Fix two errors in the n**5 coefficients. Add sixth order coefficients. Fix rounding problems (#222)

git-svn-id: http://svn.osgeo.org/metacrs/proj/trunk@2379 4e78687f-474d-0410-85f9-8d5e500ac6b2

2 files changed, 207 insertions, 165 deletions

diff --git a/ChangeLog b/ChangeLog
index 8b42efed..6a421e1b 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -1,3 +1,8 @@
+2013-07-21  Frank Warmerdam  <warmerdam@pobox.com>
+
+	* src/proj_etmerc.c: Fix two errors in the n**5 coefficients.  Add 
+	sixth order coefficients.  Fix rounding problems (#222)
+
 2013-07-19  Frank Warmerdam  <warmerdam@pobox.com>
 
 	* src/PJ_healpix.c: major update for polar scaling and parms (#219)
diff --git a/src/proj_etmerc.c b/src/proj_etmerc.c
index 3a389600..1097eb3c 100644
--- a/src/proj_etmerc.c
+++ b/src/proj_etmerc.c
@@ -37,14 +37,13 @@
  * Division, Kort og Matrikelstyrelsen (KMS), Copenhagen, Denmark
 */
 
-
 #define PROJ_PARMS__ \
-	double    Qn;    /* Merid. quad., scaled to the projection */ \
-	double    Zb;    /* Radius vector in polar coord. systems  */ \
-	double    cgb[5]; /* Constants for Gauss -> Geo lat */ \
-	double    cbg[5]; /* Constants for Geo lat -> Gauss */ \
-	double    utg[5]; /* Constants for transv. merc. -> geo */ \
-	double    gtu[5]; /* Constants for geo -> transv. merc. */
+    double    Qn;    /* Merid. quad., scaled to the projection */ \
+    double    Zb;    /* Radius vector in polar coord. systems  */ \
+    double    cgb[6]; /* Constants for Gauss -> Geo lat */ \
+    double    cbg[6]; /* Constants for Geo lat -> Gauss */ \
+    double    utg[6]; /* Constants for transv. merc. -> geo */ \
+    double    gtu[6]; /* Constants for geo -> transv. merc. */
 
 #define PROJ_LIB__
 #define PJ_LIB__
@@ -54,199 +53,237 @@
 PROJ_HEAD(etmerc, "Extended Transverse Mercator")
     "\n\tCyl, Sph\n\tlat_ts=(0)\nlat_0=(0)";
 
-#define FABS(x) ((x)<0?-(x):(x))
+#define PROJ_ETMERC_ORDER 6
+
+#ifdef _GNU_SOURCE
+    inline
+#endif
+    static double
+log1py(double x) {              /* Compute log(1+x) accurately */
+    volatile double
+      y = 1 + x,
+      z = y - 1;
+    /* Here's the explanation for this magic: y = 1 + z, exactly, and z
+     * approx x, thus log(y)/z (which is nearly constant near z = 0) returns
+     * a good approximation to the true log(1 + x)/x.  The multiplication x *
+     * (log(y)/z) introduces little additional error. */
+    return z == 0 ? x : x * log(y) / z;
+}
+
+#ifdef _GNU_SOURCE
+    inline
+#endif
+    static double
+asinhy(double x) {              /* Compute asinh(x) accurately */
+    double y = fabs(x);         /* Enforce odd parity */
+    y = log1py(y * (1 + y/(hypot(1.0, y) + 1)));
+    return x < 0 ? -y : y;
+}
 
 #ifdef _GNU_SOURCE
     inline
 #endif
-	static double
+    static double
 gatg(double *p1, int len_p1, double B) {
-	double *p;
-	double h = 0., h1, h2 = 0., cos_2B;
+    double *p;
+    double h = 0, h1, h2 = 0, cos_2B;
 
-	cos_2B = 2.*cos(2.0*B);
-   	for (p = p1 + len_p1, h1 = *--p; p - p1; h2 = h1, h1 = h)
-   		h = -h2 + cos_2B*h1 + *--p;
-   	return (B + h*sin(2.0*B));
+    cos_2B = 2*cos(2*B);
+    for (p = p1 + len_p1, h1 = *--p; p - p1; h2 = h1, h1 = h)
+        h = -h2 + cos_2B*h1 + *--p;
+    return (B + h*sin(2*B));
 }
 
