;+ ; NAME: ; WCSSPH2XY ; PURPOSE: ; Convert spherical coordinates to x and y (map) angular coordinates ; EXPLANATION: ; Convert spherical (longitude and latitude -- sky) coordinates to x ; and y (map) angular coordinates. This procedure is the inverse of ; WCSXY2SPH. See WCS_DEMO for example of use. ; ; This is a lower level procedure -- given a FITS header, the user will ; usually use ADXY which will then call WCSSPH2XY with the appropriate ; parameters. ; CATEGORY: ; Mapping and Auxiliary FITS Routine ; ; CALLING SEQUENCE: ; wcssph2xy, longitude, latitude, x, y, [ map_type , CTYPE = , ; FACE =,PV2= , CRVAL = , CRXY = , LONGPOLE = , ; LATPOLE = , NORTH_OFFSET =, SOUTH_OFFSET =, BADINDEX =] ; ; INPUT PARAMETERS: ; longitude - longitude of data, scalar or vector, in degrees ; latitude - latitude of data, same number of elements as longitude, ; in degrees ; map_type - optional positional parameter, numeric scalar (0-26) ; corresponding to a particular map projection. This is not a ; FITS standard, it is simply put in to allow function similar ; to that of less general map projection procedures (eg AITOFF). ; The following list gives the map projection types and their ; respective numbers. ; ; FITS Number Name Comments ; code code ; ---- ------ ----------------------- ----------------------------------- ; DEF 0 Default = Cartesian ; AZP 1 Zenithal perspective PV2_1 required ; TAN 2 Gnomic AZP w/ mu = 0 ; SIN 3 Orthographic PV2_1,PV2_2 optional ; STG 4 Stereographic AZP w/ mu = 1 ; ARC 5 Zenithal Equidistant ; ZPN 6 Zenithal polynomial PV2_0, PV2_1....PV2_20 possible ; ZEA 7 Zenithal equal area ; AIR 8 Airy PV2_1 required ; CYP 9 Cylindrical perspective PV2_1 and PV2_2 required ; CAR 10 Cartesian ; MER 11 Mercator ; CEA 12 Cylindrical equal area PV2_1 required ; COP 13 Conical perspective PV2_1 and PV2_2 required ; COD 14 Conical equidistant PV2_1 and PV2_2 required ; COE 15 Conical equal area PV2_1 and PV2_2 required ; COO 16 Conical orthomorphic PV2_1 and PV2_2 required ; BON 17 Bonne's equal area PV2_1 required ; PCO 18 Polyconic ; SFL 19 Sanson-Flamsteed ; PAR 20 Parabolic ; AIT 21 Hammer-Aitoff ; MOL 22 Mollweide ; CSC 23 Cobe Quadrilateralized convergence of inverse is poor ; Spherical Cube ; QSC 24 Quadrilateralized ; Spherical Cube ; TSC 25 Tangential Spherical Cube ; SZP 26 Slant Zenithal Projection PV2_1,PV2_2, PV2_3 optional ; ; OPTIONAL INPUT KEYWORD PARAMETERS: ; ; CTYPE - One, two, or three element vector containing 8 character ; strings corresponding to the CTYPE1, CTYPE2, and CTYPE3 ; FITS keywords: ; ; CTYPE[0] - first four characters specify standard system ; ('RA--','GLON' or 'ELON' for right ascension, Galactic ; longitude or ecliptic longitude respectively), second four ; letters specify the type of map projection (eg '-AIT' for ; Aitoff projection) ; CTYPE[1] - first four characters specify standard system ; ('DEC-','GLAT' or 'ELAT' for declination, galactic latitude ; or ecliptic latitude respectively; these must match ; the appropriate system of ctype1), second four letters of ; ctype2 must match second four letters of ctype1. ; CTYPE[2] - if present must be the 8 character string,'CUBEFACE', ; only used for spherical cube projections to identify an axis ; as containing the face on which each x and y pair of ; coordinates lie. ; PV2 - Vector of projection parameter associated with latitude axis ; PV2 will have up to 21 elements for the ZPN projection, up to 3 ; for the SIN projection and no more than 2 for any other ; projection. The first element corresponds to PV2_1, the ; second to PV2_2, etc. ; CRVAL - 2 element vector containing standard system coordinates (the ; longitude and latitude) of the reference point ; CRXY - 2 element vector giving the x and y coordinates of the ; reference point, if this is not set the offset is [0,0] ; This is not a FITS standard -- it is similar to CRPIX but in ; angular X,Y coordinates (degrees) rather than pixel coordinates ; LATPOLE - native latitude of the standard system's North Pole ; LONGPOLE - native longitude of standard system's North Pole, default ; is 180 degrees for Zenithal systems ; NORTH_OFFSET - offset (radians) added to input points near north pole. ; SOUTH_OFFSET - offset (radians) added to input points