IR-eߵ dZddlZddlmZmZddlZddlmZ ddlm Z ddl m Z ddl mZmZdd lmZmZgd Zd gZeeZd eDZgd eedjeddgeezZGddeZGdde ZGddeZGddeZGddeZ Gddee Z!Gddee Z"Gdd ee Z#Gd!d"ee Z$Gd#d$ee Z%Gd%d&ee Z&Gd'd(ee Z'Gd)d*ee Z(Gd+d,ee Z)Gd-d.ee Z*Gd/d0ee Z+Gd1d2ee Z,Gd3d4ee Z-Gd5d6ee Z.Gd7d8ee Z/Gd9d:ee Z0Gd;dee1Z2Gd?d@ee1Z3GdAdBee1Z4GdCdDee1Z5GdEdFee1Z6GdGdHee1Z7GdIdJee1Z8GdKdLee1Z9GdMdNeZ:GdOdPee:Z;GdQdRee:Z<GdSdTee:Z=GdUdVee:Z>GdWdXee:Z?GdYdZee:Z@Gd[d\ee:ZAGd]d^ee:ZBGd_d`eZCGdadbeeCZDGdcddeeCZEGdedfeeCZFGdgdheeCZGGdidjeeCZHGdkdleeCZIGdmdneeCZJGdodpeeCZKGdqdreZLGdsdteeLZMGdudveeLZNGdwdxeeLZOGdydzeeLZPGd{d|eZQGd}d~eeQZRGddeeQZSGddeeQZTGddeeQZUGddeeQZVGddeeQZWGddeZXGddeeXZYGddeeXZZGddeeXZ[GddeeXZ\Gdde Z]eD]\Z^Z_e`de^ze`de_z<e`de^ze`de_z<e`de^ze`de^z_ae`de^ze`de^z_adS)aK Implements projections--particularly sky projections defined in WCS Paper II [1]_. All angles are set and and displayed in degrees but internally computations are performed in radians. All functions expect inputs and outputs degrees. References ---------- .. [1] Calabretta, M.R., Greisen, E.W., 2002, A&A, 395, 1077 (Paper II) N)chainproduct)units)wcs)Model)InputParameterError Parameter) _to_orig_unit _to_radian))ZenithalPerspectiveAZP)SlantZenithalPerspectiveSZP)GnomonicTAN) StereographicSTG)SlantOrthographicSIN)ZenithalEquidistantARC)ZenithalEqualAreaZEA)AiryAIR)CylindricalPerspectiveCYP)CylindricalEqualAreaCEA) PlateCarreeCAR)MercatorMER)SansonFlamsteedSFL) ParabolicPAR) MolleweideMOL) HammerAitoffAIT)ConicPerspectiveCOP)ConicEqualAreaCOE)ConicEquidistantCOD)ConicOrthomorphicCOO)BonneEqualAreaBON) PolyconicPCO)TangentialSphericalCubeTSC)COBEQuadSphericalCubeCSC)QuadSphericalCubeQSC)HEALPixHPX) HEALPixPolarXPHZPNcg|]\}}|SrE).0_codes rJ@s 1 1 1gaT 1 1 1) ProjectionPix2SkyProjectionSky2PixProjectionZenithal CylindricalPseudoCylindricalConic PseudoConicQuadCuber?AffineTransformation2D projcodesrGPix2SkySky2Pixc,eZdZdZfdZfdZxZS) _ParameterDSai Same as `Parameter` but can indicate its modified status via the ``dirty`` property. This flag also gets set automatically when a parameter is modified. This ability to track parameter's modified status is needed for automatic update of WCSLIB's prjprm structure (which may be a more-time intensive operation) *only as required*. cHtj|i|d|_dSNT)super__init__dirtyselfargskwargs __class__s rIr^z_ParameterDS.__init___s*$)&))) rKcXt|d|_dSr\)r]validater_)ravaluerds rIrfz_ParameterDS.validatecs&  rK)__name__ __module__ __qualname____doc__r^rf __classcell__rds@rIrZrZSs[  rKrZceZdZdZdejzejz ZdZ fdZ e e j dZe dZdZdZd ZxZS) rLz#Base class for all sky projections.Fcjtj|i|tj|_dSN)r]r^rPrjprm_prjr`s rIr^zProjection.__init__qs0$)&)))JLL rKcdS)zT Inverse projection--all projection models must provide an inverse. NrEras rIinversezProjection.inverseusrKc8||jS)zWCSLIB ``prjprm`` structure.) _update_prjrsrus rIprjprmzProjection.prjprm|s yrKc|jsdSg}d}|jD]J}t||}|t|j||jz}d|_K|r+dg|R|j_|jdSdS)a A default updater for projection's pv. .. warning:: This method assumes that PV0 is never modified. If a projection that uses PV0 is ever implemented in this module, that projection class should override this method. .. warning:: This method assumes that the order in which PVi values (i>0) are to be assigned is identical to the order of model parameters in ``param_names``. That is, pv[1] = model.parameters[0], ... NF) param_namesgetattrappendfloatrgr_rspvset)rarr_pparams rIrxzProjection._update_prjs  F !  AD!$$E IIeEK(( ) ) ) U[ EEKK  9"99DIL IMMOOOOO  rKc8|j|j|j|jdS)N)rfixedtiedbounds) parametersrrrrus rI __getstate__zProjection.__getstate__s%ZIk    rKcF|d}|j|i|S)Nr)popr^)rastateparamss rI __setstate__zProjection.__setstate__s(3t}f....rK)rhrirjrkudegnppir0 _separabler^propertyabcabstractmethodrvryrxrrrlrms@rIrLrLhs-- quru BJ!!!!!  X X >   ///////rKrLceZdZdZdZdZdZdZfdZfdZ e dZ e dZ dZ e d ZxZS) rMz'Base class for all Pix2Sky projections.Tc|jdd}t||_t |SNrGrnamesplit_PROJ_NAME_CODE_MAPprj_coder]__new__clsrbrc long_namerds rIrzPix2SkyProjection.