 #ifdef _GNU_SOURCE
     inline
 #endif
-	static double
+    static double
 clenS(double *a, int size, double arg_r, double arg_i, double *R, double *I) {
-	double      *p, r, i, hr, hr1, hr2, hi, hi1, hi2;
-	double      sin_arg_r, cos_arg_r, sinh_arg_i, cosh_arg_i;
-	double      exp_arg_i, pxe_arg_i;
+    double      *p, r, i, hr, hr1, hr2, hi, hi1, hi2;
+    double      sin_arg_r, cos_arg_r, sinh_arg_i, cosh_arg_i;
 
-	/* arguments */
-	p = a + size;
+    /* arguments */
+    p = a + size;
 #ifdef _GNU_SOURCE
-	sincos(arg_r, &sin_arg_r, &cos_arg_r);
+    sincos(arg_r, &sin_arg_r, &cos_arg_r);
 #else
-	sin_arg_r  = sin(arg_r);
-	cos_arg_r  = cos(arg_r);
+    sin_arg_r  = sin(arg_r);
+    cos_arg_r  = cos(arg_r);
 #endif
-    exp_arg_i  = exp( arg_i);
-    pxe_arg_i  = exp(-arg_i);
-	sinh_arg_i = (exp_arg_i-pxe_arg_i)/2;
-	cosh_arg_i = (exp_arg_i+pxe_arg_i)/2;
-	r          =  2.0*cos_arg_r*cosh_arg_i;
-	i          = -2.0*sin_arg_r*sinh_arg_i;
-	/* summation loop */
-	for (hi1 = hr1 = hi = 0.0, hr = *--p; a - p;) {
-		hr2 = hr1;
-		hi2 = hi1;
-		hr1 = hr;
-		hi1 = hi;
-		hr  = -hr2 + r*hr1 - i*hi1 + *--p;
-		hi  = -hi2 + i*hr1 + r*hi1;
-	}
-	r   = sin_arg_r*cosh_arg_i;
-	i   = cos_arg_r*sinh_arg_i;
-	*R  = r*hr - i*hi;
-	*I  = r*hi + i*hr;
-	return(*R);
+    sinh_arg_i = sinh(arg_i);
+    cosh_arg_i = cosh(arg_i);
+    r          =  2*cos_arg_r*cosh_arg_i;
+    i          = -2*sin_arg_r*sinh_arg_i;
+    /* summation loop */
+    for (hi1 = hr1 = hi = 0, hr = *--p; a - p;) {
+        hr2 = hr1;
+        hi2 = hi1;
+        hr1 = hr;
+        hi1 = hi;
+        hr  = -hr2 + r*hr1 - i*hi1 + *--p;
+        hi  = -hi2 + i*hr1 + r*hi1;
+    }
+    r   = sin_arg_r*cosh_arg_i;
+    i   = cos_arg_r*sinh_arg_i;
+    *R  = r*hr - i*hi;
+    *I  = r*hi + i*hr;
+    return(*R);
 }
-	static double
+    static double
 clens(double *a, int size, double arg_r) {
-	double      *p, r, hr, hr1, hr2, cos_arg_r;
-
-	p = a + size;
-	cos_arg_r  = cos(arg_r);
-	r          =  2.0*cos_arg_r;
-	/* summation loop */
-	for (hr1 = 0.0, hr = *--p; a - p;) {
-		hr2 = hr1;
-		hr1 = hr;
-		hr  = -hr2 + r*hr1 + *--p;
-	}
-	return(sin(arg_r)*hr);
-}
+    double      *p, r, hr, hr1, hr2, cos_arg_r;
 
+    p = a + size;
+    cos_arg_r  = cos(arg_r);
+    r          =  2*cos_arg_r;
+    /* summation loop */
+    for (hr1 = 0, hr = *--p; a - p;) {
+        hr2 = hr1;
+        hr1 = hr;
+        hr  = -hr2 + r*hr1 + *--p;
+    }
+    return(sin(arg_r)*hr);
+}
 