near south pole. ; BADINDEX - vector, list of transformed points too close to poles. ; ; ; OUTPUT PARAMETERS: ; ; x - x coordinate of data, same number of elements as longitude, in ; degrees; if CRXY is set, then x will be returned offset by ; crxy(0). NOTE: x in all map projections increases to the ; left, not the right. ; y - y coordinate of data, same number of elements as longitude, in ; degrees; if CRXY is set, y will be returned offset by crxy[1] ; bad - vector returning index to transformed points close to pole. ; ; OPTIONAL OUTPUT KEYWORD PARAMETERS: ; FACE - a output variable used for spherical cube projections to ; designate the face of the cube on which the x and y ; coordinates lie. Will contain the same number of elements as ; X and Y. Must contain at least 1 arbitrary element on input ; If FACE is NOT defined on input, it is assumed that the ; spherical cube projection is laid out over the whole sky ; in the "sideways T" configuration. ; NOTES: ; The conventions followed here are described in more detail in ; "Representations of Celestial Coordinates in FITS" by Mark Calabretta ; and Eric Greisen (2002, A&A, 395, 1077; also see ; http://www.aoc.nrao.edu/~egreisen). The general ; scheme outlined in that article is to first use WCS_ROTATE to convert ; coordinates in one of three standard systems (celestial, galactic, ; or ecliptic) into a "native system" of latitude and longitude. The ; latitude and longitude are then converted into x and y coordinates ; which depend on the map projection which is performed. The rotation ; from standard to native coordinates can be skipped if one so desires. ; This procedure necessitates two basic sections. The first converts ; "standard" coordinates to "native" coordinates while the second converts ; "native" coordinates to x and y coordinates. The first section is ; simply a call to WCS_ROTATE, while the second contains the guts of ; the code in which all of the map projection is done. This procedure ; can be called in a form similar to AITOFF, EQPOLE, or QDCB by calling ; wcssph2xy with a fifth parameter specifying the map projection by ; number and by not using any of the keywords related to the map ; projection type (e.g. CTYPE). ; ; PROCEDURE: ; ; The first task of the procedure is to do general error-checking to ; make sure the procedure was called correctly and none of the ; parameters or keywords conflict. This is particularly important ; because the procedure can be called in two ways (either using ; FITS-type keywords or using a number corresponding to a map projection ; type). All variables are converted into double precision values and ; angular measurements are converted from degrees into radians. ; If necessary, longitude values are converted into the range -pi to pi. ; Any latitude points close to the of the poles are mapped to a specific ; latitude of from the pole so that the map transformations become ; completely invertible. The magnitude of this correction is given by ; the keywords NORTH_OFFSET and SOUTH_OFFSET and a list of affected ; points is optionally returned in the "badindex" output parameter. ; The next task of the procedure is to convert the "standard" ; coordinates to "native" coordinates by rotating the coordinate system. ; This rotation is performed by the procedure WCS_ROTATE and is governed ; by the keywords CRVAL and LONGPOLE. The final task of the WCSSPH2XY ; is to take "native" latitude and longitude coordinates and convert ; them into x and y coordinates. Any map specific error-checking is ; done at this time. All of the equations were obtained from ; "Representations of Celestial Coordinates in FITS" and cases needing ; special attention are handled appropriately (see the comments with ; individual map projections for more information on special cases). ; ; Note that a further transformation (using the CD matrix) is required ; to convert the (x,y) coordinates to pixel coordinates. ; COMMON BLOCKS: ; ; none ; ; PROCEDURES CALLED: ; WCS_ROTATE ; ; COPYRIGHT NOTICE: ; ; Copyright 1993, The Regents of the University of California. This ; software was produced under U.S. Government contract (W-7405-ENG-36) ; by Los Alamos National Laboratory, which is operated by the ; University of California for