__new__=HNN3''* *95 wws###rKctj|i||j|j_||js|jd|_d|_ dS)Nxyphitheta r]r^rrsrHrxr{rinputsoutputsr`s rIr^zPix2SkyProjection.__init__sj$)&)))    IMMOOO  ' rKcb|jdtj|jdtjiSNrrrrrrus rI input_unitszPix2SkyProjection.input_units! At{1~qu==rKcb|jdtj|jdtjiSrrrrrus rI return_unitszPix2SkyProjection.return_units! Q Q??rKc`||j||Srq)rxrsprjx2s)rarrrbrcs rIevaluatezPix2SkyProjection.evaluates, y1%%%rKc>fdjD}j|S)Nc:g|]}t|jSrEr|rgrFrras rIrJz-Pix2SkyProjection.inverse..& G G GUgdE""( G G GrKr{_inv_clsrars` rIrvzPix2SkyProjection.inverse. G G G Gd6F G G Gt}b!!rKrhrirjrkn_inputs n_outputs_input_units_strict _input_units_allow_dimensionlessrr^rrrrrvrlrms@rIrMrMs11HI'+$$$$$$ ( ( ( ( (>>X>@@X@&&&""X"""""rKrMceZdZdZdZdZdZdZfdZfdZ e dZ e dZ dZ e d ZxZS) rNz'Base class for all Sky2Pix projections.rTc|jdd}t||_t |Srrrs rIrzSky2PixProjection.__new__rrKctj|i||j|j_||js|jd|_d|_ dS)Nrrrr`s rIr^zSky2PixProjection.__init__sj$)&)))    IMMOOO& ! rKcb|jdtj|jdtjiSrrrus rIrzSky2PixProjection.input_unitsrrKcb|jdtj|jdtjiSrrrus rIrzSky2PixProjection.return_unitsrrKc`||j||Srq)rxrsprjs2x)rarrrbrcs rIrzSky2PixProjection.evaluates, yU+++rKc>fdjD}j|S)Nc:g|]}t|jSrErrs rIrJz-Sky2PixProjection.inverse..rrKrrs` rIrvzSky2PixProjection.inverserrKrrms@rIrNrNs11HI'+$$$$$$ " " " " ">>X>@@X@,,,""X"""""rKrNceZdZdZdS)rOaBase class for all Zenithal projections. Zenithal (or azimuthal) projections map the sphere directly onto a plane. All zenithal projections are specified by defining the radius as a function of native latitude, :math:`R_\theta`. The pixel-to-sky transformation is defined as: .. math:: \phi &= \arg(-y, x) \\ R_\theta &= \sqrt{x^2 + y^2} and the inverse (sky-to-pixel) is defined as: .. math:: x &= R_\theta \sin \phi \\ y &= R_\theta \cos \phi NrhrirjrkrErKrIrOrOsrKrOc^eZdZdZeddZedeedZdZ e e_ dS) Pix2Sky_ZenithalPerspectiveu Zenithal perspective projection - pixel to sky. Corresponds to the ``AZP`` projection in FITS WCS. .. math:: \phi &= \arg(-y \cos \gamma, x) \\ \theta &= \left\{\genfrac{}{}{0pt}{}{\psi - \omega}{\psi + \omega + 180^{\circ}}\right. where: .. math:: \psi &= \arg(\rho, 1) \\ \omega &= \sin^{-1}\left(\frac{\rho \mu}{\sqrt{\rho^2 + 1}}\right) \\ \rho &= \frac{R}{\frac{180^{\circ}}{\pi}(\mu + 1) + y \sin \gamma} \\ R &= \sqrt{x^2 + y^2 \cos^2 \gamma} Parameters ---------- mu : float Distance from point of projection to center of sphere in spherical radii, μ. Default is 0. gamma : float Look angle γ in degrees. Default is 0°. 5Distance from point of projection to center of spheredefault descriptionu(Look angle γ in degrees (Default = 0°)rgettersetterrcrtjtj|drtddSNgz:Zenithal perspective projection is not defined for mu = -1ranyequalr rargs rI _mu_validatorz)Pix2Sky_ZenithalPerspective._mu_validatorD? 6"(5$'' ( ( %L   rKN rhrirjrkrZmur r gammar _validatorrErKrIrrsr8 !X   B L>    E "BMMMrKrc^eZdZdZeddZedeedZdZ e e_ dS) Sky2Pix_ZenithalPerspectiveuF Zenithal perspective projection - sky to pixel. Corresponds to the ``AZP`` projection in FITS WCS. .. math:: x &= R \sin \phi \\ y &= -R \sec \gamma \cos \theta where: .. math:: R = \frac{180^{\circ}}{\pi} \frac{(\mu + 1) \cos \theta} {(\mu + \sin \theta) + \cos \theta \cos \phi \tan \gamma} Parameters ---------- mu : float Distance from point of projection to center of sphere in spherical radii, μ. Default is 0. gamma : float Look angle γ in degrees. Default is 0°. rrru&Look angle γ in degrees (Default=0°)rcrtjtj|drtddSrrrs rIrz)Sky2Pix_ZenithalPerspective._mu_validatorrrrKNrrErKrIrrMsr4 !X   B L<    E "BMMMrKrc|eZdZdZeddZedeedZedeedZ d Z e e_ d S) Pix2Sky_SlantZenithalPerspectiveu Slant zenithal perspective projection - pixel to sky. Corresponds to the ``SZP`` projection in FITS WCS. Parameters ---------- mu : float Distance from point of projection to center of sphere in spherical radii, μ. Default is 0. phi0 : float The longitude φ₀ of the reference point, in degrees. Default is 0°. theta0 : float