 FORWARD(e_forward); /* ellipsoid */
-	double sin_Cn, cos_Cn, cos_Ce, sin_Ce, dCn, dCe;
-	double Cn = lp.phi, Ce = lp.lam;
+    double sin_Cn, cos_Cn, cos_Ce, sin_Ce, dCn, dCe;
+    double Cn = lp.phi, Ce = lp.lam;
 
-	/* ell. LAT, LNG -> Gaussian LAT, LNG */
-	Cn  = gatg(P->cbg, 5, Cn);
-	/* Gaussian LAT, LNG -> compl. sph. LAT */
+    /* ell. LAT, LNG -> Gaussian LAT, LNG */
+    Cn  = gatg(P->cbg, PROJ_ETMERC_ORDER, Cn);
+    /* Gaussian LAT, LNG -> compl. sph. LAT */
 #ifdef _GNU_SOURCE
-	sincos(Cn, &sin_Cn, &cos_Cn);
-	sincos(Ce, &sin_Ce, &cos_Ce);
+    sincos(Cn, &sin_Cn, &cos_Cn);
+    sincos(Ce, &sin_Ce, &cos_Ce);
 #else
-	sin_Cn = sin(Cn);
-	cos_Cn = cos(Cn);
-	sin_Ce = sin(Ce);
-	cos_Ce = cos(Ce);
+    sin_Cn = sin(Cn);
+    cos_Cn = cos(Cn);
+    sin_Ce = sin(Ce);
+    cos_Ce = cos(Ce);
 #endif
 
-	Cn     = atan2(sin_Cn, cos_Ce*cos_Cn);
-	Ce     = atan2(sin_Ce*cos_Cn, hypot(sin_Cn, cos_Cn*cos_Ce));
-	/* compl. sph. N, E -> ell. norm. N, E */
-	Ce  = log(tan(FORTPI + Ce*0.5));
-	Cn += clenS(P->gtu, 5, 2.*Cn, 2.*Ce, &dCn, &dCe);
-	Ce += dCe;
-	if (FABS(Ce) <= 2.623395162778) {
-		xy.y  = P->Qn * Cn + P->Zb;  /* Northing */
-		xy.x  = P->Qn * Ce;  /* Easting  */
-	} else
-		xy.x = xy.y = HUGE_VAL;
-	return (xy);
+    Cn     = atan2(sin_Cn, cos_Ce*cos_Cn);
+    Ce     = atan2(sin_Ce*cos_Cn, hypot(sin_Cn, cos_Cn*cos_Ce));
+    /* compl. sph. N, E -> ell. norm. N, E */
+    Ce  = asinhy(tan(Ce));     /* Replaces: Ce  = log(tan(FORTPI + Ce*0.5)); */
+    Cn += clenS(P->gtu, PROJ_ETMERC_ORDER, 2*Cn, 2*Ce, &dCn, &dCe);
+    Ce += dCe;
+    if (fabs(Ce) <= 2.623395162778) {
+        xy.y  = P->Qn * Cn + P->Zb;  /* Northing */
+        xy.x  = P->Qn * Ce;  /* Easting  */
+    } else
+        xy.x = xy.y = HUGE_VAL;
+    return (xy);
 }
 