the U.S. Department of Energy. ; The U.S. Government is licensed to use, reproduce, and distribute ; this software. Neither the Government nor the University makes ; any warranty, express or implied, or assumes any liability or ; responsibility for the use of this software. ; ; AUTHOR: ; ; Rick Balsano ; ; MODIFICATIONS/REVISION LEVEL: ; ; 1.1 8/31/93 ; 2.3 9/15/93 W. Landsman (HSTX) Update quad cube coords, vectorize ; keywords ; 2.4 12/29/93 I. Freedman (HSTX) Eliminated LU decomposition ; 2.5 1/5/93 I. Freedman (HSTX) Offset keywords / bad point index ; 2.6 Dec 94 Compute pole for transformations where the reference ; pixel is at the native origin W. Landsman (HSTX) ; 2.7 May 95 Change internal variable BETA for V4.0 compatibility ; 2.8 June 95 Change loop indices from integer to long ; 2.9 3/18/96 Change FACE usage for cube projections to match WCSLIB ; C/FORTRAN software library. ; 2.10 02/18/99 Fixed implementation of ARC algorithm ; 2.11 June 2003 Update conic projections, add LATPOLE keyword ; 2.12 Aug 2003, N.Rich - Fix pre-V5.5 bug from previous update ; 2.13 Sep 2003, W. Landsman CTYPE keywords need not be 8 characters ; 2.14 Jan 2004, W. Landsman don't modify scalars, fix PARabolic code ; 2.15 Feb 2004, W. Landsman Fix AZP and AIR algorithms ; 3.0 May 2004 W. Landsman Support extended SIN (=NCP), slant zenithal ; (SZP), and zenithal polynomail (ZPN) projections, use ; PV2 keyword vector instead of PROJP1, PROJP2 ; 3.1 Jul 2005 W.Landsman/C. Markwardt Set unprojectable points in ; tangent projection to NaN ; 3.1.1 Jul 2005 Fixed 3.1 mod to work for scalars ; 3.2 Dec 2005 Fixed Airy projection for latitude centered at 90 deg ; 3.3 Aug 2007 R. Munoz, W.Landsman Correct treatment of PV1_2 and ; PV2_2 parameters ; 3.4 Oct 2007 Sergey Koposov Support HEALPIX projection ;- PRO wcssph2xy,longitude,latitude,x,y,map_type, ctype=ctype,$ face=face, pv2 = pv2,$ crval=crval,crxy=crxy,longpole=longpole, latpole= latpole, $ north_offset=north_offset, south_offset=south_offset, $ badindex=badindex ; DEFINE ANGLE CONSTANTS pi = !DPI pi2 = pi/2.d0 radeg = 57.295779513082323d0 map_types=['DEF','AZP','TAN','SIN','STG','ARC','ZPN','ZEA','AIR','CYP',$ 'CAR','MER','CEA','COP','COD','COE','COO','BON','PCO','SFL',$ 'PAR','AIT','MOL','CSC','QSC','TSC', 'SZP'] ; check to see that enough parameters (at least 4) were sent if (N_params() lt 4) then begin print,'Syntax - WCSSPH2XY, longitude, latitude, x, y, [ map_type,' print,' CTYPE= ,FACE=, PV2_1=, PV2_2=, CRVAL=, CRXY=, LATPOLE=' print,' LONGPOLE= ,NORTH_OFFSET=, SOUTH_OFFSET=, BADINDEX=]' return endif ; GENERAL ERROR CHECKING ; find the number of elements in each of the data arrays n_long = N_elements( longitude ) n_lat = N_elements( latitude ) ; check to see that the data arrays have the same size if (n_long ne n_lat) then begin message,$ 'LONGITUDE and LATITUDE must have the same number of elements.' endif if (N_params() eq 5) then begin if (keyword_set(ctype1) or keyword_set(ctype2)) then message,$ 'Use either the MAP_TYPE positional parameter or set the projection type with CRVAL1 and CRVAL2, but not both.' ; set projection_type string using map_type parameter (a number) if ((map_type ge 0) and (map_type le 25)) then begin projection_type=map_types[map_type] endif else message,'MAP_TYPE must be >= 0 and <= 25, it was set to '+map_type endif else if (n_params() eq 4) then begin ; check to see that CTYPE is set correctly if N_elements( ctype ) GE 1 then begin ctype1 = ctype[0] projection_type = strupcase(strmid(ctype1,5,3)) endif if N_elements( ctype ) GE 2 then begin ctype2 = ctype[1] if (projection_type ne strupcase(strmid(ctype2,5,3))) then begin message,'The same map projection type must be in characters',/continue message,' 5-8 of both CTYPE1 and CTYPE2.',/CON return endif if (((strupcase(strmid(ctype1,1,2)) eq 'RA') and $ (strupcase(strmid(ctype2,1,3)) ne 'DEC')) or $ ((strupcase(strmid(ctype1,1,4)) eq 'GLON') and $ (strupcase(strmid(ctype2,1,4)) ne 'GLAT')) or $ ((strupcase(strmid(ctype1,1,4)) eq 'ELON') and $ (strupcase(strmid(ctype2,1,4)) ne 'ELAT'))) $ then begin message,'The same standard system must be in the first 4',$ /continue print,'characters of both CTYPE1 and CTYPE2.' return endif endif else projection_type = 'DEF' endif ; this sets the default map projection type for the cases when map_type or ; projection_type is set to 'DEF' or if projection_type is not set at this ; point. As suggested in 'Representations of Celestial Coordinates in FITS' ; the default type is set to CAR (Cartesian) the simplest of all projections. if ((n_elements(projection_type) eq 0) or $ (projection_type eq 'DEF') ) then begin projection_type='CAR' message,'Projection type not supplied, set to default (Cartesian)',/INF endif ; Check to make sure all the correct parameters and keywords are set for ; spherical projections. if (keyword_set(ctype3) or keyword_set(face) or (projection_type eq 'CSC') or $ (projection_type eq 'QSC') or (projection_type eq 'TSC')) then begin if (n_elements(face) eq 0) then noface=1 else noface=0 endif ; check to see if the x and y offsets are set properly. If not, break out ; of program. If the x and y offsets are not set then assume they are zero. if (((n_elements(crx) eq 1) and (n_elements(cry) eq 0)) or $ ((n_elements(crx) eq 0) and (n_elements(cry) eq 1))) then $ message,'If either CRX or CRY is set, the other must also be set' if (((n_elements(crval1) eq 1) and (n_elements(crval2) eq 0)) or $ ((n_elements(crval1) eq 0) and (n_elements(crval1) eq 1))) then $ message,'If either CRVAL1 or CRVAL2 is set, the other must also be set' ; Convert all longitude values into the range -180 to 180 so that equations ; work properly. lng = double( longitude ) & lat = double( latitude ) temp = where(lng ge 180.0, Ntemp) if Ntemp GT 0 then lng[temp] = lng[temp] - 360.0d0 ; Make small offsets at poles to allow the transformations to be ; completely invertible. These generally introduce a small fractional error ; but only at the poles. They are necessary since all maps ; lose information at the poles when a rotation is applied, because all points ; within NORTH_ or SOUTH_OFFSET of the poles are mapped to the same points. IF N_elements(north_offset) EQ 0 then north_offset = 1.d-7 IF N_elements(south_offset) EQ 0 then south_offset = 1.d-7 bad = where(abs(lat - 90.0) lt north_offset*radeg) IF (bad[0] ne -1) THEN BEGIN ;; commented out this message ;;MESSAGE,/INFORM,'Some input points are too close to the NORTH pole.' lat[bad] = 90.0 - north_offset*RADEG IF KEYWORD_SET(badindex) THEN badindex = bad ENDIF bad = where(abs(lat + 90.0) lt south_offset*radeg) IF (bad[0] ne -1) THEN BEGIN ;; commented out this message ;;MESSAGE,/INFORM,'Some input points are too close to the SOUTH pole.' lat[bad] = south_offset*radeg - 90.0 IF KEYWORD_SET(badindex) THEN BEGIN badindex = [badindex, bad] badindex = badindex[sort(badindex)] ENDIF ENDIF ; Convert from standard coordinate system to "native" coordinate system ; if the CRVAL keyword is set. Otherwise, assume the latitude and longitude ; given are in "native" coordinates already (this is essentially what is done ; in the procedure AITOFF). PV2_1 = N_elements(pv2) GT 0 ? pv2[0] : 0 PV2_2 = N_elements(pv2) GT 1 ? pv2[1] : 0 if N_elements(crval) GE 2 then begin if N_elements(map_type) EQ 0 then begin wmt = where(projection_type EQ map_types) map_type = wmt[0] endif conic = (map_type GE 13) and (map_type LE 16) zenithal = ((map_type GE 1) and (map_type LE 8)) or (map_type EQ 26) ; Rotate from standard celestial coordinates into the native system. if conic then theta0 = PV2_1 else if zenithal then theta0 = 90 $ else theta0 = 0 wcs_rotate, lng, lat, phi, theta, crval, $ latpole = latpole, longpole=longpole, theta0 = theta0 phi = phi/radeg theta = theta/radeg endif else begin phi = lng/radeg theta = lat/radeg endelse ; BRANCH BY MAP PROJECTION TYPE case strupcase(projection_type) of 'AZP':begin if (PV2_1 lt 0) then message,$ 'AZP map projection requires the keyword PV2_1 >= 0' gamma = PV2_2/radeg mu = PV2_1 r_theta = radeg*cos(theta)*(mu + 1.d0)/ $ ( (mu + sin(theta)) + cos(theta)*cos(phi)*tan(gamma)) x = r_theta*sin(phi) y = -r_theta*cos(phi)/cos(gamma) end 'SZP': begin mu = N_elements(PV2) GT 0 ? PV2[0] : 0 phi_c = N_elements(PV2) GT 1 ? PV2[1] : 0 theta_c = N_elements(PV2) GT 1 ? PV2[2] : 90 phi_c = phi_c/radeg & theta_c = theta_c/radeg xp = -mu*cos(theta_c)*sin(phi_c) yp = mu*cos(theta_c)*cos(phi_c) zp = mu*sin(theta_c) + 1. denom = zp - (1-sin(theta)) x = radeg*( zp*cos(theta)*sin(phi) - xp*(1-sin(theta)) )/ denom y = -radeg*( zp*cos(theta)*cos(phi) + yp*(1-sin(theta)) )/ denom end 'TAN':begin sz_theta = size(theta,/dimen) if sz_theta[0] EQ 0 then x = !Values.D_NAN else $ x = make_array(value = !values.D_NAN, dimen=sz_theta) y = x g = where(theta GT 0, Ng) if Ng GT 0 then begin r_theta = radeg/tan(theta[g]) x[g] = r_theta*sin(phi[g]) y[g] = -r_theta*cos(phi[g]) endif end 'SIN':begin if N_elements(PV2_1) EQ 0 then PV2_1 = 0 if N_elements(PV2_2) EQ 0 then PV2_2 = 0 if (PV2_1 EQ 0) and (PV2_2 EQ 0) then begin r_theta = radeg*cos(theta) x = r_theta*sin(phi) y = -r_theta*cos(phi) endif else begin ;NCP projection x = radeg*(cos(theta)*sin(phi) + PV2_1*(1-sin(theta)) ) y = -radeg*(cos(theta)*cos(phi) - PV2_2*(1-sin(theta)) ) endelse end 'STG':begin r_theta = 2.d0*radeg*tan((pi2-theta)/2.d0) x = r_theta*sin(phi) y = -r_theta*cos(phi) end 'ARC':begin r_theta = radeg*( pi2 - theta ) x = r_theta*sin(phi) y = -r_theta*cos(phi) end 'ZPN':begin z = pi2 - theta g = where(pv2 NE 0, Ng) if Ng GT 0 then np = max(g) else np =0 r_theta = radeg*poly(z, pv2[0:np]) x = r_theta*sin(phi) y = -r_theta*cos(phi) end 'ZEA':begin r_theta = 2.d0*radeg*sin((pi2 - theta)/2.d0) x = r_theta*sin(phi) y = -r_theta*cos(phi) end 'AIR':begin if not(keyword_set(PV2_1)) then begin message,/informational,$ 'PV2_1 not set, using default of PV2_1 = 90 for AIR map projection' PV2_1 = 9.d1 endif theta_b = PV2_1/radeg xi = (pi2 - theta)/2.d0 ; When theta_b (aka PV2_1 in radians) is equal to pi/2 the normal equations ; for the AIR projection produce infinities. To avoid the problem, values ; of theta_b equal to pi/2 cause a different set of equations to be used. if (theta_b eq pi2) then begin ; AIR produces the same radii for different latitudes, causing some overlap. To ; avoid this problem, if latitudes which are far enough south to be a problem ; are included in the data, the routine will stop. if (min(theta) lt -36/radeg) then begin message,'AIR produces overlap of native latitudes south of ',/continue print,'-36 with the PV2_1 = 90' return endif ; points with xi too small are labelled as bad to prevent poor behavior of the ; equation for r_theta good = where(abs(xi) ge 1.d-10, Ngood) r_theta = lng*0 if (Ngood GT 0) then $ r_theta[good] = -2*radeg*(alog(cos(xi[good]))/tan(xi[good]) - $ 0.5*tan(xi[good])) endif else begin xi_b = (pi2 - theta_b)/2.d0 a = alog(cos(xi_b))/tan(xi_b)/tan(xi_b) ; AIR produces the same radii for different latitudes, causing some overlap. To ; avoid this problem, if latitudes which are far enough south to be a problem ; are included in the data, the routine will stop. xi_temp = (findgen(90) + 1)/radeg radius=-radeg*(alog(cos(xi_temp))/tan(xi_temp)+alog(cos(xi_b))/$ tan(xi_b)*tan(xi_temp)) i = 0 repeat i = i + 1 $ until ((radius[i + 1] le radius[i]) or (i eq n_elements(radius) - 2)) if (i lt (n_elements(radius)- 2)) then min_lat = 90 - 2*radeg*xi_temp[i] $ else min_lat = -90 if (min(theta) lt min_lat[0]/radeg) then begin message,'AIR produces overlap of native latitudes south of ',/continue print,format='(i3,a21,i3)',min_lat[0],' with the PV2_1 = ',PV2_1 return endif ; points with xi too small are labelled as bad to prevent poor behavior of the ; equation for r_theta good = where(abs(xi) ge 1.d-10, Ngood) r_theta = lng*0 if (Ngood GT 0) then r_theta[good] = -2*radeg*(alog(cos(xi[good]))/$ tan(xi[good]) + a*tan(xi[good])) endelse x = r_theta*sin(phi) y = -r_theta*cos(phi) end 'CYP':begin if (n_elements(PV2_1) eq 0) then begin message,/informational,$ 'PV2_1 not set, using default of PV2_1 = 0 for CYP map projection' PV2_1 = 0.d0 endif if (n_elements(PV2_2) eq 0) then begin message,/informational,$ 'PV2_2 not set, using default of PV2_2 = 1 for CYP map projection' PV2_2 = 1.d0 endif if (PV2_1 eq -PV2_2) then message,$ 'PV2_1 = -PV2_2 is not allowed for CYP map projection.' x = PV2_2*radeg*phi y = radeg*(PV2_1 + PV2_2)*sin(theta)/(PV2_1 + cos(theta)) end 'CAR':begin x = radeg*phi y = radeg*theta end 'MER':begin x = radeg*phi y = radeg*alog(tan((pi2 + theta)/2.d0)) end 'CEA':begin if N_elements(PV2_1) EQ 0 then message,$ 'CEA map projection requires that PV2_1 keyword be set.' if ((PV2_1 le 0) or (PV2_1 gt 1)) then message,$ 'CEA map projection requires 0 < PV2_1 <= 1' x = radeg*phi