
          if isinstance(header, (str, bytes)):
              return (None, None)
  
          if dist == 'CPDIS':
              d_kw = 'DP'
              err_kw = 'CPERR'
          else:
              d_kw = 'DQ'
              err_kw = 'CQERR'
  
          tables = {}
          for i in range(1, self.naxis + 1):
              d_error_key = err_kw + str(i)
              if d_error_key in header:
                  d_error = header[d_error_key]
                  del header[d_error_key]
              else:
                  d_error = 0.0
              if d_error < err:
                  tables[i] = None
                  continue
              distortion = dist + str(i)
              if distortion in header:
                  dis = header[distortion].lower()
                  del header[distortion]
                  if dis == 'lookup':
                      if not isinstance(fobj, fits.HDUList):
                          raise ValueError('an astropy.io.fits.HDUList is '
                                  'required for Lookup table distortion.')
                      dp = (d_kw + str(i)).strip()
                      dp_extver_key = dp + '.EXTVER'
                      if dp_extver_key in header:
                          d_extver = header[dp_extver_key]
                          del header[dp_extver_key]
                      else:
                          d_extver = 1
                      dp_axis_key = dp + f'.AXIS.{i:d}'
                      if i == header[dp_axis_key]:
                          d_data = fobj['WCSDVARR', d_extver].data
                      else:
                          d_data = (fobj['WCSDVARR', d_extver].data).transpose()
                      del header[dp_axis_key]
                      d_header = fobj['WCSDVARR', d_extver].header
                      d_crpix = (d_header.get('CRPIX1', 0.0),
                                 d_header.get('CRPIX2', 0.0))
                      d_crval = (d_header.get('CRVAL1', 0.0),
                                 d_header.get('CRVAL2', 0.0))
                      d_cdelt = (d_header.get('CDELT1', 1.0),
                                 d_header.get('CDELT2', 1.0))
                      d_lookup = DistortionLookupTable(d_data, d_crpix, d_crval, d_cdelt)
                      tables[i] = d_lookup
  
                      for key in set(header):
                          if key.startswith(dp + '.'):
                              del header[key]
                  else:
                      warnings.warn('Polynomial distortion is not implemented.\n', AstropyUserWarning)
              else:
                  tables[i] = None
  
          if not tables:
              return (None, None)
          else:
              return (tables.get(1), tables.get(2))
  
      def _write_distortion_kw(self, hdulist, dist='CPDIS'):
          """
          Write out `distortion paper`_ keywords to the given
          `fits.HDUList`.
          """
          if self.cpdis1 is None and self.cpdis2 is None:
              return
  
          if dist == 'CPDIS':
              d_kw = 'DP'
              err_kw = 'CPERR'
          else:
              d_kw = 'DQ'
              err_kw = 'CQERR'
  
          def write_dist(num, cpdis):
              if cpdis is None:
                  return
  
              hdulist[0].header[f'{dist}{num:d}'] = (
                  'LOOKUP', 'Prior distortion function type')
              hdulist[0].header[f'{d_kw}{num:d}.EXTVER'] = (
                  num, 'Version number of WCSDVARR extension')
              hdulist[0].header[f'{d_kw}{num:d}.NAXES'] = (
                  len(cpdis.data.shape), 'Number of independent variables in distortion function')
  
              for i in range(cpdis.data.ndim):
                  hdulist[0].header['{}{:d}.AXIS.{:d}'.format(d_kw, num, i + 1)] = (
                      i + 1,
                      'Axis number of the jth independent variable in a distortion function')
  
              image = fits.ImageHDU(cpdis.data, name='WCSDVARR')
              header = image.header
  
              header['CRPIX1'] = (cpdis.crpix[0], 'Coordinate system reference pixel')
              header['CRPIX2'] = (cpdis.crpix[1], 'Coordinate system reference pixel')
              header['CRVAL1'] = (cpdis.crval[0], 'Coordinate system value at reference pixel')
              header['CRVAL2'] = (cpdis.crval[1], 'Coordinate system value at reference pixel')
              header['CDELT1'] = (cpdis.cdelt[0], 'Coordinate increment along axis')
              header['CDELT2'] = (cpdis.cdelt[1], 'Coordinate increment along axis')
              image.ver = int(hdulist[0].header[f'{d_kw}{num:d}.EXTVER'])
              hdulist.append(image)
  
          write_dist(1, self.cpdis1)
          write_dist(2, self.cpdis2)
  
      def _remove_sip_kw(self, header):
          """
          Remove SIP information from a header.
          """
          # Never pass SIP coefficients to wcslib
          # CTYPE must be passed with -SIP to wcslib
          for key in set(m.group() for m in map(SIP_KW.match, list(header))
                      if m is not None):
              del header[key]
  
      def _read_sip_kw(self, header, wcskey=""):
          """
          Reads `SIP`_ header keywords and returns a `~astropy.wcs.Sip`
          object.
  
          If no `SIP`_ header keywords are found, ``None`` is returned.
          """
          if isinstance(header, (str, bytes)):
              # TODO: Parse SIP from a string without pyfits around
              return None
  
          if "A_ORDER" in header and header['A_ORDER'] > 1:
              if "B_ORDER" not in header:
                  raise ValueError(
                      "A_ORDER provided without corresponding B_ORDER "
                      "keyword for SIP distortion")
  
              m = int(header["A_ORDER"])
              a = np.zeros((m + 1, m + 1), np.double)
              for i in range(m + 1):
                  for j in range(m - i + 1):
                      key = f"A_{i}_{j}"
                      if key in header:
                          a[i, j] = header[key]
                          del header[key]
  
              m = int(header["B_ORDER"])
              if m > 1:
                  b = np.zeros((m + 1, m + 1), np.double)
                  for i in range(m + 1):
                      for j in range(m - i + 1):
                          key = f"B_{i}_{j}"
                          if key in header:
                              b[i, j] = header[key]
                              del header[key]
              else:
                  a = None
                  b = None
  
              del header['A_ORDER']
              del header['B_ORDER']
  
              ctype = [header[f'CTYPE{nax}{wcskey}'] for nax in range(1, self.naxis + 1)]
              if any(not ctyp.endswith('-SIP') for ctyp in ctype):
                  message = """
                  Inconsistent SIP distortion information is present in the FITS header and the WCS object:
                  SIP coefficients were detected, but CTYPE is missing a "-SIP" suffix.
                  astropy.wcs is using the SIP distortion coefficients,
                  therefore the coordinates calculated here might be incorrect.
  
                  If you do not want to apply the SIP distortion coefficients,
                  please remove the SIP coefficients from the FITS header or the
                  WCS object.  As an example, if the image is already distortion-corrected
                  (e.g., drizzled) then distortion components should not apply and the SIP
                  coefficients should be removed.
  
                  While the SIP distortion coefficients are being applied here, if that was indeed the intent,
                  for consistency please append "-SIP" to the CTYPE in the FITS header or the WCS object.
  
                  """
                  log.info(message)
          elif "B_ORDER" in header and header['B_ORDER'] > 1:
              raise ValueError(
                  "B_ORDER provided without corresponding A_ORDER " +
                  "keyword for SIP distortion")
          else:
              a = None
              b = None
  
          if "AP_ORDER" in header and header['AP_ORDER'] > 1:
              if "BP_ORDER" not in header:
                  raise ValueError(
                      "AP_ORDER provided without corresponding BP_ORDER "
                      "keyword for SIP distortion")
  
              m = int(header["AP_ORDER"])
              ap = np.zeros((m + 1, m + 1), np.double)
              for i in range(m + 1):
                  for j in range(m - i + 1):
                      key = f"AP_{i}_{j}"
                      if key in header:
                          ap[i, j] = header[key]
                          del header[key]
  
              m = int(header["BP_ORDER"])
              if m > 1:
                  bp = np.zeros((m + 1, m + 1), np.double)
                  for i in range(m + 1):
                      for j in range(m - i + 1):
                          key = f"BP_{i}_{j}"
                          if key in header:
                              bp[i, j] = header[key]
                              del header[key]
              else:
                  ap = None
                  bp = None
  
              del header['AP_ORDER']
              del header['BP_ORDER']
          elif "BP_ORDER" in header and header['BP_ORDER'] > 1:
              raise ValueError(
                  "BP_ORDER provided without corresponding AP_ORDER "
                  "keyword for SIP distortion")
          else:
              ap = None
              bp = None
  
          if a is None and b is None and ap is None and bp is None:
              return None
  
          if f"CRPIX1{wcskey}" not in header or f"CRPIX2{wcskey}" not in header:
              raise ValueError(
                  "Header has SIP keywords without CRPIX keywords")
  
          crpix1 = header.get(f"CRPIX1{wcskey}")
          crpix2 = header.get(f"CRPIX2{wcskey}")
  
          return Sip(a, b, ap, bp, (crpix1, crpix2))
  
      def _write_sip_kw(self):
          """
          Write out SIP keywords.  Returns a dictionary of key-value
          pairs.
          """
          if self.sip is None:
              return {}
  
          keywords = {}
  
          def write_array(name, a):
              if a is None:
                  return
              size = a.shape[0]
              keywords[f'{name}_ORDER'] = size - 1
              for i in range(size):
                  for j in range(size - i):
                      if a[i, j] != 0.0:
                          keywords[
                              f'{name}_{i:d}_{j:d}'] = a[i, j]
  
          write_array('A', self.sip.a)
          write_array('B', self.sip.b)
          write_array('AP', self.sip.ap)
          write_array('BP', self.sip.bp)
  