The latitude θ₀ of the reference point, in degrees. Default is 90°. rrruCThe longitude φ₀ of the reference point in degrees (Default=0°)rV@uCThe latitude θ₀ of the reference point, in degrees (Default=0°)crtjtj|drtddSrrrs rIrz.Pix2Sky_SlantZenithalPerspective._mu_validatorrrKN rhrirjrkrZrr r phi0theta0rrrErKrIrr{s* !X   B <Y    D \Y F "BMMMrKrc|eZdZdZeddZedeedZedeedZ dZ e e_ d S) Sky2Pix_SlantZenithalPerspectiveu Zenithal perspective projection - sky to pixel. Corresponds to the ``SZP`` projection in FITS WCS. Parameters ---------- mu : float distance from point of projection to center of sphere in spherical radii, μ. Default is 0. phi0 : float The longitude φ₀ of the reference point, in degrees. Default is 0°. theta0 : float The latitude θ₀ of the reference point, in degrees. Default is 90°. rrru5The longitude φ₀ of the reference point in degreesru5The latitude θ₀ of the reference point, in degreescrtjtj|drtddSrrrs rIrz.Sky2Pix_SlantZenithalPerspective._mu_validatorrrKNrrErKrIrrs* !X   B <K    D \K F "BMMMrKrceZdZdZdS)Pix2Sky_Gnomonicz Gnomonic projection - pixel to sky. Corresponds to the ``TAN`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: \theta = \tan^{-1}\left(\frac{180^{\circ}}{\pi R_\theta}\right) NrrErKrIrr    rKrceZdZdZdS)Sky2Pix_Gnomonicz Gnomonic Projection - sky to pixel. Corresponds to the ``TAN`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: R_\theta = \frac{180^{\circ}}{\pi}\cot \theta NrrErKrIrrrrKrceZdZdZdS)Pix2Sky_Stereographica Stereographic Projection - pixel to sky. Corresponds to the ``STG`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: \theta = 90^{\circ} - 2 \tan^{-1}\left(\frac{\pi R_\theta}{360^{\circ}}\right) NrrErKrIrrrrKrceZdZdZdS)Sky2Pix_Stereographica  Stereographic Projection - sky to pixel. Corresponds to the ``STG`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: R_\theta = \frac{180^{\circ}}{\pi}\frac{2 \cos \theta}{1 + \sin \theta} NrrErKrIrrrrKrcFeZdZdZeddZeddZdS)Pix2Sky_SlantOrthographicu Slant orthographic projection - pixel to sky. Corresponds to the ``SIN`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. The following transformation applies when :math:`\xi` and :math:`\eta` are both zero. .. math:: \theta = \cos^{-1}\left(\frac{\pi}{180^{\circ}}R_\theta\right) The parameters :math:`\xi` and :math:`\eta` are defined from the reference point :math:`(\phi_c, \theta_c)` as: .. math:: \xi &= \cot \theta_c \sin \phi_c \\ \eta &= - \cot \theta_c \cos \phi_c Parameters ---------- xi : float Obliqueness parameter, ξ. Default is 0.0. eta : float Obliqueness parameter, η. Default is 0.0. rzObliqueness parameterrNrhrirjrkrZxietarErKrIrr sD< c/F G G GB ,s0G H H HCCCrKrcBeZdZdZedZedZdS)Sky2Pix_SlantOrthographica* Slant orthographic projection - sky to pixel. Corresponds to the ``SIN`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. The following transformation applies when :math:`\xi` and :math:`\eta` are both zero. .. math:: R_\theta = \frac{180^{\circ}}{\pi}\cos \theta But more specifically are: .. math:: x &= \frac{180^\circ}{\pi}[\cos \theta \sin \phi + \xi(1 - \sin \theta)] \\ y &= \frac{180^\circ}{\pi}[\cos \theta \cos \phi + \eta(1 - \sin \theta)] rrNrrErKrIrr0s>* c " " "B ,s # # #CCCrKrceZdZdZdS)Pix2Sky_ZenithalEquidistantz Zenithal equidistant projection - pixel to sky. Corresponds to the ``ARC`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: \theta = 90^\circ - R_\theta NrrErKrIr r JrrKr ceZdZdZdS)Sky2Pix_ZenithalEquidistantz Zenithal equidistant projection - sky to pixel. Corresponds to the ``ARC`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: R_\theta = 90^\circ - \theta NrrErKrIr r WrrKr ceZdZdZdS)Pix2Sky_ZenithalEqualAreaa Zenithal equidistant projection - pixel to sky. Corresponds to the ``ZEA`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: \theta = 90^\circ - 2 \sin^{-1} \left(\frac{\pi R_\theta}{360^\circ}\right) NrrErKrIr r drrKr