y = radeg*sin(theta)/PV2_1 end 'COP':begin if not(keyword_set(PV2_1)) then message,$ 'COP map projection requires that PV2_1 keyword be set.' if not(keyword_set(PV2_2)) then begin message,/informational,$ 'PV2_2 not set, using default of PV2_2 = 0 for COP map projection' PV2_2= 0 endif if ((PV2_1 lt -90) or (PV2_2 gt 90) or (PV2_1 gt 90)) then message,$ 'PV2_1 and PV2_2 must satisfy -90<=PV2_1<=90,0<=PV2_2<=90 for COP projection' if (PV2_1 eq -PV2_2) then message,$ 'COP projection with PV2_1=-PV2_2 is better done as a cylindrical projection' theta_a = PV2_1/radeg alpha = PV2_2/radeg bad = where((theta ge theta_a + pi2) or (theta le theta_a - pi2)) if (bad[0] ne -1) then begin message,/continue,$ 'COP map projection diverges for native latitude = PV2_1 +- 90.' message,'Remove these points and try again.' endif r_theta = radeg*cos(alpha)*(1.d0/tan(theta_a)-tan(theta-theta_a)) a_phi = phi*sin(theta_a) y_0 = radeg*cos(alpha)/tan(theta_a) x = r_theta*sin(a_phi) y = y_0 - r_theta*cos(a_phi) end 'COD':begin if not(keyword_set(PV2_1)) then message,$ 'COD map projection requires that PV2_1 keyword be set.' if not(keyword_set(PV2_2)) then begin message,/informational,$ 'PV2_2 not set, using default of PV2_2 = 0 for COD map projection' PV2_2 = 0 end if ((PV2_1 lt -90) or (PV2_2 gt 90) or (PV2_1 gt 90)) then message,$ 'PV2_1 and PV2_2 must satisfy -90<=PV2_1<=90,PV2_2<=90 for COD projection' if (PV2_1 eq -PV2_2) then message,$ 'COD gives divergent equations for PV2_1 = -PV2_2' theta_a = PV2_1/radeg ; when PV2_1 not = PV2_2 use regular equations if (PV2_2 NE 0) then begin alpha = PV2_2/radeg r_theta = theta_a - theta + alpha/(tan(alpha)*tan(theta_a)) a_phi = sin(theta_a)*sin(alpha)*phi/alpha y_0 = radeg*alpha/(tan(alpha)*tan(theta_a)) ; if the two parameters PV2_1 and PV2_2 are equal use the simpler set of ; equations endif else begin r_theta = theta_a - theta + 1.d0/tan(theta_a) a_phi = phi*sin(theta_a) y_0 = radeg/tan(theta_a) endelse x = radeg*r_theta*sin(a_phi) y = y_0 - radeg*r_theta*cos(a_phi) end 'COE':begin if N_elements(PV2_1) EQ 0 then message,$ 'COE map projection requires that PV2_1 keyword be set.' if N_elements(PV2_2) EQ 0 then begin message,/informational,$ 'PV2_2 not set, using default of PV2_2 = 0 for COE map projection' PV2_2 = 0 end if ((PV2_1 lt -90) or (PV2_2 gt 90) or (PV2_1 gt PV2_2)) then message,$ 'PV2_1 and PV2_2 must satisfy -90<=PV2_1<=PV2_2<=90 for COE map projection' if (PV2_1 eq -PV2_2) then message,$ 'COE gives divergent equations for PV2_1 = -PV2_2' theta_1 = (PV2_1 - PV2_2)/radeg theta_2 = (PV2_1 + PV2_2)/radeg s_1 = sin(theta_1) s_2 = sin(theta_2) stheta_a = sin(PV2_1/radeg) gamma = s_1 + s_2 r_theta=radeg*2.d0*sqrt(1.d0+ s_1*s_2-gamma*sin(theta))/gamma a_phi = phi*gamma/2.d0 y_0 = radeg*2.d0*sqrt(1.d0+ s_1*s_2-gamma*stheta_a)/gamma x = r_theta*sin(a_phi) y = y_0 - r_theta*cos(a_phi) end 'COO':begin if not(keyword_set(PV2_1)) then message,$ 'COO map projection requires that PV2_1 keyword be set.' if not(keyword_set(PV2_2)) then begin message,/informational,$ 'PV2_2 not set, using default of PV2_2 = 0 for COO map projection' PV2_2 = 0 end if ((PV2_1 lt -90) or (PV2_2 gt 90) or (PV2_1 gt 90)) then message,$ 'PV2_1 and PV2_2 must satisfy -90<=PV2_1<=90,PV2_2<=90 for COO projection' if (PV2_1 eq -PV2_2) then message,$ 'COO gives divergent equations for PV2_1 = -PV2_2' theta_1 = (PV2_1 - PV2_2)/radeg theta_2 = (PV2_1 + PV2_2)/radeg theta_a = PV2_1/radeg ; for cases where PV2_1 = 0, use a simpler formula to calculate c, ; otherwise use the regular formula if (PV2_2 eq 0) then c = sin(theta_1) else $ c = alog(cos(theta_2)/cos(theta_1))/alog(tan((pi2-theta_2)/2.d0)/$ tan((pi2-theta_1)/2.d0)) alpha = radeg*cos(theta_1)/(c*(tan((pi2-theta_1)/2.d0))^c) r_theta = alpha*(tan((pi2-theta)/2.d0))^c y_0 = alpha*tan((pi2-theta_a)/2.)