          return keywords
  
      def _denormalize_sky(self, sky):
          if self.wcs.lngtyp != 'RA':
              raise ValueError(
                  "WCS does not have longitude type of 'RA', therefore " +
                  "(ra, dec) data can not be used as input")
          if self.wcs.lattyp != 'DEC':
              raise ValueError(
                  "WCS does not have longitude type of 'DEC', therefore " +
                  "(ra, dec) data can not be used as input")
          if self.wcs.naxis == 2:
              if self.wcs.lng == 0 and self.wcs.lat == 1:
                  return sky
              elif self.wcs.lng == 1 and self.wcs.lat == 0:
                  # Reverse the order of the columns
                  return sky[:, ::-1]
              else:
                  raise ValueError(
                      "WCS does not have longitude and latitude celestial " +
                      "axes, therefore (ra, dec) data can not be used as input")
          else:
              if self.wcs.lng < 0 or self.wcs.lat < 0:
                  raise ValueError(
                      "WCS does not have both longitude and latitude "
                      "celestial axes, therefore (ra, dec) data can not be " +
                      "used as input")
              out = np.zeros((sky.shape[0], self.wcs.naxis))
              out[:, self.wcs.lng] = sky[:, 0]
              out[:, self.wcs.lat] = sky[:, 1]
              return out
  
      def _normalize_sky(self, sky):
          if self.wcs.lngtyp != 'RA':
              raise ValueError(
                  "WCS does not have longitude type of 'RA', therefore " +
                  "(ra, dec) data can not be returned")
          if self.wcs.lattyp != 'DEC':
              raise ValueError(
                  "WCS does not have longitude type of 'DEC', therefore " +
                  "(ra, dec) data can not be returned")
          if self.wcs.naxis == 2:
              if self.wcs.lng == 0 and self.wcs.lat == 1:
                  return sky
              elif self.wcs.lng == 1 and self.wcs.lat == 0:
                  # Reverse the order of the columns
                  return sky[:, ::-1]
              else:
                  raise ValueError(
                      "WCS does not have longitude and latitude celestial "
                      "axes, therefore (ra, dec) data can not be returned")
          else:
              if self.wcs.lng < 0 or self.wcs.lat < 0:
                  raise ValueError(
                      "WCS does not have both longitude and latitude celestial "
                      "axes, therefore (ra, dec) data can not be returned")
              out = np.empty((sky.shape[0], 2))
              out[:, 0] = sky[:, self.wcs.lng]
              out[:, 1] = sky[:, self.wcs.lat]
              return out
  
      def _array_converter(self, func, sky, *args, ra_dec_order=False):
          """
          A helper function to support reading either a pair of arrays
          or a single Nx2 array.
          """
  
          def _return_list_of_arrays(axes, origin):
              if any([x.size == 0 for x in axes]):
                  return axes
  
              try:
                  axes = np.broadcast_arrays(*axes)
              except ValueError:
                  raise ValueError(
                      "Coordinate arrays are not broadcastable to each other")
  
              xy = np.hstack([x.reshape((x.size, 1)) for x in axes])
  
              if ra_dec_order and sky == 'input':
                  xy = self._denormalize_sky(xy)
              output = func(xy, origin)
              if ra_dec_order and sky == 'output':
                  output = self._normalize_sky(output)
                  return (output[:, 0].reshape(axes[0].shape),
                          output[:, 1].reshape(axes[0].shape))
              return [output[:, i].reshape(axes[0].shape)
                      for i in range(output.shape[1])]
  
          def _return_single_array(xy, origin):
              if xy.shape[-1] != self.naxis:
                  raise ValueError(
                      "When providing two arguments, the array must be "
                      "of shape (N, {})".format(self.naxis))
              if 0 in xy.shape:
                  return xy
              if ra_dec_order and sky == 'input':
                  xy = self._denormalize_sky(xy)
              result = func(xy, origin)
              if ra_dec_order and sky == 'output':
                  result = self._normalize_sky(result)
              return result
  
          if len(args) == 2:
              try:
                  xy, origin = args
                  xy = np.asarray(xy)
                  origin = int(origin)
              except Exception:
                  raise TypeError(
                      "When providing two arguments, they must be "
                      "(coords[N][{}], origin)".format(self.naxis))
              if xy.shape == () or len(xy.shape) == 1:
                  return _return_list_of_arrays([xy], origin)
              return _return_single_array(xy, origin)
  
          elif len(args) == self.naxis + 1:
              axes = args[:-1]
              origin = args[-1]
              try:
                  axes = [np.asarray(x) for x in axes]
                  origin = int(origin)
              except Exception:
                  raise TypeError(
                      "When providing more than two arguments, they must be " +
                      "a 1-D array for each axis, followed by an origin.")
  
              return _return_list_of_arrays(axes, origin)
  
          raise TypeError(
              "WCS projection has {0} dimensions, so expected 2 (an Nx{0} array "
              "and the origin argument) or {1} arguments (the position in each "
              "dimension, and the origin argument). Instead, {2} arguments were "
              "given.".format(
                  self.naxis, self.naxis + 1, len(args)))
  
      def all_pix2world(self, *args, **kwargs):
          return self._array_converter(
              self._all_pix2world, 'output', *args, **kwargs)
      all_pix2world.__doc__ = """
          Transforms pixel coordinates to world coordinates.
  
          Performs all of the following in series:
  
              - Detector to image plane correction (if present in the
                FITS file)
  
              - `SIP`_ distortion correction (if present in the FITS
                file)
  
              - `distortion paper`_ table-lookup correction (if present
                in the FITS file)
  
              - `wcslib`_ "core" WCS transformation
  
          Parameters
          ----------
          {}
  
              For a transformation that is not two-dimensional, the
              two-argument form must be used.
  
          {}
  
          Returns
          -------
  
          {}
  
          Notes
          -----
          The order of the axes for the result is determined by the
          ``CTYPEia`` keywords in the FITS header, therefore it may not
          always be of the form (*ra*, *dec*).  The
          `~astropy.wcs.Wcsprm.lat`, `~astropy.wcs.Wcsprm.lng`,
          `~astropy.wcs.Wcsprm.lattyp` and `~astropy.wcs.Wcsprm.lngtyp`
          members can be used to determine the order of the axes.
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          SingularMatrixError
              Linear transformation matrix is singular.
  
          InconsistentAxisTypesError
              Inconsistent or unrecognized coordinate axis types.
  
          ValueError
              Invalid parameter value.
  
          ValueError
              Invalid coordinate transformation parameters.
  
          ValueError
              x- and y-coordinate arrays are not the same size.
  
          InvalidTransformError
              Invalid coordinate transformation parameters.
  
          InvalidTransformError
              Ill-conditioned coordinate transformation parameters.
          """.format(docstrings.TWO_OR_MORE_ARGS('naxis', 8),
                     docstrings.RA_DEC_ORDER(8),
                     docstrings.RETURNS('sky coordinates, in degrees', 8))
  
      def wcs_pix2world(self, *args, **kwargs):
          if self.wcs is None:
              raise ValueError("No basic WCS settings were created.")
          return self._array_converter(
              lambda xy, o: self.wcs.p2s(xy, o)['world'],
              'output', *args, **kwargs)
      wcs_pix2world.__doc__ = """
          Transforms pixel coordinates to world coordinates by doing
          only the basic `wcslib`_ transformation.
  
          No `SIP`_ or `distortion paper`_ table lookup correction is
          applied.  To perform distortion correction, see
          `~astropy.wcs.WCS.all_pix2world`,
          `~astropy.wcs.WCS.sip_pix2foc`, `~astropy.wcs.WCS.p4_pix2foc`,
          or `~astropy.wcs.WCS.pix2foc`.
  
          Parameters
          ----------
          {}
  
              For a transformation that is not two-dimensional, the
              two-argument form must be used.
  
          {}
  
          Returns
          -------
  
          {}
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          SingularMatrixError
              Linear transformation matrix is singular.
  
          InconsistentAxisTypesError
              Inconsistent or unrecognized coordinate axis types.
  
          ValueError
              Invalid parameter value.
  
          ValueError
              Invalid coordinate transformation parameters.
  
          ValueError
              x- and y-coordinate arrays are not the same size.
  
          InvalidTransformError
              Invalid coordinate transformation parameters.
  
          InvalidTransformError
              Ill-conditioned coordinate transformation parameters.
  
          Notes
          -----
          The order of the axes for the result is determined by the
          ``CTYPEia`` keywords in the FITS header, therefore it may not
          always be of the form (*ra*, *dec*).  The
          `~astropy.wcs.Wcsprm.lat`, `~astropy.wcs.Wcsprm.lng`,
          `~astropy.wcs.Wcsprm.lattyp` and `~astropy.wcs.Wcsprm.lngtyp`
          members can be used to determine the order of the axes.
  
          """.format(docstrings.TWO_OR_MORE_ARGS('naxis', 8),
                     docstrings.RA_DEC_ORDER(8),
                     docstrings.RETURNS('world coordinates, in degrees', 8))
  