ceZdZdZdS)Sky2Pix_ZenithalEqualAreaa^ Zenithal equidistant projection - sky to pixel. Corresponds to the ``ZEA`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: R_\theta &= \frac{180^\circ}{\pi} \sqrt{2(1 - \sin\theta)} \\ &= \frac{360^\circ}{\pi} \sin\left(\frac{90^\circ - \theta}{2}\right) NrrErKrIrrqs    rKrc*eZdZdZedZdS) Pix2Sky_AiryuA Airy projection - pixel to sky. Corresponds to the ``AIR`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. Parameters ---------- theta_b : float The latitude :math:`\theta_b` at which to minimize the error, in degrees. Default is 90°. rrNrhrirjrkrZtheta_brErKrIrrs,  l4(((GGGrKrc,eZdZdZeddZdS) Sky2Pix_AiryuR Airy - sky to pixel. Corresponds to the ``AIR`` projection in FITS WCS. See `Zenithal` for a definition of the full transformation. .. math:: R_\theta = -2 \frac{180^\circ}{\pi}\left(\frac{\ln(\cos \xi)}{\tan \xi} + \frac{\ln(\cos \xi_b)}{\tan^2 \xi_b} \tan \xi \right) where: .. math:: \xi &= \frac{90^\circ - \theta}{2} \\ \xi_b &= \frac{90^\circ - \theta_b}{2} Parameters ---------- theta_b : float The latitude :math:`\theta_b` at which to minimize the error, in degrees. Default is 90°. rz6The latitude at which to minimize the error,in degreesrNrrErKrIrrs62lLGGGrKrceZdZdZdZdS)rPzBase class for Cylindrical projections. Cylindrical projections are so-named because the surface of projection is a cylinder. TNrhrirjrkrrErKrIrPrPs JJJrKrPcjeZdZdZedZedZdZee_dZ e e_dS)Pix2Sky_CylindricalPerspectiveu[ Cylindrical perspective - pixel to sky. Corresponds to the ``CYP`` projection in FITS WCS. .. math:: \phi &= \frac{x}{\lambda} \\ \theta &= \arg(1, \eta) + \sin{-1}\left(\frac{\eta \mu}{\sqrt{\eta^2 + 1}}\right) where: .. math:: \eta = \frac{\pi}{180^{\circ}}\frac{y}{\mu + \lambda} Parameters ---------- mu : float Distance from center of sphere in the direction opposite the projected surface, in spherical radii, μ. Default is 1. lam : float Radius of the cylinder in spherical radii, λ. Default is 1. ?rc`tj||j krtddSNz.CYP projection is not defined for mu = -lambdarrlamr rs rIrz,Pix2Sky_CylindricalPerspective._mu_validator9 6%DH9$ % % X%&VWW W X XrKc`tj||j krtddSNz.CYP projection is not defined for lambda = -murrrr rs rI_lam_validatorz-Pix2Sky_CylindricalPerspective._lam_validator9 6%DG8# $ $ X%&VWW W X XrKN rhrirjrkrZrrrrr#rErKrIrrst2 c " " "B ,s # # #CXXX"BMXXX$CNNNrKrcneZdZdZeddZeddZdZee_dZ e e_dS) Sky2Pix_CylindricalPerspectiveu Cylindrical Perspective - sky to pixel. Corresponds to the ``CYP`` projection in FITS WCS. .. math:: x &= \lambda \phi \\ y &= \frac{180^{\circ}}{\pi}\left(\frac{\mu + \lambda}{\mu + \cos \theta}\right)\sin \theta Parameters ---------- mu : float Distance from center of sphere in the direction opposite the projected surface, in spherical radii, μ. Default is 0. lam : float Radius of the cylinder in spherical radii, λ. Default is 0. rz1Distance from center of sphere in spherical radiirz)Radius of the cylinder in spherical radiic`tj||j krtddSrrrs rIrz,Sky2Pix_CylindricalPerspective._mu_validatorrrKc`tj||j krtddSr!r"rs rIr#z-Sky2Pix_CylindricalPerspective._lam_validatorr$rKNr%rErKrIr'r's( !T   B ,!L   CXXX"BMXXX$CNNNrKr'c*eZdZdZedZdS)Pix2Sky_CylindricalEqualAreauU Cylindrical equal area projection - pixel to sky. Corresponds to the ``CEA`` projection in FITS WCS. .. math:: \phi &= x \\ \theta &= \sin^{-1}\left(\frac{\pi}{180^{\circ}}\lambda y\right) Parameters ---------- lam : float Radius of the cylinder in spherical radii, λ. Default is 1. rrNrhrirjrkrZrrErKrIr+r+,   ,q ! ! !CCCrKr+c*eZdZdZedZdS)Sky2Pix_CylindricalEqualAreauL Cylindrical equal area projection - sky to pixel. Corresponds to the ``CEA`` projection in FITS WCS. .. math:: x &= \phi \\ y &= \frac{180^{\circ}}{\pi}\frac{\sin \theta}{\lambda} Parameters ---------- lam : float Radius of the cylinder in spherical radii, λ. Default is 0. rrNr,rErKrIr/r/!r-rKr/c(eZdZdZedZdS)Pix2Sky_PlateCarreeu Plate carrée projection - pixel to sky. Corresponds