^c a_phi = c*phi x = r_theta*sin(a_phi) y = y_0 - r_theta*cos(a_phi) end 'BON':begin if (N_elements(PV2) LT 1) then message,$ 'BON map projection requires that PV2_1 keyword be set.' pv2_1 = pv2[0] if ((PV2_1 lt -90) or (PV2_1 gt 90)) then message,$ 'PV2_1 must satisfy -90 <= PV2_1 <= 90 for BON map projection' if (PV2_1 eq 0) then message,$ 'PV2_1 = 0 for BON map projection is better done with SFL map projection' theta_1 = PV2_1/radeg s = theta_1/abs(theta_1) y_0 = 1.d0/tan(theta_1) + theta_1 a = phi*cos(theta)/(y_0 - theta) x = radeg*(y_0 - theta)*sin(a) y = radeg*(y_0 - (y_0 - theta)*cos(a)) end 'PCO':begin ; The equations for x and y are poorly behaved for theta = 0. Avoid this by ; explicitly assigning values for x and y when theta = 0. zero_ind = where(theta eq 0, Nzero) ; create x and y with same structure as longitude x = lng*0 & y = x if (Nzero GT 0) then begin x[zero_ind] = radeg*phi[zero_ind] y[zero_ind] = 0.d0 endif good_ind = where(theta ne 0, Ngood) if Ngood GT 0 then begin x[good_ind] = radeg*sin(phi[good_ind]*sin(theta[good_ind]))/$ tan(theta[good_ind]) y[good_ind] = radeg*(theta[good_ind]+$ (1.d0 - cos(phi[good_ind]*sin(theta[good_ind])))/tan(theta[good_ind])) endif end 'SFL':begin x = radeg*phi*cos(theta) y = radeg*theta end 'PAR':begin x = radeg*phi*(2.d0*cos(2.d0*theta/3.d0) - 1.d0) y = 180.0*sin(theta/3.d0) end 'AIT':begin alpha = radeg*sqrt(2.d0/(1.d0 + cos(theta)*cos(0.5d0*phi))) x = 2.d0*alpha*cos(theta)*sin(0.5d0*phi) y = alpha*sin(theta) end 'MOL':begin ; Use Newton's method to find a numerical solution to the equation: ; alpha + 1/2*sin(2*alpha) - 1/2*pi*sin(theta) = 0 tolerance = 1.0d-14 alpha = lng*0 repeat begin alpha_old = alpha alpha = alpha_old - (alpha_old + 0.5*sin(2.d0*alpha_old) - $ 0.5*pi*sin(theta))/(1.d0 + cos(2.d0*alpha_old)) endrep until (max(abs(alpha - alpha_old)) lt tolerance) x = 2.d0^1.5*phi*radeg*cos(alpha)/pi y = sqrt(2.d0)*radeg*sin(alpha) end 'CSC':begin ; calculate direction cosines l = cos(theta)*sin(phi) m = cos(theta)*cos(phi) n = sin(theta) ; determine the face on which the x and y coordinates will reside by setting ; rho equal to the maximum of n,m,l,-m,-l,-n which corresponds to faces 0 ; through 5 respectively rho = lng*0 if size(lng,/N_dimen) EQ 0 then face = 0 else face = lonarr(n_long) ; use an array to store a remapping of the direction cosines. This way, faces ; 0 and 5 take points on their borders with faces 1-4. The reason for this is ; that if the max function sees identical values in an array, it takes the ; index of the first occurrence of that value. remap = [0,5,2,1,4,3] for i = 0l, n_long-1 do begin dir_cos = float([n[i],-n[i],l[i],m[i],-l[i],-m[i]]) rho[i] = max(dir_cos,temp) face[i] = remap[temp] endfor ; based on the face determined for each point, find the parameters alpha and ; beta1 alpha = lng*0 beta1 = alpha for i = 0l, n_long-1 do begin case face[i] of 0:begin alpha[i] = l[i]/n[i] beta1[i] = -m[i]/n[i] end 1:begin alpha[i] = l[i]/m[i] beta1[i] = n[i]/m[i] end 2:begin alpha[i] = -m[i]/l[i] beta1[i] = n[i]/l[i] end 3:begin alpha[i] = l[i]/m[i] beta1[i] = -n[i]/m[i] end 4:begin alpha[i] = -m[i]/l[i] beta1[i] = -n[i]/l[i] end 5:begin alpha[i] = -l[i]/n[i] beta1[i] = -m[i]/n[i] end endcase end ; define all of the numerical constants to use in determining x and y r_0 = 0.577350269 gam_s = 1.37484847732 em = 0.004869491981 gam = -0.13161671474 ome = -0.159596235474 d_0 = 0.0759196200467 d_1 = -0.0217762490699 c_00 = 0.141189631152 c_10 = 0.0809701286525 c_01 = -0.281528535557 c_20 = -0.178251207466 c_11 = 0.15384112876 c_02 = 0.106959469314 fconst = 45.0d0 x = fconst*(alpha*gam_s+alpha^3*(1-gam_s)+alpha*beta1^2*(1-alpha^2)*$ (gam+(em-gam)*alpha^2+(1-beta1^2)*(c_00+c_10*alpha^2+c_01*beta1^2+$ c_20*alpha^4+c_11*alpha^2*beta1^2+c_02*beta1^4))+alpha^3*(1-alpha^2)*$ (ome-(1-alpha^2)*(d_0+d_1*alpha^2))) y = fconst*(beta1*gam_s+beta1^3*(1-gam_s)+beta1*alpha^2*(1-beta1^2)*$ (gam+(em-gam)*beta1^2+(1-alpha^2)*(c_00+c_10*beta1^2+c_01*alpha^2+$ c_20*beta1^4+c_11*beta1^2*alpha^2+c_02*alpha^4))+beta1^3*(1-beta1^2)*$ (ome-(1-beta1^2)*(d_0+d_1*beta1^2))) if noface eq 1 then begin xf=fconst*[0.0d0,0.0d0,2.0d0,4.0d0,6.0d0,0.0d0] yf=fconst*[2.0d0,0.0d0,0.0d0,0.0d0,0.0d0,-2.0d0] x=x+xf[face] y=y+yf[face] endif end 'QSC':begin ; calculate direction cosines l = cos(theta)*sin(phi) m = cos(theta)*cos(phi) n = sin(theta) ; determine the face on which the x and y coordinates will reside by setting ; rho equal to the maximum of n,m,l,-m,-l,-n which corresponds to faces 0 ; through 5 respectively rho = lng*0 if size(lng,/N_dimen) EQ 0 then face = 0 else face = lonarr(n_long) ; use an array to store a remapping of the direction cosines. This way, faces ; 0 and 5 take