      def _all_world2pix(self, world, origin, tolerance, maxiter, adaptive,
                         detect_divergence, quiet):
          # ############################################################
          # #          DESCRIPTION OF THE NUMERICAL METHOD            ##
          # ############################################################
          # In this section I will outline the method of solving
          # the inverse problem of converting world coordinates to
          # pixel coordinates (*inverse* of the direct transformation
          # `all_pix2world`) and I will summarize some of the aspects
          # of the method proposed here and some of the issues of the
          # original `all_world2pix` (in relation to this method)
          # discussed in https://github.com/astropy/astropy/issues/1977
          # A more detailed discussion can be found here:
          # https://github.com/astropy/astropy/pull/2373
          #
          #
          #                  ### Background ###
          #
          #
          # I will refer here to the [SIP Paper]
          # (http://fits.gsfc.nasa.gov/registry/sip/SIP_distortion_v1_0.pdf).
          # According to this paper, the effect of distortions as
          # described in *their* equation (1) is:
          #
          # (1)   x = CD*(u+f(u)),
          #
          # where `x` is a *vector* of "intermediate spherical
          # coordinates" (equivalent to (x,y) in the paper) and `u`
          # is a *vector* of "pixel coordinates", and `f` is a vector
          # function describing geometrical distortions
          # (see equations 2 and 3 in SIP Paper.
          # However, I prefer to use `w` for "intermediate world
          # coordinates", `x` for pixel coordinates, and assume that
          # transformation `W` performs the **linear**
          # (CD matrix + projection onto celestial sphere) part of the
          # conversion from pixel coordinates to world coordinates.
          # Then we can re-write (1) as:
          #
          # (2)   w = W*(x+f(x)) = T(x)
          #
          # In `astropy.wcs.WCS` transformation `W` is represented by
          # the `wcs_pix2world` member, while the combined ("total")
          # transformation (linear part + distortions) is performed by
          # `all_pix2world`. Below I summarize the notations and their
          # equivalents in `astropy.wcs.WCS`:
          #
          # | Equation term | astropy.WCS/meaning          |
          # | ------------- | ---------------------------- |
          # | `x`           | pixel coordinates            |
          # | `w`           | world coordinates            |
          # | `W`           | `wcs_pix2world()`            |
          # | `W^{-1}`      | `wcs_world2pix()`            |
          # | `T`           | `all_pix2world()`            |
          # | `x+f(x)`      | `pix2foc()`                  |
          #
          #
          #      ### Direct Solving of Equation (2)  ###
          #
          #
          # In order to find the pixel coordinates that correspond to
          # given world coordinates `w`, it is necessary to invert
          # equation (2): `x=T^{-1}(w)`, or solve equation `w==T(x)`
          # for `x`. However, this approach has the following
          # disadvantages:
          #    1. It requires unnecessary transformations (see next
          #       section).
          #    2. It is prone to "RA wrapping" issues as described in
          # https://github.com/astropy/astropy/issues/1977
          # (essentially because `all_pix2world` may return points with
          # a different phase than user's input `w`).
          #
          #
          #      ### Description of the Method Used here ###
          #
          #
          # By applying inverse linear WCS transformation (`W^{-1}`)
          # to both sides of equation (2) and introducing notation `x'`
          # (prime) for the pixels coordinates obtained from the world
          # coordinates by applying inverse *linear* WCS transformation
          # ("focal plane coordinates"):
          #
          # (3)   x' = W^{-1}(w)
          #
          # we obtain the following equation:
          #
          # (4)   x' = x+f(x),
          #
          # or,
          #
          # (5)   x = x'-f(x)
          #
          # This equation is well suited for solving using the method
          # of fixed-point iterations
          # (http://en.wikipedia.org/wiki/Fixed-point_iteration):
          #
          # (6)   x_{i+1} = x'-f(x_i)
          #
          # As an initial value of the pixel coordinate `x_0` we take
          # "focal plane coordinate" `x'=W^{-1}(w)=wcs_world2pix(w)`.
          # We stop iterations when `|x_{i+1}-x_i|<tolerance`. We also
          # consider the process to be diverging if
          # `|x_{i+1}-x_i|>|x_i-x_{i-1}|`
          # **when** `|x_{i+1}-x_i|>=tolerance` (when current
          # approximation is close to the true solution,
          # `|x_{i+1}-x_i|>|x_i-x_{i-1}|` may be due to rounding errors
          # and we ignore such "divergences" when
          # `|x_{i+1}-x_i|<tolerance`). It may appear that checking for
          # `|x_{i+1}-x_i|<tolerance` in order to ignore divergence is
          # unnecessary since the iterative process should stop anyway,
          # however, the proposed implementation of this iterative
          # process is completely vectorized and, therefore, we may
          # continue iterating over *some* points even though they have
          # converged to within a specified tolerance (while iterating
          # over other points that have not yet converged to
          # a solution).
          #
          # In order to efficiently implement iterative process (6)
          # using available methods in `astropy.wcs.WCS`, we add and
          # subtract `x_i` from the right side of equation (6):
          #
          # (7)   x_{i+1} = x'-(x_i+f(x_i))+x_i = x'-pix2foc(x_i)+x_i,
          #
          # where `x'=wcs_world2pix(w)` and it is computed only *once*
          # before the beginning of the iterative process (and we also
          # set `x_0=x'`). By using `pix2foc` at each iteration instead
          # of `all_pix2world` we get about 25% increase in performance
          # (by not performing the linear `W` transformation at each
          # step) and we also avoid the "RA wrapping" issue described
          # above (by working in focal plane coordinates and avoiding
          # pix->world transformations).
          #
          # As an added benefit, the process converges to the correct
          # solution in just one iteration when distortions are not
          # present (compare to
          # https://github.com/astropy/astropy/issues/1977 and
          # https://github.com/astropy/astropy/pull/2294): in this case
          # `pix2foc` is the identical transformation
          # `x_i=pix2foc(x_i)` and from equation (7) we get:
          #
          # x' = x_0 = wcs_world2pix(w)
          # x_1 = x' - pix2foc(x_0) + x_0 = x' - pix2foc(x') + x' = x'
          #     = wcs_world2pix(w) = x_0
          # =>
          # |x_1-x_0| = 0 < tolerance (with tolerance > 0)
          #
          # However, for performance reasons, it is still better to
          # avoid iterations altogether and return the exact linear
          # solution (`wcs_world2pix`) right-away when non-linear
          # distortions are not present by checking that attributes
          # `sip`, `cpdis1`, `cpdis2`, `det2im1`, and `det2im2` are
          # *all* `None`.
          #
          #
          #         ### Outline of the Algorithm ###
          #
          #
          # While the proposed code is relatively long (considering
          # the simplicity of the algorithm), this is due to: 1)
          # checking if iterative solution is necessary at all; 2)
          # checking for divergence; 3) re-implementation of the
          # completely vectorized algorithm as an "adaptive" vectorized
          # algorithm (for cases when some points diverge for which we
          # want to stop iterations). In my tests, the adaptive version
          # of the algorithm is about 50% slower than non-adaptive
          # version for all HST images.
          #
          # The essential part of the vectorized non-adaptive algorithm
          # (without divergence and other checks) can be described
          # as follows:
          #
          #     pix0 = self.wcs_world2pix(world, origin)
          #     pix  = pix0.copy() # 0-order solution
          #
          #     for k in range(maxiter):
          #         # find correction to the previous solution:
          #         dpix = self.pix2foc(pix, origin) - pix0
          #
          #         # compute norm (L2) of the correction:
          #         dn = np.linalg.norm(dpix, axis=1)
          #
          #         # apply correction:
          #         pix -= dpix
          #
          #         # check convergence:
          #         if np.max(dn) < tolerance:
          #             break
          #
          #    return pix
          #
          # Here, the input parameter `world` can be a `MxN` array
          # where `M` is the number of coordinate axes in WCS and `N`
          # is the number of points to be converted simultaneously to
          # image coordinates.
          #
          #
          #                ###  IMPORTANT NOTE:  ###
          #
          # If, in the future releases of the `~astropy.wcs`,
          # `pix2foc` will not apply all the required distortion
          # corrections then in the code below, calls to `pix2foc` will
          # have to be replaced with
          # wcs_world2pix(all_pix2world(pix_list, origin), origin)
          #
  
          # ############################################################
          # #            INITIALIZE ITERATIVE PROCESS:                ##
          # ############################################################
  
          # initial approximation (linear WCS based only)
          pix0 = self.wcs_world2pix(world, origin)
  
          # Check that an iterative solution is required at all
          # (when any of the non-CD-matrix-based corrections are
          # present). If not required return the initial
          # approximation (pix0).
          if not self.has_distortion:
              # No non-WCS corrections detected so
              # simply return initial approximation:
              return pix0
  
          pix = pix0.copy()  # 0-order solution
  
          # initial correction:
          dpix = self.pix2foc(pix, origin) - pix0
  
          # Update initial solution:
          pix -= dpix
  
          # Norm (L2) squared of the correction:
          dn = np.sum(dpix*dpix, axis=1)
          dnprev = dn.copy()  # if adaptive else dn
          tol2 = tolerance**2
  
          # Prepare for iterative process
          k = 1
          ind = None
          inddiv = None
  
          # Turn off numpy runtime warnings for 'invalid' and 'over':
          old_invalid = np.geterr()['invalid']
          old_over = np.geterr()['over']
          np.seterr(invalid='ignore', over='ignore')
  
          # ############################################################
          # #                NON-ADAPTIVE ITERATIONS:                 ##
          # ############################################################
          if not adaptive:
              # Fixed-point iterations:
              while (np.nanmax(dn) >= tol2 and k < maxiter):
                  # Find correction to the previous solution:
                  dpix = self.pix2foc(pix, origin) - pix0
  
                  # Compute norm (L2) squared of the correction:
                  dn = np.sum(dpix*dpix, axis=1)
  
                  # Check for divergence (we do this in two stages
                  # to optimize performance for the most common
                  # scenario when successive approximations converge):
                  if detect_divergence:
                      divergent = (dn >= dnprev)
                      if np.any(divergent):
                          # Find solutions that have not yet converged:
                          slowconv = (dn >= tol2)
                          inddiv, = np.where(divergent & slowconv)
  
                          if inddiv.shape[0] > 0:
                              # Update indices of elements that
                              # still need correction:
                              conv = (dn < dnprev)
                              iconv = np.where(conv)
  
                              # Apply correction:
                              dpixgood = dpix[iconv]
                              pix[iconv] -= dpixgood
                              dpix[iconv] = dpixgood
  
                              # For the next iteration choose
                              # non-divergent points that have not yet
                              # converged to the requested accuracy:
                              ind, = np.where(slowconv & conv)
                              pix0 = pix0[ind]
                              dnprev[ind] = dn[ind]
                              k += 1
  
                              # Switch to adaptive iterations:
                              adaptive = True
                              break
                      # Save current correction magnitudes for later:
                      dnprev = dn
  
                  # Apply correction:
                  pix -= dpix
                  k += 1
  
          # ############################################################
          # #                  ADAPTIVE ITERATIONS:                   ##
          # ############################################################
          if adaptive:
              if ind is None:
                  ind, = np.where(np.isfinite(pix).all(axis=1))
                  pix0 = pix0[ind]
  
              # "Adaptive" fixed-point iterations:
              while (ind.shape[0] > 0 and k < maxiter):
                  # Find correction to the previous solution:
                  dpixnew = self.pix2foc(pix[ind], origin) - pix0
  
                  # Compute norm (L2) of the correction:
                  dnnew = np.sum(np.square(dpixnew), axis=1)
  
                  # Bookeeping of corrections:
                  dnprev[ind] = dn[ind].copy()
                  dn[ind] = dnnew
  
                  if detect_divergence:
                      # Find indices of pixels that are converging:
                      conv = (dnnew < dnprev[ind])
                      iconv = np.where(conv)
                      iiconv = ind[iconv]
  
