head 1.4; access; symbols; locks; strict; comment @# @; 1.4 date 2002.09.07.01.50.40; author curt; state Exp; branches; next 1.3; 1.3 date 2002.08.31.19.01.57; author curt; state Exp; branches; next 1.2; 1.2 date 2002.08.31.18.54.03; author curt; state Exp; branches; next 1.1; 1.1 date 2002.02.15.17.22.00; author curt; state Exp; branches; next ; desc @@ 1.4 log @Updated doxygen docs. @ text @
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# sg_geodesy.cxx

    00001 // sg_geodesy.cxx -- routines to convert between geodetic and geocentric
    00002 //                   coordinate systems.
    00003 //
    00004 // Copied and adapted directly from LaRCsim/ls_geodesy.c
    00005 //
    00006 // See below for the complete original LaRCsim comments.
    00007 //
    00008 // $Id: sg_geodesy.cxx,v 1.10 2001/03/24 05:02:15 curt Exp $
    00009
    00010 #include <simgear/compiler.h>
    00011
    00012 #ifdef SG_HAVE_STD_INCLUDES
    00013 # include <cmath>
    00014 # include <cerrno>
    00015 # include <cstdio>
    00016 #else
    00017 # include <math.h>
    00018 # include <errno.h>
    00019 # include <stdio.h>
    00020 #endif
    00021
    00022 #include <simgear/constants.h>
    00023 #include <simgear/debug/logstream.hxx>
    00024
    00025 #include "point3d.hxx"
    00026 #include "sg_geodesy.hxx"
    00027 #include "localconsts.hxx"
    00028
    00029
    00030 #ifndef SG_HAVE_NATIVE_SGI_COMPILERS
    00031 SG_USING_STD(cout);
    00032 #endif
    00033
    00034
    00035 #define DOMAIN_ERR_DEBUG 1
    00036
    00037
    00038 // sgGeocToGeod(lat_geoc, radius, *lat_geod, *alt, *sea_level_r)
    00039 //     INPUTS:
    00040 //         lat_geoc     Geocentric latitude, radians, + = North
    00041 //         radius       C.G. radius to earth center (meters)
    00042 //
    00043 //     OUTPUTS:
    00044 //         lat_geod     Geodetic latitude, radians, + = North
    00045 //         alt          C.G. altitude above mean sea level (meters)
    00046 //         sea_level_r  radius from earth center to sea level at
    00047 //                      local vertical (surface normal) of C.G. (meters)
    00048
    00049
    00050 void sgGeocToGeod( double lat_geoc, double radius, double
    00051                    *lat_geod, double *alt, double *sea_level_r )
    00052 {
    00053 #ifdef DOMAIN_ERR_DEBUG
    00054     errno = 0;                  // start with error zero'd
    00055 #endif
    00056
    00057     double t_lat, x_alpha, mu_alpha, delt_mu, r_alpha, l_point, rho_alpha;
    00058     double sin_mu_a, denom,delt_lambda, lambda_sl, sin_lambda_sl;
    00059
    00060     if( ( (SGD_PI_2 - lat_geoc) < SG_ONE_SECOND )        // near North pole
    00061         || ( (SGD_PI_2 + lat_geoc) < SG_ONE_SECOND ) )   // near South pole
    00062     {
    00063         *lat_geod = lat_geoc;
    00064         *sea_level_r = SG_EQUATORIAL_RADIUS_M*E;
    00065         *alt = radius - *sea_level_r;
    00066     } else {
    00067         // cout << "  lat_geoc = " << lat_geoc << endl;
    00068         t_lat = tan(lat_geoc);
    00069         // cout << "  tan(t_lat) = " << t_lat << endl;
    00070         x_alpha = E*SG_EQUATORIAL_RADIUS_M/sqrt(t_lat*t_lat + E*E);
    00071 #ifdef DOMAIN_ERR_DEBUG
    00072         if ( errno ) {
    00073             perror("fgGeocToGeod()");
    00074             SG_LOG( SG_GENERAL, SG_ALERT, "sqrt(" << t_lat*t_lat + E*E << ")" );