-
 INVERSE(e_inverse); /* ellipsoid */
-	double sin_Cn, cos_Cn, cos_Ce, sin_Ce, dCn, dCe;
-	double Cn = xy.y, Ce = xy.x;
-
-	/* normalize N, E */
-	Cn = (Cn - P->Zb)/P->Qn;
-	Ce = Ce/P->Qn;
-	if (FABS(Ce) <= 2.623395162778) { /* 150 degrees */
-	/* norm. N, E -> compl. sph. LAT, LNG */
-		Cn += clenS(P->utg, 5, 2.*Cn, 2.*Ce, &dCn, &dCe);
-		Ce += dCe;
-		Ce = 2.0*(atan(exp(Ce)) - FORTPI);
-		/* compl. sph. LAT -> Gaussian LAT, LNG */
+    double sin_Cn, cos_Cn, cos_Ce, sin_Ce, dCn, dCe;
+    double Cn = xy.y, Ce = xy.x;
+
+    /* normalize N, E */
+    Cn = (Cn - P->Zb)/P->Qn;
+    Ce = Ce/P->Qn;
+    if (fabs(Ce) <= 2.623395162778) { /* 150 degrees */
+    /* norm. N, E -> compl. sph. LAT, LNG */
+        Cn += clenS(P->utg, PROJ_ETMERC_ORDER, 2*Cn, 2*Ce, &dCn, &dCe);
+        Ce += dCe;
+        Ce = atan(sinh(Ce)); /* Replaces: Ce = 2*(atan(exp(Ce)) - FORTPI); */
+        /* compl. sph. LAT -> Gaussian LAT, LNG */
 #ifdef _GNU_SOURCE
-    	sincos(Cn, &sin_Cn, &cos_Cn);
-    	sincos(Ce, &sin_Ce, &cos_Ce);
+        sincos(Cn, &sin_Cn, &cos_Cn);
+        sincos(Ce, &sin_Ce, &cos_Ce);
 #else
-    	sin_Cn = sin(Cn);
-    	cos_Cn = cos(Cn);
-    	sin_Ce = sin(Ce);
-    	cos_Ce = cos(Ce);
+        sin_Cn = sin(Cn);
+        cos_Cn = cos(Cn);
+        sin_Ce = sin(Ce);
+        cos_Ce = cos(Ce);
 #endif
-		Ce     = atan2(sin_Ce, cos_Ce*cos_Cn);
-		Cn     = atan2(sin_Cn*cos_Ce, hypot(sin_Ce, cos_Ce*cos_Cn));
-		/* Gaussian LAT, LNG -> ell. LAT, LNG */
-		lp.phi = gatg(P->cgb,  5, Cn);
-		lp.lam = Ce;
-	}
-	else
-		lp.phi = lp.lam = HUGE_VAL;
-	return (lp);
+        Ce     = atan2(sin_Ce, cos_Ce*cos_Cn);
+        Cn     = atan2(sin_Cn*cos_Ce, hypot(sin_Ce, cos_Ce*cos_Cn));
+        /* Gaussian LAT, LNG -> ell. LAT, LNG */
+        lp.phi = gatg(P->cgb,  PROJ_ETMERC_ORDER, Cn);
+        lp.lam = Ce;
+    }
+    else
+        lp.phi = lp.lam = HUGE_VAL;
+    return (lp);
 }
 