                      # Apply correction:
                      dpixgood = dpixnew[iconv]
                      pix[iiconv] -= dpixgood
                      dpix[iiconv] = dpixgood
  
                      # Find indices of solutions that have not yet
                      # converged to the requested accuracy
                      # AND that do not diverge:
                      subind, = np.where((dnnew >= tol2) & conv)
  
                  else:
                      # Apply correction:
                      pix[ind] -= dpixnew
                      dpix[ind] = dpixnew
  
                      # Find indices of solutions that have not yet
                      # converged to the requested accuracy:
                      subind, = np.where(dnnew >= tol2)
  
                  # Choose solutions that need more iterations:
                  ind = ind[subind]
                  pix0 = pix0[subind]
  
                  k += 1
  
          # ############################################################
          # #         FINAL DETECTION OF INVALID, DIVERGING,          ##
          # #         AND FAILED-TO-CONVERGE POINTS                   ##
          # ############################################################
          # Identify diverging and/or invalid points:
          invalid = ((~np.all(np.isfinite(pix), axis=1)) &
                     (np.all(np.isfinite(world), axis=1)))
  
          # When detect_divergence==False, dnprev is outdated
          # (it is the norm of the very first correction).
          # Still better than nothing...
          inddiv, = np.where(((dn >= tol2) & (dn >= dnprev)) | invalid)
          if inddiv.shape[0] == 0:
              inddiv = None
  
          # Identify points that did not converge within 'maxiter'
          # iterations:
          if k >= maxiter:
              ind, = np.where((dn >= tol2) & (dn < dnprev) & (~invalid))
              if ind.shape[0] == 0:
                  ind = None
          else:
              ind = None
  
          # Restore previous numpy error settings:
          np.seterr(invalid=old_invalid, over=old_over)
  
          # ############################################################
          # #  RAISE EXCEPTION IF DIVERGING OR TOO SLOWLY CONVERGING  ##
          # #  DATA POINTS HAVE BEEN DETECTED:                        ##
          # ############################################################
          if (ind is not None or inddiv is not None) and not quiet:
              if inddiv is None:
                  raise NoConvergence(
                      "'WCS.all_world2pix' failed to "
                      "converge to the requested accuracy after {:d} "
                      "iterations.".format(k), best_solution=pix,
                      accuracy=np.abs(dpix), niter=k,
                      slow_conv=ind, divergent=None)
              else:
                  raise NoConvergence(
                      "'WCS.all_world2pix' failed to "
                      "converge to the requested accuracy.\n"
                      "After {:d} iterations, the solution is diverging "
                      "at least for one input point."
                      .format(k), best_solution=pix,
                      accuracy=np.abs(dpix), niter=k,
                      slow_conv=ind, divergent=inddiv)
  
          return pix
  
      def all_world2pix(self, *args, tolerance=1e-4, maxiter=20, adaptive=False,
                        detect_divergence=True, quiet=False, **kwargs):
          if self.wcs is None:
              raise ValueError("No basic WCS settings were created.")
  
          return self._array_converter(
              lambda *args, **kwargs:
              self._all_world2pix(
                  *args, tolerance=tolerance, maxiter=maxiter,
                  adaptive=adaptive, detect_divergence=detect_divergence,
                  quiet=quiet),
              'input', *args, **kwargs
          )
  
      all_world2pix.__doc__ = """
          all_world2pix(*arg, accuracy=1.0e-4, maxiter=20,
          adaptive=False, detect_divergence=True, quiet=False)
  
          Transforms world coordinates to pixel coordinates, using
          numerical iteration to invert the full forward transformation
          `~astropy.wcs.WCS.all_pix2world` with complete
          distortion model.
  
  
          Parameters
          ----------
          {0}
  
              For a transformation that is not two-dimensional, the
              two-argument form must be used.
  
          {1}
  
          tolerance : float, optional (Default = 1.0e-4)
              Tolerance of solution. Iteration terminates when the
              iterative solver estimates that the "true solution" is
              within this many pixels current estimate, more
              specifically, when the correction to the solution found
              during the previous iteration is smaller
              (in the sense of the L2 norm) than ``tolerance``.
  
          maxiter : int, optional (Default = 20)
              Maximum number of iterations allowed to reach a solution.
  
          quiet : bool, optional (Default = False)
              Do not throw :py:class:`NoConvergence` exceptions when
              the method does not converge to a solution with the
              required accuracy within a specified number of maximum
              iterations set by ``maxiter`` parameter. Instead,
              simply return the found solution.
  
          Other Parameters
          ----------------
          adaptive : bool, optional (Default = False)
              Specifies whether to adaptively select only points that
              did not converge to a solution within the required
              accuracy for the next iteration. Default is recommended
              for HST as well as most other instruments.
  
              .. note::
                 The :py:meth:`all_world2pix` uses a vectorized
                 implementation of the method of consecutive
                 approximations (see ``Notes`` section below) in which it
                 iterates over *all* input points *regardless* until
                 the required accuracy has been reached for *all* input
                 points. In some cases it may be possible that
                 *almost all* points have reached the required accuracy
                 but there are only a few of input data points for
                 which additional iterations may be needed (this
                 depends mostly on the characteristics of the geometric
                 distortions for a given instrument). In this situation
                 it may be advantageous to set ``adaptive`` = `True` in
                 which case :py:meth:`all_world2pix` will continue
                 iterating *only* over the points that have not yet
                 converged to the required accuracy. However, for the
                 HST's ACS/WFC detector, which has the strongest
                 distortions of all HST instruments, testing has
                 shown that enabling this option would lead to a about
                 50-100% penalty in computational time (depending on
                 specifics of the image, geometric distortions, and
                 number of input points to be converted). Therefore,
                 for HST and possibly instruments, it is recommended
                 to set ``adaptive`` = `False`. The only danger in
                 getting this setting wrong will be a performance
                 penalty.
  
              .. note::
                 When ``detect_divergence`` is `True`,
                 :py:meth:`all_world2pix` will automatically switch
                 to the adaptive algorithm once divergence has been
                 detected.
  
          detect_divergence : bool, optional (Default = True)
              Specifies whether to perform a more detailed analysis
              of the convergence to a solution. Normally
              :py:meth:`all_world2pix` may not achieve the required
              accuracy if either the ``tolerance`` or ``maxiter`` arguments
              are too low. However, it may happen that for some
              geometric distortions the conditions of convergence for
              the the method of consecutive approximations used by
              :py:meth:`all_world2pix` may not be satisfied, in which
              case consecutive approximations to the solution will
              diverge regardless of the ``tolerance`` or ``maxiter``
              settings.
  
              When ``detect_divergence`` is `False`, these divergent
              points will be detected as not having achieved the
              required accuracy (without further details). In addition,
              if ``adaptive`` is `False` then the algorithm will not
              know that the solution (for specific points) is diverging
              and will continue iterating and trying to "improve"
              diverging solutions. This may result in ``NaN`` or
              ``Inf`` values in the return results (in addition to a
              performance penalties). Even when ``detect_divergence``
              is `False`, :py:meth:`all_world2pix`, at the end of the
              iterative process, will identify invalid results
              (``NaN`` or ``Inf``) as "diverging" solutions and will
              raise :py:class:`NoConvergence` unless the ``quiet``
              parameter is set to `True`.
  
              When ``detect_divergence`` is `True`,
              :py:meth:`all_world2pix` will detect points for which
              current correction to the coordinates is larger than
              the correction applied during the previous iteration
              **if** the requested accuracy **has not yet been
              achieved**. In this case, if ``adaptive`` is `True`,
              these points will be excluded from further iterations and
              if ``adaptive`` is `False`, :py:meth:`all_world2pix` will
              automatically switch to the adaptive algorithm. Thus, the
              reported divergent solution will be the latest converging
              solution computed immediately *before* divergence
              has been detected.
  
              .. note::
                 When accuracy has been achieved, small increases in
                 current corrections may be possible due to rounding
                 errors (when ``adaptive`` is `False`) and such
                 increases will be ignored.
  
              .. note::
                 Based on our testing using HST ACS/WFC images, setting
                 ``detect_divergence`` to `True` will incur about 5-20%
                 performance penalty with the larger penalty
                 corresponding to ``adaptive`` set to `True`.
                 Because the benefits of enabling this
                 feature outweigh the small performance penalty,
                 especially when ``adaptive`` = `False`, it is
                 recommended to set ``detect_divergence`` to `True`,
                 unless extensive testing of the distortion models for
                 images from specific instruments show a good stability
                 of the numerical method for a wide range of
                 coordinates (even outside the image itself).
  
              .. note::
                 Indices of the diverging inverse solutions will be
                 reported in the ``divergent`` attribute of the
                 raised :py:class:`NoConvergence` exception object.
  
          Returns
          -------
  
          {2}
  
          Notes
          -----
          The order of the axes for the input world array is determined by
          the ``CTYPEia`` keywords in the FITS header, therefore it may
          not always be of the form (*ra*, *dec*).  The
          `~astropy.wcs.Wcsprm.lat`, `~astropy.wcs.Wcsprm.lng`,
          `~astropy.wcs.Wcsprm.lattyp`, and
          `~astropy.wcs.Wcsprm.lngtyp`
          members can be used to determine the order of the axes.
  
          Using the method of fixed-point iterations approximations we
          iterate starting with the initial approximation, which is
          computed using the non-distortion-aware
          :py:meth:`wcs_world2pix` (or equivalent).
  
          The :py:meth:`all_world2pix` function uses a vectorized
          implementation of the method of consecutive approximations and
          therefore it is highly efficient (>30x) when *all* data points
          that need to be converted from sky coordinates to image
          coordinates are passed at *once*. Therefore, it is advisable,
          whenever possible, to pass as input a long array of all points
          that need to be converted to :py:meth:`all_world2pix` instead
          of calling :py:meth:`all_world2pix` for each data point. Also
          see the note to the ``adaptive`` parameter.
  