    00075         }
    00076 #endif
    00077         // cout << "  x_alpha = " << x_alpha << endl;
    00078         double tmp = sqrt(SG_EQ_RAD_SQUARE_M - x_alpha * x_alpha);
    00079         if ( tmp < 0.0 ) { tmp = 0.0; }
    00080 #ifdef DOMAIN_ERR_DEBUG
    00081         if ( errno ) {
    00082             perror("fgGeocToGeod()");
    00083             SG_LOG( SG_GENERAL, SG_ALERT, "sqrt(" << SG_EQ_RAD_SQUARE_M - x_alpha * x_alpha
    00084                     << ")" );
    00085         }
    00086 #endif
    00087         mu_alpha = atan2(tmp,E*x_alpha);
    00088         if (lat_geoc < 0) mu_alpha = - mu_alpha;
    00089         sin_mu_a = sin(mu_alpha);
    00090         delt_lambda = mu_alpha - lat_geoc;
    00091         r_alpha = x_alpha/cos(lat_geoc);
    00092         l_point = radius - r_alpha;
    00093         *alt = l_point*cos(delt_lambda);
    00094
    00095         denom = sqrt(1-EPS*EPS*sin_mu_a*sin_mu_a);
    00096 #ifdef DOMAIN_ERR_DEBUG
    00097         if ( errno ) {
    00098             perror("fgGeocToGeod()");
    00099             SG_LOG( SG_GENERAL, SG_ALERT, "sqrt(" <<
    00100                     1-EPS*EPS*sin_mu_a*sin_mu_a << ")" );
    00101         }
    00102 #endif
    00103         rho_alpha = SG_EQUATORIAL_RADIUS_M*(1-EPS)/
    00104             (denom*denom*denom);
    00105         delt_mu = atan2(l_point*sin(delt_lambda),rho_alpha + *alt);
    00106         *lat_geod = mu_alpha - delt_mu;
    00107         lambda_sl = atan( E*E * tan(*lat_geod) ); // SL geoc. latitude
    00108         sin_lambda_sl = sin( lambda_sl );
    00109         *sea_level_r =
    00110             sqrt(SG_EQ_RAD_SQUARE_M / (1 + ((1/(E*E))-1)*sin_lambda_sl*sin_lambda_sl));
    00111 #ifdef DOMAIN_ERR_DEBUG
    00112         if ( errno ) {
    00113             perror("fgGeocToGeod()");
    00114             SG_LOG( SG_GENERAL, SG_ALERT, "sqrt(" <<
    00115                     SG_EQ_RAD_SQUARE_M / (1 + ((1/(E*E))-1)*sin_lambda_sl*sin_lambda_sl)
    00116                     << ")" );
    00117         }
    00118 #endif
    00119
    00120     }
    00121
    00122 }
    00123
    00124
    00125 // sgGeodToGeoc( lat_geod, alt, *sl_radius, *lat_geoc )
    00126 //     INPUTS:
    00127 //         lat_geod     Geodetic latitude, radians, + = North
    00128 //         alt          C.G. altitude above mean sea level (meters)
    00129 //
    00130 //     OUTPUTS:
    00131 //         sl_radius    SEA LEVEL radius to earth center (meters)
    00132 //                      (add Altitude to get true distance from earth center.
    00133 //         lat_geoc     Geocentric latitude, radians, + = North
    00134 //
    00135
    00136
    00137 void sgGeodToGeoc( double lat_geod, double alt, double *sl_radius,
    00138                       double *lat_geoc )
    00139 {
    00140     double lambda_sl, sin_lambda_sl, cos_lambda_sl, sin_mu, cos_mu, px, py;
    00141
    00142 #ifdef DOMAIN_ERR_DEBUG
    00143     errno = 0;
    00144 #endif
    00145
    00146     lambda_sl = atan( E*E * tan(lat_geod) ); // sea level geocentric latitude
    00147     sin_lambda_sl = sin( lambda_sl );
    00148     cos_lambda_sl = cos( lambda_sl );
    00149     sin_mu = sin(lat_geod);                  // Geodetic (map makers') latitude
    00150     cos_mu = cos(lat_geod);
    00151     *sl_radius =