-
 FREEUP; if (P) free(P); }
 
-
 ENTRY0(etmerc)
-	double f, n, np, Z;
-
-	if (P->es <= 0.) E_ERROR(-34);
-	f = 1. - sqrt(1. - P->es);
-	/* third flattening */
-	np = n = f/(2.0 - f);
-
-	/* COEF. OF TRIG SERIES GEO <-> GAUSS */
-	/* cgb := Gaussian -> Geodetic, KW p190 - 191 (61) - (62) */
-	/* cbg := Geodetic -> Gaussian, KW p186 - 187 (51) - (52) */
-	/* 5 degree : Engsager and Poder: ICC2007 */
-	P->cgb[0] = n*( 2.0 + n*(-2.0/3.0  + n*(-2.0      + n*(116.0/45.0 + n*(26.0/45.0)))));
-	P->cbg[0] = n*(-2.0 + n*( 2.0/3.0  + n*( 4.0/3.0  + n*(-82.0/45.0 + n*(32.0/45.0)))));
-	np     *= n;
-	P->cgb[1] = np*(7.0/3.0     + n*( -8.0/5.0  + n*(-227.0/45.0 + n*(2704.0/315.0))));
-	P->cbg[1] = np*(5.0/3.0     + n*(-16.0/15.0 + n*( -13.0/ 9.0 + n*( 904.0/315.0))));
-	np     *= n;
-	P->cgb[2] = np*( 56.0/15.0  + n*(-136.0/35.0 + n*(1262.0/105.0)));
-	P->cbg[2] = np*(-26.0/15.0  + n*(  34.0/21.0 + n*(   8.0/  5.0)));
-	np     *= n;
-	P->cgb[3] = np*(4279.0/630.0 + n*(-322.0/35.0));
-	P->cbg[3] = np*(1237.0/630.0 + n*( -12.0/ 5.0));
-	np     *= n;
-	P->cgb[4] = np*(4174.0/315.0);
-	P->cbg[4] = np*(-734.0/315.0);
-
-	/* Constants of the projections */
-	/* Transverse Mercator (UTM, ITM, etc) */
-	np = n*n;
-	/* Norm. mer. quad, K&W p.50 (96), p.19 (38b), p.5 (2) */
-	P->Qn = P->k0/(1 + n) * (1. + np*(1./4.0 + np*(1./64.0 + np/256.0)));
-	/* coef of trig series */
-	/* utg := ell. N, E -> sph. N, E,  KW p194 (65) */
-	/* gtu := sph. N, E -> ell. N, E,  KW p196 (69) */
-	P->utg[0] = n*(-0.5  + n*( 2.0/3.0 + n*(-37.0/96.0 + n*( 1.0/360.0 + n*(  81.0/512.0)))));
-	P->gtu[0] = n*( 0.5  + n*(-2.0/3.0 + n*(  5.0/16.0 + n*(41.0/180.0 + n*(-127.0/288.0)))));
-	P->utg[1] = np*(-1.0/48.0 + n*(-1.0/15.0 + n*(437.0/1440.0 + n*(-46.0/105.0))));
-	P->gtu[1] = np*(13.0/48.0 + n*(-3.0/5.0  + n*(557.0/1440.0 + n*(281.0/630.0))));
-	np      *= n;
-	P->utg[2] = np*(-17.0/480.0 + n*(  37.0/840.0 + n*(  209.0/ 4480.0)));
-	P->gtu[2] = np*( 61.0/240.0 + n*(-103.0/140.0 + n*(15061.0/26880.0)));
-	np      *= n;
-	P->utg[3] = np*(-4397.0/161280.0 + n*(  11.0/504.0));
-	P->gtu[3] = np*(49561.0/161280.0 + n*(-179.0/168.0));
-	np     *= n;
-	P->utg[4] = np*(-4583.0/161280.0);
-	P->gtu[4] = np*(34729.0/ 80640.0);
-	/* Gaussian latitude value of the origin latitude */
-	Z = gatg(P->cbg, 5, P->phi0);
-	/* Origin northing minus true northing at the origin latitude */
-	/* i.e. true northing = N - P->Zb                         */
-	P->Zb  = - P->Qn*(Z + clens(P->gtu, 5, 2.0*Z));
-	P->inv = e_inverse;
-	P->fwd = e_forward;
+    double f, n, np, Z;
+
+    if (P->es <= 0) E_ERROR(-34);
+    f = P->es / (1 + sqrt(1 -  P->es)); /* Replaces: f = 1 - sqrt(1-P->es); */
+    /* third flattening */
+    np = n = f/(2 - f);
+
+    /* COEF. OF TRIG SERIES GEO <-> GAUSS */
+    /* cgb := Gaussian -> Geodetic, KW p190 - 191 (61) - (62) */
+    /* cbg := Geodetic -> Gaussian, KW p186 - 187 (51) - (52) */
+    /* PROJ_ETMERC_ORDER = 6th degree : Engsager and Poder: ICC2007 */
+    P->cgb[0] = n*( 2 + n*(-2/3.0  + n*(-2      + n*(116/45.0 + n*(26/45.0 +