          Raises
          ------
          NoConvergence
              The method did not converge to a
              solution to the required accuracy within a specified
              number of maximum iterations set by the ``maxiter``
              parameter. To turn off this exception, set ``quiet`` to
              `True`. Indices of the points for which the requested
              accuracy was not achieved (if any) will be listed in the
              ``slow_conv`` attribute of the
              raised :py:class:`NoConvergence` exception object.
  
              See :py:class:`NoConvergence` documentation for
              more details.
  
          MemoryError
              Memory allocation failed.
  
          SingularMatrixError
              Linear transformation matrix is singular.
  
          InconsistentAxisTypesError
              Inconsistent or unrecognized coordinate axis types.
  
          ValueError
              Invalid parameter value.
  
          ValueError
              Invalid coordinate transformation parameters.
  
          ValueError
              x- and y-coordinate arrays are not the same size.
  
          InvalidTransformError
              Invalid coordinate transformation parameters.
  
          InvalidTransformError
              Ill-conditioned coordinate transformation parameters.
  
          Examples
          --------
          >>> import astropy.io.fits as fits
          >>> import astropy.wcs as wcs
          >>> import numpy as np
          >>> import os
  
          >>> filename = os.path.join(wcs.__path__[0], 'tests/data/j94f05bgq_flt.fits')
          >>> hdulist = fits.open(filename)
          >>> w = wcs.WCS(hdulist[('sci',1)].header, hdulist)
          >>> hdulist.close()
  
          >>> ra, dec = w.all_pix2world([1,2,3], [1,1,1], 1)
          >>> print(ra)  # doctest: +FLOAT_CMP
          [ 5.52645627  5.52649663  5.52653698]
          >>> print(dec)  # doctest: +FLOAT_CMP
          [-72.05171757 -72.05171276 -72.05170795]
          >>> radec = w.all_pix2world([[1,1], [2,1], [3,1]], 1)
          >>> print(radec)  # doctest: +FLOAT_CMP
          [[  5.52645627 -72.05171757]
           [  5.52649663 -72.05171276]
           [  5.52653698 -72.05170795]]
          >>> x, y = w.all_world2pix(ra, dec, 1)
          >>> print(x)  # doctest: +FLOAT_CMP
          [ 1.00000238  2.00000237  3.00000236]
          >>> print(y)  # doctest: +FLOAT_CMP
          [ 0.99999996  0.99999997  0.99999997]
          >>> xy = w.all_world2pix(radec, 1)
          >>> print(xy)  # doctest: +FLOAT_CMP
          [[ 1.00000238  0.99999996]
           [ 2.00000237  0.99999997]
           [ 3.00000236  0.99999997]]
          >>> xy = w.all_world2pix(radec, 1, maxiter=3,
          ...                      tolerance=1.0e-10, quiet=False)
          Traceback (most recent call last):
          ...
          NoConvergence: 'WCS.all_world2pix' failed to converge to the
          requested accuracy. After 3 iterations, the solution is
          diverging at least for one input point.
  
          >>> # Now try to use some diverging data:
          >>> divradec = w.all_pix2world([[1.0, 1.0],
          ...                             [10000.0, 50000.0],
          ...                             [3.0, 1.0]], 1)
          >>> print(divradec)  # doctest: +FLOAT_CMP
          [[  5.52645627 -72.05171757]
           [  7.15976932 -70.8140779 ]
           [  5.52653698 -72.05170795]]
  
          >>> # First, turn detect_divergence on:
          >>> try:  # doctest: +FLOAT_CMP
          ...   xy = w.all_world2pix(divradec, 1, maxiter=20,
          ...                        tolerance=1.0e-4, adaptive=False,
          ...                        detect_divergence=True,
          ...                        quiet=False)
          ... except wcs.wcs.NoConvergence as e:
          ...   print("Indices of diverging points: {{0}}"
          ...         .format(e.divergent))
          ...   print("Indices of poorly converging points: {{0}}"
          ...         .format(e.slow_conv))
          ...   print("Best solution:\\n{{0}}".format(e.best_solution))
          ...   print("Achieved accuracy:\\n{{0}}".format(e.accuracy))
          Indices of diverging points: [1]
          Indices of poorly converging points: None
          Best solution:
          [[  1.00000238e+00   9.99999965e-01]
           [ -1.99441636e+06   1.44309097e+06]
           [  3.00000236e+00   9.99999966e-01]]
          Achieved accuracy:
          [[  6.13968380e-05   8.59638593e-07]
           [  8.59526812e+11   6.61713548e+11]
           [  6.09398446e-05   8.38759724e-07]]
          >>> raise e
          Traceback (most recent call last):
          ...
          NoConvergence: 'WCS.all_world2pix' failed to converge to the
          requested accuracy.  After 5 iterations, the solution is
          diverging at least for one input point.
  
          >>> # This time turn detect_divergence off:
          >>> try:  # doctest: +FLOAT_CMP
          ...   xy = w.all_world2pix(divradec, 1, maxiter=20,
          ...                        tolerance=1.0e-4, adaptive=False,
          ...                        detect_divergence=False,
          ...                        quiet=False)
          ... except wcs.wcs.NoConvergence as e:
          ...   print("Indices of diverging points: {{0}}"
          ...         .format(e.divergent))
          ...   print("Indices of poorly converging points: {{0}}"
          ...         .format(e.slow_conv))
          ...   print("Best solution:\\n{{0}}".format(e.best_solution))
          ...   print("Achieved accuracy:\\n{{0}}".format(e.accuracy))
          Indices of diverging points: [1]
          Indices of poorly converging points: None
          Best solution:
          [[ 1.00000009  1.        ]
           [        nan         nan]
           [ 3.00000009  1.        ]]
          Achieved accuracy:
          [[  2.29417358e-06   3.21222995e-08]
           [             nan              nan]
           [  2.27407877e-06   3.13005639e-08]]
          >>> raise e
          Traceback (most recent call last):
          ...
          NoConvergence: 'WCS.all_world2pix' failed to converge to the
          requested accuracy.  After 6 iterations, the solution is
          diverging at least for one input point.
  
          """.format(docstrings.TWO_OR_MORE_ARGS('naxis', 8),
                     docstrings.RA_DEC_ORDER(8),
                     docstrings.RETURNS('pixel coordinates', 8))
  
      def wcs_world2pix(self, *args, **kwargs):
          if self.wcs is None:
              raise ValueError("No basic WCS settings were created.")
          return self._array_converter(
              lambda xy, o: self.wcs.s2p(xy, o)['pixcrd'],
              'input', *args, **kwargs)
      wcs_world2pix.__doc__ = """
          Transforms world coordinates to pixel coordinates, using only
          the basic `wcslib`_ WCS transformation.  No `SIP`_ or
          `distortion paper`_ table lookup transformation is applied.
  
          Parameters
          ----------
          {}
  
              For a transformation that is not two-dimensional, the
              two-argument form must be used.
  
          {}
  
          Returns
          -------
  
          {}
  
          Notes
          -----
          The order of the axes for the input world array is determined by
          the ``CTYPEia`` keywords in the FITS header, therefore it may
          not always be of the form (*ra*, *dec*).  The
          `~astropy.wcs.Wcsprm.lat`, `~astropy.wcs.Wcsprm.lng`,
          `~astropy.wcs.Wcsprm.lattyp` and `~astropy.wcs.Wcsprm.lngtyp`
          members can be used to determine the order of the axes.
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          SingularMatrixError
              Linear transformation matrix is singular.
  
          InconsistentAxisTypesError
              Inconsistent or unrecognized coordinate axis types.
  
          ValueError
              Invalid parameter value.
  
          ValueError
              Invalid coordinate transformation parameters.
  
          ValueError
              x- and y-coordinate arrays are not the same size.
  
          InvalidTransformError
              Invalid coordinate transformation parameters.
  
          InvalidTransformError
              Ill-conditioned coordinate transformation parameters.
          """.format(docstrings.TWO_OR_MORE_ARGS('naxis', 8),
                     docstrings.RA_DEC_ORDER(8),
                     docstrings.RETURNS('pixel coordinates', 8))
  
      def pix2foc(self, *args):
          return self._array_converter(self._pix2foc, None, *args)
      pix2foc.__doc__ = """
          Convert pixel coordinates to focal plane coordinates using the
          `SIP`_ polynomial distortion convention and `distortion
          paper`_ table-lookup correction.
  
          The output is in absolute pixel coordinates, not relative to
          ``CRPIX``.
  
          Parameters
          ----------
  
          {}
  
          Returns
          -------
  
          {}
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          ValueError
              Invalid coordinate transformation parameters.
          """.format(docstrings.TWO_OR_MORE_ARGS('2', 8),
                     docstrings.RETURNS('focal coordinates', 8))
  
      def p4_pix2foc(self, *args):
          return self._array_converter(self._p4_pix2foc, None, *args)
      p4_pix2foc.__doc__ = """
          Convert pixel coordinates to focal plane coordinates using
          `distortion paper`_ table-lookup correction.
  
          The output is in absolute pixel coordinates, not relative to
          ``CRPIX``.
  
          Parameters
          ----------
  
          {}
  
          Returns
          -------
  
          {}
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          ValueError
              Invalid coordinate transformation parameters.
          """.format(docstrings.TWO_OR_MORE_ARGS('2', 8),
                     docstrings.RETURNS('focal coordinates', 8))
  
      def det2im(self, *args):
          return self._array_converter(self._det2im, None, *args)
      det2im.__doc__ = """
          Convert detector coordinates to image plane coordinates using
          `distortion paper`_ table-lookup correction.
  
          The output is in absolute pixel coordinates, not relative to
          ``CRPIX``.
  
          Parameters
          ----------
  
          {}
  
          Returns
          -------
  
          {}
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          ValueError
              Invalid coordinate transformation parameters.
          """.format(docstrings.TWO_OR_MORE_ARGS('2', 8),
                     docstrings.RETURNS('pixel coordinates', 8))
  
      def sip_pix2foc(self, *args):
          if self.sip is None:
              if len(args) == 2:
                  return args[0]
              elif len(args) == 3:
                  return args[:2]
              else:
                  raise TypeError("Wrong number of arguments")
          return self._array_converter(self.sip.pix2foc, None, *args)
      sip_pix2foc.__doc__ = """
          Convert pixel coordinates to focal plane coordinates using the
          `SIP`_ polynomial distortion convention.
  