points on their borders with faces 1-4. The reason for this is ; that if the max function sees identical values in an array, it takes the ; index of the first occurrence of that value. remap = [0,5,2,1,4,3] for i = 0l, n_long-1 do begin dir_cos = float([n[i],-n[i],l[i],m[i],-l[i],-m[i]]) rho[i] = max(dir_cos,temp) face[i] = remap[temp] endfor ; based on the face determined for each point, find the parameters alpha and ; beta1 alpha = lng*0 beta1 = alpha for i = 0l, n_long-1 do begin case face[i] of 0:begin alpha[i] = l[i]/n[i] beta1[i] = -m[i]/n[i] end 1:begin alpha[i] = l[i]/m[i] beta1[i] = n[i]/m[i] end 2:begin alpha[i] = -m[i]/l[i] beta1[i] = n[i]/l[i] end 3:begin alpha[i] = l[i]/m[i] beta1[i] = -n[i]/m[i] end 4:begin alpha[i] = -m[i]/l[i] beta1[i] = -n[i]/l[i] end 5:begin alpha[i] = -l[i]/n[i] beta1[i] = -m[i]/n[i] end endcase end x = lng*0 y = x & xi = y s = 2.d0*(((alpha gt abs(beta1)) or (beta1 ge abs(alpha))) - 0.5d0) case_1 = where(abs(alpha) gt abs(beta1)) case_2 = where((abs(alpha) le abs(beta1)) and (beta1 ne 0.d0)) case_3 = where((alpha eq 0.d0) and (beta1 eq 0.d0)) if (case_1[0] ne -1) then xi[case_1] = beta1[case_1]/alpha[case_1] if (case_2[0] ne -1) then xi[case_2] = alpha[case_2]/beta1[case_2] if (case_3[0] ne -1) then xi[case_3] = 0.d0 fconst=45.0d0 u = fconst*s*sqrt((1.d0 - rho)/(1.d0 - 1.d0/sqrt(2.d0 + xi^2))) v = (u/1.5d1)*radeg*(atan(xi) - asin(xi/sqrt(2.d0*(1.d0 + xi^2)))) if (case_1[0] ne -1) then begin x[case_1] = u[case_1] y[case_1] = v[case_1] endif if (case_2[0] ne -1) then begin x[case_2] = v[case_2] y[case_2] = u[case_2] endif if (case_3[0] ne -1) then begin x[case_3] = 0.d0 y[case_3] = 0.d0 endif if noface eq 1 then begin xf=fconst*[0.0d0,0.0d0,2.0d0,4.0d0,6.0d0,0.0d0] yf=fconst*[2.0d0,0.0d0,0.0d0,0.0d0,0.0d0,-2.0d0] x=(x+xf[face]) y=(y+yf[face]) endif end 'TSC':begin ; calculate direction cosines l = cos(theta)*sin(phi) m = cos(theta)*cos(phi) n = sin(theta) ; determine the face on which the x and y coordinates will reside by setting ; rho equal to the maximum of n,m,l,-m,-l,-n which corresponds to faces 0 ; through 5 respectively rho = lng*0 if size(lng,/N_dimen) EQ 0 then face = 0 else face = lonarr(n_long) ; use an array to store a remapping of the direction cosines. This way, faces ; 0 and 5 take points on their borders with faces 1-4. The reason for this is ; that if the max function sees identical values in an array, it takes the ; index of the first occurrence of that value. remap = [0,5,2,1,4,3] for i = 0l, n_long-1 do begin dir_cos = float([n[i],-n[i],l[i],m[i],-l[i],-m[i]]) rho[i] = max(dir_cos,temp) face[i] = remap[temp] endfor ; based on the face determined for each point, find the parameters eta and xi eta = lng*0 xi = eta for i = 0l, n_long-1 do begin case face[i] of 0:begin eta[i] = -m[i] xi[i] = l[i] end 1:begin eta[i] = n[i] xi[i] = l[i] end 2:begin eta[i] = n[i] xi[i] = -m[i] end 3:begin eta[i] = n[i] xi[i] = -l[i] end 4:begin eta[i] = n[i] xi[i] = m[i] end 5:begin eta[i] = m[i] xi[i] = l[i] end endcase endfor fconst = 45.0d0 r_theta = fconst/tan(asin(rho)) a_phi = atan(xi,-eta) x = r_theta*sin(a_phi) y = -r_theta*cos(a_phi) if noface eq 1 then begin xf=fconst*[0.0d0,0.0d0,2.0d0,4.0d0,6.0d0,0.0d0] yf=fconst*[2.0d0,0.0d0,0.0d0,0.0d0,0.0d0,-2.0d0] x=(x+xf[face]) y=(y+yf[face]) endif end 'HPX':begin hpx_k = 3.D ; The main HEALPIX parameters (see Calabretta 2007, MNRAS) hpx_h = 4.D ; thetalim = asin((hpx_k-1)/hpx_k) eqfaces = where( abs(theta) le thetalim, complement=polfaces) x=phi*0.D y=theta*0.D ; equatorial region if eqfaces[0] ne -1 then begin x[eqfaces] = phi[eqfaces]*radeg y[eqfaces] = 90 * hpx_k/ hpx_h * sin( theta[eqfaces]) endif ;polar regions if polfaces[0] ne -1 then begin hpx_sig = sqrt ( hpx_k * (1 - abs(sin(theta[polfaces])))) hpx_omega = ((hpx_k mod 2 eq 1) or theta[polfaces] gt 0)*1.D hpx_phic = -180 + (2*floor((phi[polfaces]*radeg+180)*hpx_h/360+(1-hpx_omega)/2.)+hpx_omega)*180/hpx_h x[polfaces] = hpx_phic + (phi[polfaces]*radeg-hpx_phic) * hpx_sig y[polfaces] = 180./hpx_h *((theta[polfaces] gt 0)*2-1)* ((hpx_k+1)/2 - hpx_sig) endif end else:message,strupcase(projection_type)+$ ' is not a valid projection type. Reset CTYPE1 and CTYPE2' endcase if keyword_set(crxy) then begin x = x - crxy[0] y = y - crxy[1] endif END