    00152         sqrt(SG_EQ_RAD_SQUARE_M / (1 + ((1/(E*E))-1)*sin_lambda_sl*sin_lambda_sl));
    00153 #ifdef DOMAIN_ERR_DEBUG
    00154         if ( errno ) {
    00155             perror("fgGeodToGeoc()");
    00156             SG_LOG( SG_GENERAL, SG_ALERT, "sqrt(" <<
    00157                     SG_EQ_RAD_SQUARE_M / (1 + ((1/(E*E))-1)*sin_lambda_sl*sin_lambda_sl)
    00158                     << ")" );
    00159         }
    00160 #endif
    00161     py = *sl_radius*sin_lambda_sl + alt*sin_mu;
    00162     px = *sl_radius*cos_lambda_sl + alt*cos_mu;
    00163     *lat_geoc = atan2( py, px );
    00164 }
    00165
    00166
    00167 // Direct and inverse distance functions
    00168 //
    00169 // Proceedings of the 7th International Symposium on Geodetic
    00170 // Computations, 1985
    00171 //
    00172 // "The Nested Coefficient Method for Accurate Solutions of Direct and
    00173 // Inverse Geodetic Problems With Any Length"
    00174 //
    00175 // Zhang Xue-Lian
    00176 // pp 747-763
    00177 //
    00178 // modified for FlightGear to use WGS84 only -- Norman Vine
    00179
    00180 #define GEOD_INV_PI SGD_PI
    00181
    00182 // s == distance
    00183 // az = azimuth
    00184
    00185 // for WGS_84 a = 6378137.000, rf = 298.257223563;
    00186
    00187 static double M0( double e2 ) {
    00188     //double e4 = e2*e2;
    00189     return GEOD_INV_PI*(1.0 - e2*( 1.0/4.0 + e2*( 3.0/64.0 +
    00190                                                   e2*(5.0/256.0) )))/2.0;
    00191 }
    00192
    00193
    00194 // given, alt, lat1, lon1, az1 and distance (s), calculate lat2, lon2
    00195 // and az2.  Lat, lon, and azimuth are in degrees.  distance in meters
    00196 int geo_direct_wgs_84 ( double alt, double lat1, double lon1, double az1,
    00197                         double s, double *lat2, double *lon2,  double *az2 )
    00198 {
    00199     double a = 6378137.000, rf = 298.257223563;
    00200     double RADDEG = (GEOD_INV_PI)/180.0, testv = 1.0E-10;
    00201     double f = ( rf > 0.0 ? 1.0/rf : 0.0 );
    00202     double b = a*(1.0-f);
    00203     double e2 = f*(2.0-f);
    00204     double phi1 = lat1*RADDEG, lam1 = lon1*RADDEG;
    00205     double sinphi1 = sin(phi1), cosphi1 = cos(phi1);
    00206     double azm1 = az1*RADDEG;
    00207     double sinaz1 = sin(azm1), cosaz1 = cos(azm1);
    00208
    00209
    00210     if( fabs(s) < 0.01 ) {      // distance < centimeter => congruency
    00211         *lat2 = lat1;
    00212         *lon2 = lon1;
    00213         *az2 = 180.0 + az1;
    00214         if( *az2 > 360.0 ) *az2 -= 360.0;
    00215         return 0;
    00216     } else if( cosphi1 ) {      // non-polar origin
    00217         // u1 is reduced latitude
    00218         double tanu1 = sqrt(1.0-e2)*sinphi1/cosphi1;
    00219         double sig1 = atan2(tanu1,cosaz1);
    00220         double cosu1 = 1.0/sqrt( 1.0 + tanu1*tanu1 ), sinu1 = tanu1*cosu1;
    00221         double sinaz =  cosu1*sinaz1, cos2saz = 1.0-sinaz*sinaz;
    00222         double us = cos2saz*e2/(1.0-e2);
    00223
    00224         // Terms
    00225         double  ta = 1.0+us*(4096.0+us*(-768.0+us*(320.0-175.0*us)))/16384.0,
    00226             tb = us*(256.0+us*(-128.0+us*(74.0-47.0*us)))/1024.0,
    00227             tc = 0;
    00228
    00229         // FIRST ESTIMATE OF SIGMA (SIG)
    00230         double first = s/(b*ta);  // !!