          The output is in pixel coordinates, relative to ``CRPIX``.
  
          FITS WCS `distortion paper`_ table lookup correction is not
          applied, even if that information existed in the FITS file
          that initialized this :class:`~astropy.wcs.WCS` object.  To
          correct for that, use `~astropy.wcs.WCS.pix2foc` or
          `~astropy.wcs.WCS.p4_pix2foc`.
  
          Parameters
          ----------
  
          {}
  
          Returns
          -------
  
          {}
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          ValueError
              Invalid coordinate transformation parameters.
          """.format(docstrings.TWO_OR_MORE_ARGS('2', 8),
                     docstrings.RETURNS('focal coordinates', 8))
  
      def sip_foc2pix(self, *args):
          if self.sip is None:
              if len(args) == 2:
                  return args[0]
              elif len(args) == 3:
                  return args[:2]
              else:
                  raise TypeError("Wrong number of arguments")
          return self._array_converter(self.sip.foc2pix, None, *args)
      sip_foc2pix.__doc__ = """
          Convert focal plane coordinates to pixel coordinates using the
          `SIP`_ polynomial distortion convention.
  
          FITS WCS `distortion paper`_ table lookup distortion
          correction is not applied, even if that information existed in
          the FITS file that initialized this `~astropy.wcs.WCS` object.
  
          Parameters
          ----------
  
          {}
  
          Returns
          -------
  
          {}
  
          Raises
          ------
          MemoryError
              Memory allocation failed.
  
          ValueError
              Invalid coordinate transformation parameters.
          """.format(docstrings.TWO_OR_MORE_ARGS('2', 8),
                     docstrings.RETURNS('pixel coordinates', 8))
  
      def to_fits(self, relax=False, key=None):
          """
          Generate an `astropy.io.fits.HDUList` object with all of the
          information stored in this object.  This should be logically identical
          to the input FITS file, but it will be normalized in a number of ways.
  
          See `to_header` for some warnings about the output produced.
  
          Parameters
          ----------
  
          relax : bool or int, optional
              Degree of permissiveness:
  
              - `False` (default): Write all extensions that are
                considered to be safe and recommended.
  
              - `True`: Write all recognized informal extensions of the
                WCS standard.
  
              - `int`: a bit field selecting specific extensions to
                write.  See :ref:`relaxwrite` for details.
  
          key : str
              The name of a particular WCS transform to use.  This may be
              either ``' '`` or ``'A'``-``'Z'`` and corresponds to the ``"a"``
              part of the ``CTYPEia`` cards.
  
          Returns
          -------
          hdulist : `astropy.io.fits.HDUList`
          """
  
          header = self.to_header(relax=relax, key=key)
  
          hdu = fits.PrimaryHDU(header=header)
          hdulist = fits.HDUList(hdu)
  
          self._write_det2im(hdulist)
          self._write_distortion_kw(hdulist)
  
          return hdulist
  
      def to_header(self, relax=None, key=None):
          """Generate an `astropy.io.fits.Header` object with the basic WCS
          and SIP information stored in this object.  This should be
          logically identical to the input FITS file, but it will be
          normalized in a number of ways.
  
          .. warning::
  
            This function does not write out FITS WCS `distortion
            paper`_ information, since that requires multiple FITS
            header data units.  To get a full representation of
            everything in this object, use `to_fits`.
  
          Parameters
          ----------
          relax : bool or int, optional
              Degree of permissiveness:
  
              - `False` (default): Write all extensions that are
                considered to be safe and recommended.
  
              - `True`: Write all recognized informal extensions of the
                WCS standard.
  
              - `int`: a bit field selecting specific extensions to
                write.  See :ref:`relaxwrite` for details.
  
              If the ``relax`` keyword argument is not given and any
              keywords were omitted from the output, an
              `~astropy.utils.exceptions.AstropyWarning` is displayed.
              To override this, explicitly pass a value to ``relax``.
  
          key : str
              The name of a particular WCS transform to use.  This may be
              either ``' '`` or ``'A'``-``'Z'`` and corresponds to the ``"a"``
              part of the ``CTYPEia`` cards.
  
          Returns
          -------
          header : `astropy.io.fits.Header`
  
          Notes
          -----
          The output header will almost certainly differ from the input in a
          number of respects:
  
            1. The output header only contains WCS-related keywords.  In
               particular, it does not contain syntactically-required
               keywords such as ``SIMPLE``, ``NAXIS``, ``BITPIX``, or
               ``END``.
  
            2. Deprecated (e.g. ``CROTAn``) or non-standard usage will
               be translated to standard (this is partially dependent on
               whether ``fix`` was applied).
  
            3. Quantities will be converted to the units used internally,
               basically SI with the addition of degrees.
  
            4. Floating-point quantities may be given to a different decimal
               precision.
  
            5. Elements of the ``PCi_j`` matrix will be written if and
               only if they differ from the unit matrix.  Thus, if the
               matrix is unity then no elements will be written.
  
            6. Additional keywords such as ``WCSAXES``, ``CUNITia``,
               ``LONPOLEa`` and ``LATPOLEa`` may appear.
  
            7. The original keycomments will be lost, although
               `to_header` tries hard to write meaningful comments.
  
            8. Keyword order may be changed.
  
          """
          # default precision for numerical WCS keywords
          precision = WCSHDO_P14
          display_warning = False
          if relax is None:
              display_warning = True
              relax = False
  
          if relax not in (True, False):
              do_sip = relax & WCSHDO_SIP
              relax &= ~WCSHDO_SIP
          else:
              do_sip = relax
              relax = WCSHDO_all if relax is True else WCSHDO_safe
  
          relax = precision | relax
  
          if self.wcs is not None:
              if key is not None:
                  orig_key = self.wcs.alt
                  self.wcs.alt = key
              header_string = self.wcs.to_header(relax)
              header = fits.Header.fromstring(header_string)
              keys_to_remove = ["", " ", "COMMENT"]
              for kw in keys_to_remove:
                  if kw in header:
                      del header[kw]
          else:
              header = fits.Header()
  
          if do_sip and self.sip is not None:
              if self.wcs is not None and any(not ctyp.endswith('-SIP') for ctyp in self.wcs.ctype):
                  self._fix_ctype(header, add_sip=True)
  
              for kw, val in self._write_sip_kw().items():
                  header[kw] = val
  
          if not do_sip and self.wcs is not None and any(self.wcs.ctype) and self.sip is not None:
              # This is called when relax is not False or WCSHDO_SIP
              # The default case of ``relax=None`` is handled further in the code.
              header = self._fix_ctype(header, add_sip=False)
  
          if display_warning:
              full_header = self.to_header(relax=True, key=key)
              missing_keys = []
              for kw, val in full_header.items():
                  if kw not in header:
                      missing_keys.append(kw)
  
              if len(missing_keys):
                  warnings.warn(
                      "Some non-standard WCS keywords were excluded: {} "
                      "Use the ``relax`` kwarg to control this.".format(
                          ', '.join(missing_keys)),
                      AstropyWarning)
              # called when ``relax=None``
              # This is different from the case of ``relax=False``.
              if any(self.wcs.ctype) and self.sip is not None:
                  header = self._fix_ctype(header, add_sip=False, log_message=False)
          # Finally reset the key. This must be called after ``_fix_ctype``.
          if key is not None:
              self.wcs.alt = orig_key
          return header
  
      def _fix_ctype(self, header, add_sip=True, log_message=True):
          """
          Parameters
          ----------
          header : `~astropy.io.fits.Header`
              FITS header.
          add_sip : bool
              Flag indicating whether "-SIP" should be added or removed from CTYPE keywords.
  
              Remove "-SIP" from CTYPE when writing out a header with relax=False.
              This needs to be done outside ``to_header`` because ``to_header`` runs
              twice when ``relax=False`` and the second time ``relax`` is set to ``True``
              to display the missing keywords.
  
              If the user requested SIP distortion to be written out add "-SIP" to
              CTYPE if it is missing.
          """
  
          _add_sip_to_ctype = """
          Inconsistent SIP distortion information is present in the current WCS:
          SIP coefficients were detected, but CTYPE is missing "-SIP" suffix,
          therefore the current WCS is internally inconsistent.
  
          Because relax has been set to True, the resulting output WCS will have
          "-SIP" appended to CTYPE in order to make the header internally consistent.
  
          However, this may produce incorrect astrometry in the output WCS, if
          in fact the current WCS is already distortion-corrected.
  
          Therefore, if current WCS is already distortion-corrected (eg, drizzled)
          then SIP distortion components should not apply. In that case, for a WCS
          that is already distortion-corrected, please remove the SIP coefficients
          from the header.
  
          """
          if log_message:
              if add_sip:
                  log.info(_add_sip_to_ctype)
          for i in range(1, self.naxis+1):
              # strip() must be called here to cover the case of alt key= " "
              kw = f'CTYPE{i}{self.wcs.alt}'.strip()
              if kw in header:
                  if add_sip:
                      val = header[kw].strip("-SIP") + "-SIP"
                  else:
                      val = header[kw].strip("-SIP")
                  header[kw] = val
              else:
                  continue
          return header
  
      def to_header_string(self, relax=None):
          """
          Identical to `to_header`, but returns a string containing the
          header cards.
          """
          return str(self.to_header(relax))
  
      def footprint_to_file(self, filename='footprint.reg', color='green',
                            width=2, coordsys=None):
          """
          Writes out a `ds9`_ style regions file. It can be loaded
          directly by `ds9`_.
  