to the ``CAR`` projection in FITS WCS. .. math:: \phi &= x \\ \theta &= y cZtj|}tj|}||fSrqrarray)rrrrs rIrzPix2Sky_PlateCarree.evaluate?s(hqkk EzrKNrhrirjrk staticmethodrrErKrIr1r14s9\rKr1c(eZdZdZedZdS)Sky2Pix_PlateCarreeu Plate carrée projection - sky to pixel. Corresponds to the ``CAR`` projection in FITS WCS. .. math:: x &= \phi \\ y &= \theta cZtj|}tj|}||fSrqr3)rrrrs rIrzSky2Pix_PlateCarree.evaluateRs' HSMM HUOO!t rKNr5rErKrIr8r8Gs9\rKr8ceZdZdZdS)Pix2Sky_Mercatorz Mercator - pixel to sky. Corresponds to the ``MER`` projection in FITS WCS. .. math:: \phi &= x \\ \theta &= 2 \tan^{-1}\left(e^{y \pi / 180^{\circ}}\right)-90^{\circ} NrrErKrIr;r;ZrKr;ceZdZdZdS)Sky2Pix_Mercatorz Mercator - sky to pixel. Corresponds to the ``MER`` projection in FITS WCS. .. math:: x &= \phi \\ y &= \frac{180^{\circ}}{\pi}\ln \tan \left(\frac{90^{\circ} + \theta}{2}\right) NrrErKrIr>r>fr<rKr>ceZdZdZdZdS)rQa7Base class for pseudocylindrical projections. Pseudocylindrical projections are like cylindrical projections except the parallels of latitude are projected at diminishing lengths toward the polar regions in order to reduce lateral distortion there. Consequently, the meridians are curved. TNrrErKrIrQrQrsJJJrKrQceZdZdZdS)Pix2Sky_SansonFlamsteedz Sanson-Flamsteed projection - pixel to sky. Corresponds to the ``SFL`` projection in FITS WCS. .. math:: \phi &= \frac{x}{\cos y} \\ \theta &= y NrrErKrIrArA~r<rKrAceZdZdZdS)Sky2Pix_SansonFlamsteedz Sanson-Flamsteed projection - sky to pixel. Corresponds to the ``SFL`` projection in FITS WCS. .. math:: x &= \phi \cos \theta \\ y &= \theta NrrErKrIrCrCr<rKrCceZdZdZdS)Pix2Sky_Parabolicz Parabolic projection - pixel to sky. Corresponds to the ``PAR`` projection in FITS WCS. .. math:: \phi &= \frac{180^\circ}{\pi} \frac{x}{1 - 4(y / 180^\circ)^2} \\ \theta &= 3 \sin^{-1}\left(\frac{y}{180^\circ}\right) NrrErKrIrErEr<rKrEceZdZdZdS)Sky2Pix_Parabolicz Parabolic projection - sky to pixel. Corresponds to the ``PAR`` projection in FITS WCS. .. math:: x &= \phi \left(2\cos\frac{2\theta}{3} - 1\right) \\ y &= 180^\circ \sin \frac{\theta}{3} NrrErKrIrGrGr<rKrGceZdZdZdS)Pix2Sky_Molleweidea Molleweide's projection - pixel to sky. Corresponds to the ``MOL`` projection in FITS WCS. .. math:: \phi &= \frac{\pi x}{2 \sqrt{2 - \left(\frac{\pi}{180^\circ}y\right)^2}} \\ \theta &= \sin^{-1}\left( \frac{1}{90^\circ}\sin^{-1}\left(\frac{\pi}{180^\circ}\frac{y}{\sqrt{2}}\right) + \frac{y}{180^\circ}\sqrt{2 - \left(\frac{\pi}{180^\circ}y\right)^2} \right) NrrErKrIrIrIs    rKrIceZdZdZdS)Sky2Pix_Molleweidea Molleweide's projection - sky to pixel. Corresponds to the ``MOL`` projection in FITS WCS. .. math:: x &= \frac{2 \sqrt{2}}{\pi} \phi \cos \gamma \\ y &= \sqrt{2} \frac{180^\circ}{\pi} \sin \gamma where :math:`\gamma` is defined as the solution of the transcendental equation: .. math:: \sin \theta = \frac{\gamma}{90^\circ} + \frac{\sin 2 \gamma}{\pi} NrrErKrIrKrKsrKrKceZdZdZdS)Pix2Sky_HammerAitoffa  Hammer-Aitoff projection - pixel to sky. Corresponds to the ``AIT`` projection in FITS WCS. .. math:: \phi &= 2 \arg \left(2Z^2 - 1, \frac{\pi}{180^\circ} \frac{Z}{2}x\right) \\ \theta &= \sin^{-1}\left(\frac{\pi}{180^\circ}yZ\right) NrrErKrIrMrMr<rKrMceZdZdZdS)Sky2Pix_HammerAitoffaD Hammer-Aitoff projection - sky to pixel. Corresponds to the ``AIT`` projection in FITS WCS. .. math:: x &= 2 \gamma \cos \theta \sin \frac{\phi}{2} \\ y &= \gamma \sin \theta where: .. math:: \gamma = \frac{180^\circ}{\pi} \sqrt{\frac{2}{1 + \cos \theta \cos(\phi / 2)}} NrrErKrIrOrO    rKrOcJeZdZdZedeeZedeeZdS)rRaBase class for conic projections. In conic projections, the sphere is thought to be projected onto the surface of a cone which is then opened out. In a general sense, the pixel-to-sky transformation is defined as: .. math:: \phi &= \arg\left(\frac{Y_0 - y}{R_\theta}, \frac{x}{R_\theta}\right) / C \\ R_\theta &= \mathrm{sign} \theta_a \sqrt{x^2 + (Y_0 - y)^2} and