+                n*(-2854/675.0 ))))));
+    P->cbg[0] = n*(-2 + n*( 2/3.0  + n*( 4/3.0  + n*(-82/45.0 + n*(32/45.0 +
+                n*( 4642/4725.0))))));
+    np     *= n;
+    P->cgb[1] = np*(7/3.0 + n*( -8/5.0  + n*(-227/45.0 + n*(2704/315.0 +
+                n*( 2323/945.0)))));
+    P->cbg[1] = np*(5/3.0 + n*(-16/15.0 + n*( -13/9.0  + n*( 904/315.0 +
+                n*(-1522/945.0)))));
+    np     *= n;
+    /* n^5 coeff corrected from 1262/105 -> -1262/105 */
+    P->cgb[2] = np*( 56/15.0  + n*(-136/35.0 + n*(-1262/105.0 +
+                n*( 73814/2835.0))));
+    P->cbg[2] = np*(-26/15.0  + n*(  34/21.0 + n*(    8/5.0   +
+                n*(-12686/2835.0))));
+    np     *= n;
+    /* n^5 coeff corrected from 322/35 -> 332/35 */
+    P->cgb[3] = np*(4279/630.0 + n*(-332/35.0 + n*(-399572/14175.0)));
+    P->cbg[3] = np*(1237/630.0 + n*( -12/5.0  + n*( -24832/14175.0)));
+    np     *= n;
+    P->cgb[4] = np*(4174/315.0 + n*(-144838/6237.0 ));
+    P->cbg[4] = np*(-734/315.0 + n*( 109598/31185.0));
+    np     *= n;
+    P->cgb[5] = np*(601676/22275.0 );
+    P->cbg[5] = np*(444337/155925.0);
+
+    /* Constants of the projections */
+    /* Transverse Mercator (UTM, ITM, etc) */
+    np = n*n;
+    /* Norm. mer. quad, K&W p.50 (96), p.19 (38b), p.5 (2) */
+    P->Qn = P->k0/(1 + n) * (1 + np*(1/4.0 + np*(1/64.0 + np/256.0)));
+    /* coef of trig series */
+    /* utg := ell. N, E -> sph. N, E,  KW p194 (65) */
+    /* gtu := sph. N, E -> ell. N, E,  KW p196 (69) */
+    P->utg[0] = n*(-0.5  + n*( 2/3.0 + n*(-37/96.0 + n*( 1/360.0 +
+                n*(  81/512.0 + n*(-96199/604800.0))))));
+    P->gtu[0] = n*( 0.5  + n*(-2/3.0 + n*(  5/16.0 + n*(41/180.0 +
+                n*(-127/288.0 + n*(  7891/37800.0 ))))));
+    P->utg[1] = np*(-1/48.0 + n*(-1/15.0 + n*(437/1440.0 + n*(-46/105.0 +
+                n*( 1118711/3870720.0)))));
+    P->gtu[1] = np*(13/48.0 + n*(-3/5.0  + n*(557/1440.0 + n*(281/630.0 +
+                n*(-1983433/1935360.0)))));
+    np      *= n;
+    P->utg[2] = np*(-17/480.0 + n*(  37/840.0 + n*(  209/4480.0  +
+                n*( -5569/90720.0 ))));
+    P->gtu[2] = np*( 61/240.0 + n*(-103/140.0 + n*(15061/26880.0 +
+                n*(167603/181440.0))));
+    np      *= n;
+    P->utg[3] = np*(-4397/161280.0 + n*(  11/504.0 + n*( 830251/7257600.0)));
+    P->gtu[3] = np*(49561/161280.0 + n*(-179/168.0 + n*(6601661/7257600.0)));
+    np     *= n;
+    P->utg[4] = np*(-4583/161280.0 + n*(  108847/3991680.0));
+    P->gtu[4] = np*(34729/80640.0  + n*(-3418889/1995840.0));
+    np     *= n;
+    P->utg[5] = np*(-20648693/638668800.0);
+    P->gtu[5] = np*(212378941/319334400.0);
+    /* Gaussian latitude value of the origin latitude */
+    Z = gatg(P->cbg, PROJ_ETMERC_ORDER, P->phi0);
+    /* Origin northing minus true northing at the origin latitude */
+    /* i.e. true northing = N - P->Zb                         */
+    P->Zb  = - P->Qn*(Z + clens(P->gtu, PROJ_ETMERC_ORDER, 2*Z));
+    P->inv = e_inverse;
+    P->fwd = e_forward;
 ENDENTRY(P)