          Parameters
          ----------
          filename : str, optional
              Output file name - default is ``'footprint.reg'``
  
          color : str, optional
              Color to use when plotting the line.
  
          width : int, optional
              Width of the region line.
  
          coordsys : str, optional
              Coordinate system. If not specified (default), the ``radesys``
              value is used. For all possible values, see
              http://ds9.si.edu/doc/ref/region.html#RegionFileFormat
  
          """
          comments = ('# Region file format: DS9 version 4.0 \n'
                      '# global color=green font="helvetica 12 bold '
                      'select=1 highlite=1 edit=1 move=1 delete=1 '
                      'include=1 fixed=0 source\n')
  
          coordsys = coordsys or self.wcs.radesys
  
          if coordsys not in ('PHYSICAL', 'IMAGE', 'FK4', 'B1950', 'FK5',
                              'J2000', 'GALACTIC', 'ECLIPTIC', 'ICRS', 'LINEAR',
                              'AMPLIFIER', 'DETECTOR'):
              raise ValueError("Coordinate system '{}' is not supported. A valid"
                               " one can be given with the 'coordsys' argument."
                               .format(coordsys))
  
          with open(filename, mode='w') as f:
              f.write(comments)
              f.write(f'{coordsys}\n')
              f.write('polygon(')
              ftpr = self.calc_footprint()
              if ftpr is not None:
                  ftpr.tofile(f, sep=',')
                  f.write(f') # color={color}, width={width:d} \n')
  
      def _get_naxis(self, header=None):
          _naxis = []
          if (header is not None and
                  not isinstance(header, (str, bytes))):
              for naxis in itertools.count(1):
                  try:
                      _naxis.append(header[f'NAXIS{naxis}'])
                  except KeyError:
                      break
          if len(_naxis) == 0:
              _naxis = [0, 0]
          elif len(_naxis) == 1:
              _naxis.append(0)
          self._naxis = _naxis
  
      def printwcs(self):
          print(repr(self))
  
      def __repr__(self):
          '''
          Return a short description. Simply porting the behavior from
          the `printwcs()` method.
          '''
          description = ["WCS Keywords\n",
                         f"Number of WCS axes: {self.naxis!r}"]
          sfmt = ' : ' + "".join(["{"+f"{i}"+"!r}  " for i in range(self.naxis)])
  
          keywords = ['CTYPE', 'CRVAL', 'CRPIX']
          values = [self.wcs.ctype, self.wcs.crval, self.wcs.crpix]
          for keyword, value in zip(keywords, values):
              description.append(keyword+sfmt.format(*value))
  
          if hasattr(self.wcs, 'pc'):
              for i in range(self.naxis):
                  s = ''
                  for j in range(self.naxis):
                      s += ''.join(['PC', str(i+1), '_', str(j+1), ' '])
                  s += sfmt
                  description.append(s.format(*self.wcs.pc[i]))
              s = 'CDELT' + sfmt
              description.append(s.format(*self.wcs.cdelt))
          elif hasattr(self.wcs, 'cd'):
              for i in range(self.naxis):
                  s = ''
                  for j in range(self.naxis):
                      s += "".join(['CD', str(i+1), '_', str(j+1), ' '])
                  s += sfmt
                  description.append(s.format(*self.wcs.cd[i]))
  
          description.append('NAXIS : {}'.format('  '.join(map(str, self._naxis))))
          return '\n'.join(description)
  
      def get_axis_types(self):
          """
          Similar to `self.wcsprm.axis_types <astropy.wcs.Wcsprm.axis_types>`
          but provides the information in a more Python-friendly format.
  
          Returns
          -------
          result : list of dicts
  
              Returns a list of dictionaries, one for each axis, each
              containing attributes about the type of that axis.
  
              Each dictionary has the following keys:
  
              - 'coordinate_type':
  
                - None: Non-specific coordinate type.
  
                - 'stokes': Stokes coordinate.
  
                - 'celestial': Celestial coordinate (including ``CUBEFACE``).
  
                - 'spectral': Spectral coordinate.
  
              - 'scale':
  
                - 'linear': Linear axis.
  
                - 'quantized': Quantized axis (``STOKES``, ``CUBEFACE``).
  
                - 'non-linear celestial': Non-linear celestial axis.
  
                - 'non-linear spectral': Non-linear spectral axis.
  
                - 'logarithmic': Logarithmic axis.
  
                - 'tabular': Tabular axis.
  
              - 'group'
  
                - Group number, e.g. lookup table number
  
              - 'number'
  
                - For celestial axes:
  
                  - 0: Longitude coordinate.
  
                  - 1: Latitude coordinate.
  
                  - 2: ``CUBEFACE`` number.
  
                - For lookup tables:
  
                  - the axis number in a multidimensional table.
  
              ``CTYPEia`` in ``"4-3"`` form with unrecognized algorithm code will
              generate an error.
          """
          if self.wcs is None:
              raise AttributeError(
                  "This WCS object does not have a wcsprm object.")
  
          coordinate_type_map = {
              0: None,
              1: 'stokes',
              2: 'celestial',
              3: 'spectral'}
  
          scale_map = {
              0: 'linear',
              1: 'quantized',
              2: 'non-linear celestial',
              3: 'non-linear spectral',
              4: 'logarithmic',
              5: 'tabular'}
  
          result = []
          for axis_type in self.wcs.axis_types:
              subresult = {}
  
              coordinate_type = (axis_type // 1000) % 10
              subresult['coordinate_type'] = coordinate_type_map[coordinate_type]
  
              scale = (axis_type // 100) % 10
              subresult['scale'] = scale_map[scale]
  
              group = (axis_type // 10) % 10
              subresult['group'] = group
  
              number = axis_type % 10
              subresult['number'] = number
  
              result.append(subresult)
  
          return result
  
      def __reduce__(self):
          """
          Support pickling of WCS objects.  This is done by serializing
          to an in-memory FITS file and dumping that as a string.
          """
  
          hdulist = self.to_fits(relax=True)
  
          buffer = io.BytesIO()
          hdulist.writeto(buffer)
  
          return (__WCS_unpickle__,
                  (self.__class__, self.__dict__, buffer.getvalue(),))
  
      def dropaxis(self, dropax):
          """
          Remove an axis from the WCS.
  
          Parameters
          ----------
          wcs : `~astropy.wcs.WCS`
              The WCS with naxis to be chopped to naxis-1
          dropax : int
              The index of the WCS to drop, counting from 0 (i.e., python convention,
              not FITS convention)
  
          Returns
          -------
          A new `~astropy.wcs.WCS` instance with one axis fewer
          """
          inds = list(range(self.wcs.naxis))
          inds.pop(dropax)
  
          # axis 0 has special meaning to sub
          # if wcs.wcs.ctype == ['RA','DEC','VLSR'], you want
          # wcs.sub([1,2]) to get 'RA','DEC' back
          return self.sub([i+1 for i in inds])
  
      def swapaxes(self, ax0, ax1):
          """
          Swap axes in a WCS.
  
          Parameters
          ----------
          wcs : `~astropy.wcs.WCS`
              The WCS to have its axes swapped
          ax0 : int
          ax1 : int
              The indices of the WCS to be swapped, counting from 0 (i.e., python
              convention, not FITS convention)
  
          Returns
          -------
          A new `~astropy.wcs.WCS` instance with the same number of axes, but two
          swapped
          """
          inds = list(range(self.wcs.naxis))
          inds[ax0], inds[ax1] = inds[ax1], inds[ax0]
  
          return self.sub([i+1 for i in inds])
  
      def reorient_celestial_first(self):
          """
          Reorient the WCS such that the celestial axes are first, followed by
          the spectral axis, followed by any others.
          Assumes at least celestial axes are present.
          """
          return self.sub([WCSSUB_CELESTIAL, WCSSUB_SPECTRAL, WCSSUB_STOKES])
  
      def slice(self, view, numpy_order=True):
          """
          Slice a WCS instance using a Numpy slice. The order of the slice should
          be reversed (as for the data) compared to the natural WCS order.
  
          Parameters
          ----------
          view : tuple
              A tuple containing the same number of slices as the WCS system.
              The ``step`` method, the third argument to a slice, is not
              presently supported.
          numpy_order : bool
              Use numpy order, i.e. slice the WCS so that an identical slice
              applied to a numpy array will slice the array and WCS in the same
              way. If set to `False`, the WCS will be sliced in FITS order,
              meaning the first slice will be applied to the *last* numpy index
              but the *first* WCS axis.
  
          Returns
          -------
          wcs_new : `~astropy.wcs.WCS`
              A new resampled WCS axis
          """
          if hasattr(view, '__len__') and len(view) > self.wcs.naxis:
              raise ValueError("Must have # of slices <= # of WCS axes")
          elif not hasattr(view, '__len__'):  # view MUST be an iterable
              view = [view]
  
          if not all(isinstance(x, slice) for x in view):
              # We need to drop some dimensions, but this may not always be
              # possible with .sub due to correlated axes, so instead we use the
              # generalized slicing infrastructure from astropy.wcs.wcsapi.
              return SlicedFITSWCS(self, view)
  
          # NOTE: we could in principle use SlicedFITSWCS as above for all slicing,
          # but in the simple case where there are no axes dropped, we can just
          # create a full WCS object with updated WCS parameters which is faster
          # for this specific case and also backward-compatible.
  
          wcs_new = self.deepcopy()
          if wcs_new.sip is not None:
              sip_crpix = wcs_new.sip.crpix.tolist()
  
          for i, iview in enumerate(view):
              if iview.step is not None and iview.step < 0:
                  raise NotImplementedError("Reversing an axis is not "
                                            "implemented.")
  