the inverse (sky-to-pixel) is defined as: .. math:: x &= R_\theta \sin (C \phi) \\ y &= R_\theta \cos (C \phi) + Y_0 where :math:`C` is the "constant of the cone": .. math:: C = \frac{180^\circ \cos \theta}{\pi R_\theta} rrrrrN) rhrirjrkrZr r sigmadeltarErKrIrRrRsF0 LmJ O O OE L]: N N NEEErKrRceZdZdZdS)Pix2Sky_ConicPerspectiveaB Colles' conic perspective projection - pixel to sky. Corresponds to the ``COP`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \sin \theta_a \\ R_\theta &= \frac{180^\circ}{\pi} \cos \eta [ \cot \theta_a - \tan(\theta - \theta_a)] \\ Y_0 &= \frac{180^\circ}{\pi} \cos \eta \cot \theta_a Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIrVrV rKrVceZdZdZdS)Sky2Pix_ConicPerspectiveaB Colles' conic perspective projection - sky to pixel. Corresponds to the ``COP`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \sin \theta_a \\ R_\theta &= \frac{180^\circ}{\pi} \cos \eta [ \cot \theta_a - \tan(\theta - \theta_a)] \\ Y_0 &= \frac{180^\circ}{\pi} \cos \eta \cot \theta_a Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIrYrY'rWrKrYceZdZdZdS)Pix2Sky_ConicEqualAreaa Alber's conic equal area projection - pixel to sky. Corresponds to the ``COE`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \gamma / 2 \\ R_\theta &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin \theta} \\ Y_0 &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin((\theta_1 + \theta_2)/2)} where: .. math:: \gamma = \sin \theta_1 + \sin \theta_2 Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIr[r[D    rKr[ceZdZdZdS)Sky2Pix_ConicEqualAreaa Alber's conic equal area projection - sky to pixel. Corresponds to the ``COE`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \gamma / 2 \\ R_\theta &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin \theta} \\ Y_0 &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin((\theta_1 + \theta_2)/2)} where: .. math:: \gamma = \sin \theta_1 + \sin \theta_2 Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIr^r^hr\rKr^ceZdZdZdS)Pix2Sky_ConicEquidistanta1 Conic equidistant projection - pixel to sky. Corresponds to the ``COD`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \frac{180^\circ}{\pi} \frac{\sin\theta_a\sin\eta}{\eta} \\ R_\theta &= \theta_a - \theta + \eta\cot\eta\cot\theta_a \\ Y_0 = \eta\cot\eta\cot\theta_a Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIr`r`rKr`ceZdZdZdS)Sky2Pix_ConicEquidistanta1 Conic equidistant projection - sky to pixel. Corresponds to the ``COD`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \frac{180^\circ}{\pi} \frac{\sin\theta_a\sin\eta}{\eta} \\ R_\theta &= \theta_a - \theta + \eta\cot\eta\cot\theta_a \\ Y_0 = \eta\cot\eta\cot\theta_a Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIrcrcrarKrcceZdZdZdS)Pix2Sky_ConicOrthomorphica Conic orthomorphic projection - pixel to sky. Corresponds to the ``COO`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \frac{\ln \left( \frac{\cos\theta_2}{\cos\theta_1} \right)} {\ln \left[ \frac{\tan\left(\frac{90^\circ-\theta_2}{2}\right)} {\tan\left(\frac{90^\circ-\theta_1}{2}\right)} \right] } \\ R_\theta &= \psi \left[ \tan \left( \frac{90^\circ - \theta}{2} \right) \right]^C \\ Y_0 &= \psi \left[ \tan \left( \frac{90^\circ - \theta_a}{2} \right) \right]^C where: .. math:: \psi = \frac{180^\circ}{\pi} \frac{\cos \theta} {C\left[\tan\left(\frac{90^\circ-\theta}{2}\right)\right]^C} Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIrere####rKreceZdZdZdS)Sky2Pix_ConicOrthomorphica Conic orthomorphic projection - sky to pixel. Corresponds to the ``COO`` projection in FITS WCS. See `Conic` for a description of the entire equation. The projection formulae are: .. math:: C &= \frac{\ln \left( \frac{\cos\theta_2}{\cos\theta_1} \right)} {\ln \left[ \frac{\tan\left(\frac{90^\circ-\theta_2}{2}\right)} {\tan\left(\frac{90^\circ-\theta_1}{2}\right)} \right] } \\ R_\theta &= \psi \left[ \tan \left( \frac{90^\circ - \theta}{2} \right) \right]^C \\ Y_0 &= \psi \left[ \tan \left( \frac{90^\circ - \theta_a}{2} \right) \right]^C where: .. math:: \psi = \frac{180^\circ}{\pi} \frac{\cos \theta} {C\left[\tan\left(\frac{90^\circ-\theta}{2}\right)\right]^C} Parameters ---------- sigma : float :math:`(\theta_1 + \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 90. delta : float :math:`(\theta_1 - \theta_2) / 2`, where :math:`\theta_1` and :math:`\theta_2` are the latitudes of the standard parallels, in degrees. Default is 0. NrrErKrIrhrhrfrKrhceZdZdZdS)rSzrBase class for pseudoconic projections. Pseudoconics are a subclass of conics with concentric parallels. NrrErKrIrSrSsrKrSc2eZdZdZdZedeeZdS)Pix2Sky_BonneEqualAreaa Bonne's equal area pseudoconic projection - pixel to sky. Corresponds to the ``BON`` projection in FITS WCS. .. math:: \phi &= \frac{\pi}{180^\circ} A_\phi R_\theta / \cos \theta \\ \theta &= Y_0 - R_\theta where: .. math:: R_\theta &= \mathrm{sign} \theta_1 \sqrt{x^2 + (Y_0 - y)^2} \\ A_\phi &= \arg\left(\frac{Y_0 - y}{R_\theta}, \frac{x}{R_\theta}\right) Parameters ---------- theta1 : float Bonne conformal latitude, in degrees. TrrRN rhrirjrkrrZr r theta1rErKrIrkrks5.J \#mJ O O OFFFrKrkc4eZdZdZdZedeedZdS)Sky2Pix_BonneEqualAreaa Bonne's equal area pseudoconic projection - sky to pixel. Corresponds to the ``BON`` projection in FITS WCS. .. math:: x &= R_\theta \sin A_\phi \\ y &= -R_\theta \cos A_\phi + Y_0 where: .. math:: A_\phi &= \frac{180^\circ}{\pi R_\theta} \phi \cos \theta \\ R_\theta &= Y_0 - \theta \\ Y_0 &= \frac{180^\circ}{\pi} \cot \theta_1 + \theta_1 Parameters ---------- theta1 : float Bonne conformal latitude, in degrees. Trz$Bonne conformal latitude, in degreesrNrlrErKrIroro:sA,J \: FFFrKroceZdZdZdS)Pix2Sky_Polyconiczf Polyconic projection - pixel to sky. Corresponds to the ``PCO`` projection in FITS WCS. NrrErKrIrqrq[rKrqceZdZdZdS)Sky2Pix_Polyconiczf Polyconic projection - sky to pixel. Corresponds to the ``PCO`` projection in FITS WCS. NrrErKrIrtrtcrrrKrtceZdZdZdS)rTawBase class for quad cube projections. Quadrilateralized spherical cube (quad-cube) projections belong to the class of polyhedral projections in which the sphere is projected onto the surface of an enclosing polyhedron. The six faces of the quad-cube projections are numbered and laid out as:: 0 4 3 2 1 4 3 2 5 NrrErKrIrTrTkrPrKrTceZdZdZdS)Pix2Sky_TangentialSphericalCubezv Tangential spherical cube projection - pixel to sky. Corresponds to the ``TSC`` projection in FITS WCS. NrrErKrIrwrw|rrrKrwceZdZdZdS)Sky2Pix_TangentialSphericalCubezv Tangential spherical cube projection - sky to pixel. Corresponds to the ``TSC`` projection in FITS WCS. NrrErKrIryryrrrKryceZdZdZdS)Pix2Sky_COBEQuadSphericalCubez COBE quadrilateralized spherical cube projection - pixel to sky. Corresponds to the ``CSC`` projection in FITS WCS. NrrErKrIr{r{rrrKr{ceZdZdZdS)Sky2Pix_COBEQuadSphericalCubez COBE quadrilateralized spherical cube projection - sky to pixel. Corresponds to the ``CSC`` projection in FITS WCS. NrrErKrIr}r}rrrKr}ceZdZdZdS)Pix2Sky_QuadSphericalCubez} Quadrilateralized spherical cube projection - pixel to sky. Corresponds to the ``QSC`` projection in FITS WCS. NrrErKrIrrrrrKrceZdZdZdS)Sky2Pix_QuadSphericalCubez} Quadrilateralized spherical cube projection - sky to pixel. Corresponds to the ``QSC`` projection in FITS WCS. NrrErKrIrrrrrKrceZdZdZdS)r?z#Base class for HEALPix projections.NrrErKrIr?r?s....rKr?cJeZdZdZdZeddZeddZdS) Pix2Sky_HEALPixz HEALPix - pixel to sky. Corresponds to the ``HPX`` projection in FITS WCS. Parameters ---------- H : float The number of facets in longitude direction. X : float The number of facets in latitude direction. T@,The number of facets in longitude direction.r@+The number of facets in latitude direction.NrhrirjrkrrZHXrErKrIrrY  J !O   A  !N   AAArKrcJeZdZdZdZeddZeddZdS) Sky2Pix_HEALPixa  HEALPix projection - sky to pixel. Corresponds to the ``HPX`` projection in FITS WCS. Parameters ---------- H : float The number of facets in longitude direction. X : float The number of facets in latitude direction. TrrrrrNrrErKrIrrrrKrceZdZdZdS)Pix2Sky_HEALPixPolar{ HEALPix polar, aka "butterfly" projection - pixel to sky. Corresponds to the ``XPH`` projection in FITS WCS. NrrErKrIrrrrrKrceZdZdZdS)Sky2Pix_HEALPixPolarrNrrErKrIrrrrrKrceZdZdZdZdZdZdZeddgddggZ eddgZ dZ e e _ dZ e e _ e e ffd Zed Zed Zed Zed ZxZS)rUaQ Perform an affine transformation in 2 dimensions. Parameters ---------- matrix : array A 2x2 matrix specifying the linear transformation to apply to the inputs translation : array A 2D vector (given as either a 2x1 or 1x2 array) specifying a translation to apply to the inputs rFrrrcTtj|dkrtddS)z2Validates that the input matrix is a 2x2 2D array.)rrz0Expected transformation matrix to be a 2x2 arrayN)rshaper rs rI_matrix_validatorz(AffineTransformation2D._matrix_validators2 8E??f $ $%B  % $rKctj|dkrtj|dks?tj|dkrtj|dkstddSdS)z Validates that the translation vector is a 2D vector. This allows either a "row" vector or a "column" vector where in the latter case the resultant Numpy array has ``ndim=2`` but the shape is ``(1, 2)``. r)rr)rrzHExpected translation vector to be a 2 element row or column vector arrayN)rndimrr rs rI_translation_validatorz-AffineTransformation2D._translation_validatorssWU^^q RXe__%<%<!##6(A(A% &=%<(A(ArKc \tjd||d|d|_d|_dS)Nmatrix translationrrE)r]r^rr)rarrrcrds rIr^zAffineTransformation2D.__init__(s9JKJJ6JJJ  ! rKctj|jj}|dkrt d|jjdtj|jj}|jj ||jj z}tj ||j j }|||S)z Inverse transformation. Raises `~astropy.modeling.InputParameterError` if the transformation cannot be inverted. rz#Transformation matrix is singular; z model does not have an inverseNr) rlinalgdetrrgr rdrhinvunitdotr)rarrrs rIrvzAffineTransformation2D.inverse-simmDK-.. !88%,dn6M,,,  t{011 ;  'dk..Fvfd&6&<=== ~~V~EEErKc:|j|jkrtd|jpd}tjtj|tj|tj|j|jg}|jddks |j dkrtd| ||}tj ||}|d|d}}|x|_|_||fS)a, Apply the transformation to a set of 2D Cartesian coordinates given as two lists--one for the x coordinates and one for a y coordinates--or a single coordinate pair. Parameters ---------- x, y : array, float x and y coordinates z,Expected input arrays to have the same shape)rrrzIncompatible input shapesr) r ValueErrorrvstackasarrayravelonessizedtyper_create_augmented_matrixr) rrrrrrinarraugmented_matrixresults rIrzAffineTransformation2D.evaluateCs 7ag  KLL L4 Z]] " "BJqMM$7$7$9$9271617;S;S T   ;q>Q  %*//899 977 LL(%00ay&)1!!!'!t rKcd}tt|dt|dgrott|dt|dgstd|j}|j|jz t jkstdtjdt}||ddddf<||ddddf_ gd|d<|||zS|S) NrzXTo use AffineTransformation with quantities, both matrix and unit need to be quantities.z0matrix and translation must have the same units.)rr)rrr)rrr) rhasattrallrrrdimensionless_unscaledremptyr~flat)rrrrs rIrz/AffineTransformation2D._create_augmented_matrixbs  V,,gff.E.EF G G U V44gff6M6MNOO  B#DK+"22q7OOO !STTT8F%888%+1ac")41abb!&'ii  #d* *rKc|jj}|jj}||dS|&tt |j|gdzStt |j|gdzS)Nr)r input_unitrdictzipr)ratranslation_unit matrix_units rIrz"AffineTransformation2D.input_unitsxsr+6k,  # (;4  )DK*:);a)?@@AA ADK+):;;<< rQrArCrErGrIrKrMrOrRrVrYr[r^r`rcrerhrSrkrorqrtrTrwryr{r}rrr?rrrrrUr short_nameglobalsrrErKrIrs    $$$$$$$$66666666,,,,,,,,<#Gd?++ 1 1 1 1 1     DSXww 95uuo7NOO P PQQ R 9*C/C/C/C/C/C/C/C/L*"*"*"*"*" *"*"*"Z*"*"*"*"*" *"*"*"Zz*-"-"-"-"-""3X-"-"-"`+"+"+"+"+""3X+"+"+"\,",",",","'8(,",","^,",",",","'8(,",","^     ((        ((        -x        -x    I I I I I 18 I I IF$$$$$ 18$$$4     "3X        "3X         18         18   )))))$h)))$$h@*'$'$'$'$'$%6 '$'$'$T&$&$&$&$&$%6 &$&$&$R"""""#4k"""&"""""#4k"""&+[&+[&     (+        (+                /1B        /1B        )+<        )+<        *,=   *,=&     ,.?   ,.?"OOOOOJOOO:0%:0%:!!!!!.!!!H!!!!!.!!!H0%<0%<$$$$$ 15$$$N$$$$$ 15$$$N*PPPPP. PPP:. B););z"&7&7$5x$5x 18 18/////j///'4'4,g,gN=N=N=N=N=UN=N=N=b-SSIz)0: 3I)JGGIIj:%&)0: 3I)JGGIIj:%&18: ;Q1RGGIIj9$%.18: ;Q1RGGIIj9$%.. SSrK