    00231         double sig = first;
    00232         double c2sigm, sinsig,cossig, temp,denom,rnumer, dlams, dlam;
    00233         do {
    00234             c2sigm = cos(2.0*sig1+sig);
    00235             sinsig = sin(sig); cossig = cos(sig);
    00236             temp = sig;
    00237             sig = first +
    00238                 tb*sinsig*(c2sigm+tb*(cossig*(-1.0+2.0*c2sigm*c2sigm) -
    00239                                       tb*c2sigm*(-3.0+4.0*sinsig*sinsig)
    00240                                       *(-3.0+4.0*c2sigm*c2sigm)/6.0)
    00241                            /4.0);
    00242         } while( fabs(sig-temp) > testv);
    00243
    00244         // LATITUDE OF POINT 2
    00245         // DENOMINATOR IN 2 PARTS (TEMP ALSO USED LATER)
    00246         temp = sinu1*sinsig-cosu1*cossig*cosaz1;
    00247         denom = (1.0-f)*sqrt(sinaz*sinaz+temp*temp);
    00248
    00249         // NUMERATOR
    00250         rnumer = sinu1*cossig+cosu1*sinsig*cosaz1;
    00251         *lat2 = atan2(rnumer,denom)/RADDEG;
    00252
    00253         // DIFFERENCE IN LONGITUDE ON AUXILARY SPHERE (DLAMS )
    00254         rnumer = sinsig*sinaz1;
    00255         denom = cosu1*cossig-sinu1*sinsig*cosaz1;
    00256         dlams = atan2(rnumer,denom);
    00257
    00258         // TERM C
    00259         tc = f*cos2saz*(4.0+f*(4.0-3.0*cos2saz))/16.0;
    00260
    00261         // DIFFERENCE IN LONGITUDE
    00262         dlam = dlams-(1.0-tc)*f*sinaz*(sig+tc*sinsig*
    00263                                        (c2sigm+
    00264                                         tc*cossig*(-1.0+2.0*
    00265                                                    c2sigm*c2sigm)));
    00266         *lon2 = (lam1+dlam)/RADDEG;
    00267         if (*lon2 > 180.0  ) *lon2 -= 360.0;
    00268         if (*lon2 < -180.0 ) *lon2 += 360.0;
    00269
    00270         // AZIMUTH - FROM NORTH
    00271         *az2 = atan2(-sinaz,temp)/RADDEG;
    00272         if ( fabs(*az2) < testv ) *az2 = 0.0;
    00273         if( *az2 < 0.0) *az2 += 360.0;
    00274         return 0;
    00275     } else {                    // phi1 == 90 degrees, polar origin
    00276         double dM = a*M0(e2) - s;
    00277         double paz = ( phi1 < 0.0 ? 180.0 : 0.0 );
    00278         return geo_direct_wgs_84( alt, 0.0, lon1, paz, dM,lat2,lon2,az2 );
    00279     }
    00280 }
    00281
    00282
    00283 // given alt, lat1, lon1, lat2, lon2, calculate starting and ending
    00284 // az1, az2 and distance (s).  Lat, lon, and azimuth are in degrees.
    00285 // distance in meters
    00286 int geo_inverse_wgs_84( double alt, double lat1, double lon1, double lat2,
    00287                         double lon2, double *az1, double *az2, double *s )
    00288 {
    00289     double a = 6378137.000, rf = 298.257223563;
    00290     int iter=0;
    00291     double RADDEG = (GEOD_INV_PI)/180.0, testv = 1.0E-10;
    00292     double f = ( rf > 0.0 ? 1.0/rf : 0.0 );
    00293     double b = a*(1.0-f);
    00294     // double e2 = f*(2.0-f); // unused in this routine
    00295     double phi1 = lat1*RADDEG, lam1 = lon1*RADDEG;
    00296     double sinphi1 = sin(phi1), cosphi1 = cos(phi1);
    00297     double phi2 = lat2*RADDEG, lam2 = lon2*RADDEG;
    00298     double sinphi2 = sin(phi2), cosphi2 = cos(phi2);
    00299
    00300     if( (fabs(lat1-lat2) < testv &&