              if numpy_order:
                  wcs_index = self.wcs.naxis - 1 - i
              else:
                  wcs_index = i
  
              if iview.step is not None and iview.start is None:
                  # Slice from "None" is equivalent to slice from 0 (but one
                  # might want to downsample, so allow slices with
                  # None,None,step or None,stop,step)
                  iview = slice(0, iview.stop, iview.step)
  
              if iview.start is not None:
                  if iview.step not in (None, 1):
                      crpix = self.wcs.crpix[wcs_index]
                      cdelt = self.wcs.cdelt[wcs_index]
                      # equivalently (keep this comment so you can compare eqns):
                      # wcs_new.wcs.crpix[wcs_index] =
                      # (crpix - iview.start)*iview.step + 0.5 - iview.step/2.
                      crp = ((crpix - iview.start - 1.)/iview.step
                             + 0.5 + 1./iview.step/2.)
                      wcs_new.wcs.crpix[wcs_index] = crp
                      if wcs_new.sip is not None:
                          sip_crpix[wcs_index] = crp
                      wcs_new.wcs.cdelt[wcs_index] = cdelt * iview.step
                  else:
                      wcs_new.wcs.crpix[wcs_index] -= iview.start
                      if wcs_new.sip is not None:
                          sip_crpix[wcs_index] -= iview.start
  
              try:
                  # range requires integers but the other attributes can also
                  # handle arbitrary values, so this needs to be in a try/except.
                  nitems = len(builtins.range(self._naxis[wcs_index])[iview])
              except TypeError as exc:
                  if 'indices must be integers' not in str(exc):
                      raise
                  warnings.warn("NAXIS{} attribute is not updated because at "
                                "least one index ('{}') is no integer."
                                "".format(wcs_index, iview), AstropyUserWarning)
              else:
                  wcs_new._naxis[wcs_index] = nitems
  
          if wcs_new.sip is not None:
              wcs_new.sip = Sip(self.sip.a, self.sip.b, self.sip.ap, self.sip.bp,
                                sip_crpix)
  
          return wcs_new
  
      def __getitem__(self, item):
          # "getitem" is a shortcut for self.slice; it is very limited
          # there is no obvious and unambiguous interpretation of wcs[1,2,3]
          # We COULD allow wcs[1] to link to wcs.sub([2])
          # (wcs[i] -> wcs.sub([i+1])
          return self.slice(item)
  
      def __iter__(self):
          # Having __getitem__ makes Python think WCS is iterable. However,
          # Python first checks whether __iter__ is present, so we can raise an
          # exception here.
          raise TypeError(f"'{self.__class__.__name__}' object is not iterable")
  
      @property
      def axis_type_names(self):
          """
          World names for each coordinate axis
  
          Returns
          -------
          A list of names along each axis
          """
          names = list(self.wcs.cname)
          types = self.wcs.ctype
          for i in range(len(names)):
              if len(names[i]) > 0:
                  continue
              names[i] = types[i].split('-')[0]
          return names
  
      @property
      def celestial(self):
          """
          A copy of the current WCS with only the celestial axes included
          """
          return self.sub([WCSSUB_CELESTIAL])
  
      @property
      def is_celestial(self):
          return self.has_celestial and self.naxis == 2
  
      @property
      def has_celestial(self):
          try:
              return self.wcs.lng >= 0 and self.wcs.lat >= 0
          except InconsistentAxisTypesError:
              return False
  
      @property
      def has_distortion(self):
          """
          Returns `True` if any distortion terms are present.
          """
          return (self.sip is not None or
                  self.cpdis1 is not None or self.cpdis2 is not None or
                  self.det2im1 is not None and self.det2im2 is not None)
  
      @property
      def pixel_scale_matrix(self):
  
          try:
              cdelt = np.diag(self.wcs.get_cdelt())
              pc = self.wcs.get_pc()
          except InconsistentAxisTypesError:
              try:
                  # for non-celestial axes, get_cdelt doesn't work
                  with warnings.catch_warnings():
                      warnings.filterwarnings('ignore', 'cdelt will be ignored since cd is present', RuntimeWarning)
                      cdelt = np.dot(self.wcs.cd, np.diag(self.wcs.cdelt))
              except AttributeError:
                  cdelt = np.diag(self.wcs.cdelt)
  
              try:
                  pc = self.wcs.pc
              except AttributeError:
                  pc = 1
  
          pccd = np.array(np.dot(cdelt, pc))
  
          return pccd
  
      def footprint_contains(self, coord, **kwargs):
          """
          Determines if a given SkyCoord is contained in the wcs footprint.
  
          Parameters
          ----------
          coord : `~astropy.coordinates.SkyCoord`
              The coordinate to check if it is within the wcs coordinate.
          **kwargs :
             Additional arguments to pass to `~astropy.coordinates.SkyCoord.to_pixel`
  
          Returns
          -------
          response : bool
             True means the WCS footprint contains the coordinate, False means it does not.
          """
  
          return coord.contained_by(self, **kwargs)
  
  
  def __WCS_unpickle__(cls, dct, fits_data):
      """
      Unpickles a WCS object from a serialized FITS string.
      """
  
      self = cls.__new__(cls)
      self.__dict__.update(dct)
  
      buffer = io.BytesIO(fits_data)
      hdulist = fits.open(buffer)
  
      WCS.__init__(self, hdulist[0].header, hdulist)
  
      return self
  
  
  def find_all_wcs(header, relax=True, keysel=None, fix=True,
                   translate_units='',
                   _do_set=True):
      """
      Find all the WCS transformations in the given header.
  
      Parameters
      ----------
      header : str or astropy.io.fits header object.
  
      relax : bool or int, optional
          Degree of permissiveness:
  
          - `True` (default): Admit all recognized informal extensions of the
            WCS standard.
  
          - `False`: Recognize only FITS keywords defined by the
            published WCS standard.
  
          - `int`: a bit field selecting specific extensions to accept.
            See :ref:`relaxread` for details.
  
      keysel : sequence of flags, optional
          A list of flags used to select the keyword types considered by
          wcslib.  When ``None``, only the standard image header
          keywords are considered (and the underlying wcspih() C
          function is called).  To use binary table image array or pixel
          list keywords, *keysel* must be set.
  
          Each element in the list should be one of the following strings:
  
              - 'image': Image header keywords
  
              - 'binary': Binary table image array keywords
  
              - 'pixel': Pixel list keywords
  
          Keywords such as ``EQUIna`` or ``RFRQna`` that are common to
          binary table image arrays and pixel lists (including
          ``WCSNna`` and ``TWCSna``) are selected by both 'binary' and
          'pixel'.
  
      fix : bool, optional
          When `True` (default), call `~astropy.wcs.Wcsprm.fix` on
          the resulting objects to fix any non-standard uses in the
          header.  `FITSFixedWarning` warnings will be emitted if any
          changes were made.
  
      translate_units : str, optional
          Specify which potentially unsafe translations of non-standard
          unit strings to perform.  By default, performs none.  See
          `WCS.fix` for more information about this parameter.  Only
          effective when ``fix`` is `True`.
  
      Returns
      -------
      wcses : list of `WCS` objects
      """
  
      if isinstance(header, (str, bytes)):
          header_string = header
      elif isinstance(header, fits.Header):
          header_string = header.tostring()
      else:
          raise TypeError(
              "header must be a string or astropy.io.fits.Header object")
  
      keysel_flags = _parse_keysel(keysel)
  
      if isinstance(header_string, str):
          header_bytes = header_string.encode('ascii')
      else:
          header_bytes = header_string
  
      wcsprms = _wcs.find_all_wcs(header_bytes, relax, keysel_flags)
  
      result = []
      for wcsprm in wcsprms:
          subresult = WCS(fix=False, _do_set=False)
          subresult.wcs = wcsprm
          result.append(subresult)
  
          if fix:
              subresult.fix(translate_units)
  
          if _do_set:
              subresult.wcs.set()
  
      return result
  
  
  def validate(source):
      """
      Prints a WCS validation report for the given FITS file.
  
      Parameters
      ----------
      source : str path, readable file-like object or `astropy.io.fits.HDUList` object
          The FITS file to validate.
  
      Returns
      -------
      results : WcsValidateResults instance
          The result is returned as nested lists.  The first level
          corresponds to the HDUs in the given file.  The next level has
          an entry for each WCS found in that header.  The special
          subclass of list will pretty-print the results as a table when
          printed.
      """
      class _WcsValidateWcsResult(list):
          def __init__(self, key):
              self._key = key
  
          def __repr__(self):
              result = ["  WCS key '{}':".format(self._key or ' ')]
              if len(self):
                  for entry in self:
                      for i, line in enumerate(entry.splitlines()):
                          if i == 0:
                              initial_indent = '    - '
                          else:
                              initial_indent = '      '
                          result.extend(
                              textwrap.wrap(
                                  line,
                                  initial_indent=initial_indent,
                                  subsequent_indent='      '))
              else:
                  result.append("    No issues.")
              return '\n'.join(result)
  
      class _WcsValidateHduResult(list):
          def __init__(self, hdu_index, hdu_name):
              self._hdu_index = hdu_index
              self._hdu_name = hdu_name
              list.__init__(self)
  
          def __repr__(self):
              if len(self):
                  if self._hdu_name:
                      hdu_name = f' ({self._hdu_name})'
                  else:
                      hdu_name = ''
                  result = [f'HDU {self._hdu_index}{hdu_name}:']
                  for wcs in self:
                      result.append(repr(wcs))
                  return '\n'.join(result)
              return ''
  
      class _WcsValidateResults(list):
          def __repr__(self):
              result = []
              for hdu in self:
                  content = repr(hdu)
                  if len(content):
                      result.append(content)
              return '\n\n'.join(result)
  
      global __warningregistry__
  
      if isinstance(source, fits.HDUList):
          hdulist = source
      else:
          hdulist = fits.open(source)
  
      results = _WcsValidateResults()
  
      for i, hdu in enumerate(hdulist):
          hdu_results = _WcsValidateHduResult(i, hdu.name)
          results.append(hdu_results)
  
          with warnings.catch_warnings(record=True) as warning_lines:
              wcses = find_all_wcs(
                  hdu.header, relax=_wcs.WCSHDR_reject,
                  fix=False, _do_set=False)
  
          for wcs in wcses:
              wcs_results = _WcsValidateWcsResult(wcs.wcs.alt)
              hdu_results.append(wcs_results)
  
              try:
                  del __warningregistry__
              except NameError:
                  pass
  
              with warnings.catch_warnings(record=True) as warning_lines:
                  warnings.resetwarnings()
                  warnings.simplefilter(
                      "always", FITSFixedWarning, append=True)
  
                  try:
                      WCS(hdu.header,
                          key=wcs.wcs.alt or ' ',
                          relax=_wcs.WCSHDR_reject,
                          fix=True, _do_set=False)
                  except WcsError as e:
                      wcs_results.append(str(e))
  
                  wcs_results.extend([str(x.message) for x in warning_lines])
  
      return results
  