    00301          ( fabs(lon1-lon2) < testv) || fabs(lat1-90.0) < testv ) )
    00302     {
    00303         // TWO STATIONS ARE IDENTICAL : SET DISTANCE & AZIMUTHS TO ZERO */
    00304         *az1 = 0.0; *az2 = 0.0; *s = 0.0;
    00305         return 0;
    00306     } else if(  fabs(cosphi1) < testv ) {
    00307         // initial point is polar
    00308         int k = geo_inverse_wgs_84( alt, lat2,lon2,lat1,lon1, az1,az2,s );
    00309         k = k; // avoid compiler error since return result is unused
    00310         b = *az1; *az1 = *az2; *az2 = b;
    00311         return 0;
    00312     } else if( fabs(cosphi2) < testv ) {
    00313         // terminal point is polar
    00314         int k = geo_inverse_wgs_84( alt, lat1,lon1,lat1,lon1+180.0,
    00315                                     az1,az2,s );
    00316         k = k; // avoid compiler error since return result is unused
    00317         *s /= 2.0;
    00318         *az2 = *az1 + 180.0;
    00319         if( *az2 > 360.0 ) *az2 -= 360.0;
    00320         return 0;
    00321     } else if( (fabs( fabs(lon1-lon2) - 180 ) < testv) &&
    00322                (fabs(lat1+lat2) < testv) )
    00323     {
    00324         // Geodesic passes through the pole (antipodal)
    00325         double s1,s2;
    00326         geo_inverse_wgs_84( alt, lat1,lon1, lat1,lon2, az1,az2, &s1 );
    00327         geo_inverse_wgs_84( alt, lat2,lon2, lat1,lon2, az1,az2, &s2 );
    00328         *az2 = *az1;
    00329         *s = s1 + s2;
    00330         return 0;
    00331     } else {
    00332         // antipodal and polar points don't get here
    00333         double dlam = lam2 - lam1, dlams = dlam;
    00334         double sdlams,cdlams, sig,sinsig,cossig, sinaz,
    00335             cos2saz, c2sigm;
    00336         double tc,temp, us,rnumer,denom, ta,tb;
    00337         double cosu1,sinu1, sinu2,cosu2;
    00338
    00339         // Reduced latitudes
    00340         temp = (1.0-f)*sinphi1/cosphi1;
    00341         cosu1 = 1.0/sqrt(1.0+temp*temp);
    00342         sinu1 = temp*cosu1;
    00343         temp = (1.0-f)*sinphi2/cosphi2;
    00344         cosu2 = 1.0/sqrt(1.0+temp*temp);
    00345         sinu2 = temp*cosu2;
    00346
    00347         do {
    00348             sdlams = sin(dlams), cdlams = cos(dlams);
    00349             sinsig = sqrt(cosu2*cosu2*sdlams*sdlams+
    00350                           (cosu1*sinu2-sinu1*cosu2*cdlams)*
    00351                           (cosu1*sinu2-sinu1*cosu2*cdlams));
    00352             cossig = sinu1*sinu2+cosu1*cosu2*cdlams;
    00353
    00354             sig = atan2(sinsig,cossig);
    00355             sinaz = cosu1*cosu2*sdlams/sinsig;
    00356             cos2saz = 1.0-sinaz*sinaz;
    00357             c2sigm = (sinu1 == 0.0 || sinu2 == 0.0 ? cossig :
    00358                       cossig-2.0*sinu1*sinu2/cos2saz);
    00359             tc = f*cos2saz*(4.0+f*(4.0-3.0*cos2saz))/16.0;
    00360             temp = dlams;
    00361             dlams = dlam+(1.0-tc)*f*sinaz*
    00362                 (sig+tc*sinsig*
    00363                  (c2sigm+tc*cossig*(-1.0+2.0*c2sigm*c2sigm)));
    00364             if (fabs(dlams) > GEOD_INV_PI && iter++ > 50) {
    00365                 return iter;
    00366             }
    00367         } while ( fabs(temp-dlams) > testv);
    00368
    00369         us = cos2saz*(a*a-b*b)/(b*b); // !!
    00370         // BACK AZIMUTH FROM NORTH
    00371         rnumer = -(cosu1*sdlams);
    00372         denom = sinu1*cosu2-cosu1*sinu2*cdlams;
    00373         *az2 = atan2(rnumer,denom)/RADDEG;
    00374         if( fabs(*az2) < testv ) *az2 = 0.0;
    00375         if(*az2 < 0.0) *az2 += 360.0;
    00376
    00377         // FORWARD AZIMUTH FROM NORTH
    00378         rnumer = cosu2*sdlams;
    00379         denom = cosu1*sinu2-sinu1*cosu2*cdlams;
    00380         *az1 = atan2(rnumer,denom)/RADDEG;
    00381         if( fabs(*az1) < testv ) *az1 = 0.0;
    00382         if(*az1 < 0.0) *az1 += 360.0;
    00383
    00384         // Terms a & b
    00385         ta = 1.0+us*(4096.0+us*(-768.0+us*(320.0-175.0*us)))/
    00386             16384.0;
    00387         tb = us*(256.0+us*(-128.0+us*(74.0-47.0*us)))/1024.0;
    00388
    00389         // GEODETIC DISTANCE
    00390         *s = b*ta*(sig-tb*sinsig*
    00391                    (c2sigm+tb*(cossig*(-1.0+2.0*c2sigm*c2sigm)-tb*
    00392                                c2sigm*(-3.0+4.0*sinsig*sinsig)*
    00393                                (-3.0+4.0*c2sigm*c2sigm)/6.0)/
    00394                     4.0));
    00395         return 0;
    00396     }
    00397 }
    00398
    00399
    00400 /***************************************************************************
    00401
    00402         TITLE:  ls_geodesy
    00403
    00404 ----------------------------------------------------------------------------
    00405
    00406         FUNCTION:       Converts geocentric coordinates to geodetic positions
    00407
    00408 ----------------------------------------------------------------------------
    00409
    00410         MODULE STATUS:  developmental
    00411
    00412 ----------------------------------------------------------------------------
    00413
    00414         GENEALOGY:      Written as part of LaRCSim project by E. B. Jackson
    00415
    00416 ----------------------------------------------------------------------------
    00417
    00418         DESIGNED BY:    E. B. Jackson
    00419
    00420         CODED BY:       E. B. Jackson
    00421
    00422         MAINTAINED BY:  E. B. Jackson
    00423
    00424 ----------------------------------------------------------------------------
    00425
    00426         MODIFICATION HISTORY:
    00427
    00428         DATE    PURPOSE                                         BY
    00429
    00430         930208  Modified to avoid singularity near polar region.        EBJ
    00431         930602  Moved backwards calcs here from ls_step.                EBJ
    00432         931214  Changed erroneous Latitude and Altitude variables to
    00433                 *lat_geod and *alt in routine ls_geoc_to_geod.          EBJ
    00434         940111  Changed header files from old ls_eom.h style to ls_types,
    00435                 and ls_constants.  Also replaced old DATA type with new
    00436                 SCALAR type.                                            EBJ
    00437
    00438         CURRENT RCS HEADER:
    00439
    00440 $Header: /h/curt/CVS/SimGear-0.0/SimGear/simgear/math/sg_geodesy.cxx,v 1.10 2001/03/24 05:02:15 curt Exp $
    00441  * Revision 1.5  1994/01/11  18:47:05  bjax
    00442  * Changed include files to use types and constants, not ls_eom.h
    00443  * Also changed DATA type to SCALAR type.
    00444  *
    00445  * Revision 1.4  1993/12/14  21:06:47  bjax
    00446  * Removed global variable references Altitude and Latitude.   EBJ
    00447  *
    00448  * Revision 1.3  1993/06/02  15:03:40  bjax
    00449  * Made new subroutine for calculating geodetic to geocentric; changed name
    00450  * of forward conversion routine from ls_geodesy to ls_geoc_to_geod.
    00451  *
    00452
    00453 ----------------------------------------------------------------------------
    00454
    00455         REFERENCES:
    00456
    00457                 [ 1]    Stevens, Brian L.; and Lewis, Frank L.: "Aircraft
    00458                         Control and Simulation", Wiley and Sons, 1992.
    00459                         ISBN 0-471-61397-5
    00460
    00461
    00462 ----------------------------------------------------------------------------
    00463
    00464         CALLED BY:      ls_aux
    00465
    00466 ----------------------------------------------------------------------------
    00467
    00468         CALLS TO:
    00469
    00470 ----------------------------------------------------------------------------
    00471
    00472         INPUTS:
    00473                 lat_geoc        Geocentric latitude, radians, + = North
    00474                 radius          C.G. radius to earth center, ft
    00475
    00476 ----------------------------------------------------------------------------
    00477
    00478         OUTPUTS:
    00479                 lat_geod        Geodetic latitude, radians, + = North
    00480                 alt             C.G. altitude above mean sea level, ft
    00481                 sea_level_r     radius from earth center to sea level at
    00482                                 local vertical (surface normal) of C.G.
    00483
    00484 --------------------------------------------------------------------------*/
    00485
    00486

@ 1.3 log @Oops, try again. @ text @d495 1 a495 1