text: /******************************************************************************
   * Project:  PROJ.4
   * Purpose:  Implementation of the aitoff (Aitoff) and wintri (Winkel Tripel)
   *           projections.
   * Author:   Gerald Evenden (1995)
   *           Drazen Tutic, Lovro Gradiser (2015) - add inverse
   *           Thomas Knudsen (2016) - revise/add regression tests
   *
   ******************************************************************************
   * Copyright (c) 1995, Gerald Evenden
   *
   * Permission is hereby granted, free of charge, to any person obtaining a
   * copy of this software and associated documentation files (the "Software"),
   * to deal in the Software without restriction, including without limitation
   * the rights to use, copy, modify, merge, publish, distribute, sublicense,
   * and/or sell copies of the Software, and to permit persons to whom the
   * Software is furnished to do so, subject to the following conditions:
   *
   * The above copyright notice and this permission notice shall be included
   * in all copies or substantial portions of the Software.
   *
   * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
   * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
   * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
   * THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
   * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
   * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
   * DEALINGS IN THE SOFTWARE.
   *****************************************************************************/
  
  #define PJ_LIB__
  #include <projects.h>
  
  
  struct pj_opaque {
  	double	cosphi1;
  	int		mode;
  };
  
  
  PROJ_HEAD(aitoff, "Aitoff") "\n\tMisc Sph";
  PROJ_HEAD(wintri, "Winkel Tripel") "\n\tMisc Sph\n\tlat_1";
  
  
  
  #if 0
  FORWARD(s_forward); /* spheroid */
  #endif
  
  
  static XY s_forward (LP lp, PJ *P) {           /* Spheroidal, forward */
      XY xy = {0.0,0.0};
      struct pj_opaque *Q = P->opaque;
  	double c, d;
  
  	if((d = acos(cos(lp.phi) * cos(c = 0.5 * lp.lam)))) {/* basic Aitoff */
  		xy.x = 2. * d * cos(lp.phi) * sin(c) * (xy.y = 1. / sin(d));
  		xy.y *= d * sin(lp.phi);
  	} else
  		xy.x = xy.y = 0.;
  	if (Q->mode) { /* Winkel Tripel */
  		xy.x = (xy.x + lp.lam * Q->cosphi1) * 0.5;
  		xy.y = (xy.y + lp.phi) * 0.5;
  	}
  	return (xy);
  }
  
  /***********************************************************************************
  *
  * Inverse functions added by Drazen Tutic and Lovro Gradiser based on paper:
  *
  * I.Özbug Biklirici and Cengizhan Ipbüker. A General Algorithm for the Inverse
  * Transformation of Map Projections Using Jacobian Matrices. In Proceedings of the
  * Third International Symposium Mathematical & Computational Applications,
  * pages 175{182, Turkey, September 2002.
  *
  * Expected accuracy is defined by EPSILON = 1e-12. Should be appropriate for
  * most applications of Aitoff and Winkel Tripel projections.
  *
  * Longitudes of 180W and 180E can be mixed in solution obtained.
  *
  * Inverse for Aitoff projection in poles is undefined, longitude value of 0 is assumed.
  *
  * Contact : dtutic@geof.hr
  * Date: 2015-02-16
  *
  ************************************************************************************/
  
  static LP s_inverse (XY xy, PJ *P) {           /* Spheroidal, inverse */
      LP lp = {0.0,0.0};
      struct pj_opaque *Q = P->opaque;
      int iter, MAXITER = 10, round = 0, MAXROUND = 20;
  	double EPSILON = 1e-12, D, C, f1, f2, f1p, f1l, f2p, f2l, dp, dl, sl, sp, cp, cl, x, y;
  
  	if ((fabs(xy.x) < EPSILON) && (fabs(xy.y) < EPSILON )) { lp.phi = 0.; lp.lam = 0.; return lp; }
  
  	/* intial values for Newton-Raphson method */
  	lp.phi = xy.y; lp.lam = xy.x;
  	do {
  		iter = 0;
  		do {
  			sl = sin(lp.lam * 0.5); cl = cos(lp.lam * 0.5);
  			sp = sin(lp.phi); cp = cos(lp.phi);
  			D = cp * cl;
   		      	C = 1. - D * D;
  			D = acos(D) / pow(C, 1.5);
  	       		f1 = 2. * D * C * cp * sl;
  		       	f2 = D * C * sp;
  		       	f1p = 2.* (sl * cl * sp * cp / C - D * sp * sl);
  		       	f1l = cp * cp * sl * sl / C + D * cp * cl * sp * sp;
  		    	f2p = sp * sp * cl / C + D * sl * sl * cp;
  		      	f2l = 0.5 * (sp * cp * sl / C - D * sp * cp * cp * sl * cl);
  		      	if (Q->mode) { /* Winkel Tripel */
  				f1 = 0.5 * (f1 + lp.lam * Q->cosphi1);
  				f2 = 0.5 * (f2 + lp.phi);
  				f1p *= 0.5;
  				f1l = 0.5 * (f1l + Q->cosphi1);
  				f2p = 0.5 * (f2p + 1.);
  				f2l *= 0.5;
  			}
  			f1 -= xy.x; f2 -= xy.y;
  			dl = (f2 * f1p - f1 * f2p) / (dp = f1p * f2l - f2p * f1l);
  			dp = (f1 * f2l - f2 * f1l) / dp;
  			while (dl > M_PI) dl -= M_PI; /* set to interval [-M_PI, M_PI]  */
  			while (dl < -M_PI) dl += M_PI; /* set to interval [-M_PI, M_PI]  */
  			lp.phi -= dp;	lp.lam -= dl;
  		} while ((fabs(dp) > EPSILON || fabs(dl) > EPSILON) && (iter++ < MAXITER));
  		if (lp.phi > M_PI_2) lp.phi -= 2.*(lp.phi-M_PI_2); /* correct if symmetrical solution for Aitoff */
  		if (lp.phi < -M_PI_2) lp.phi -= 2.*(lp.phi+M_PI_2); /* correct if symmetrical solution for Aitoff */
  		if ((fabs(fabs(lp.phi) - M_PI_2) < EPSILON) && (!Q->mode)) lp.lam = 0.; /* if pole in Aitoff, return longitude of 0 */
  
  		/* calculate x,y coordinates with solution obtained */
  		if((D = acos(cos(lp.phi) * cos(C = 0.5 * lp.lam)))) {/* Aitoff */
  			x = 2. * D * cos(lp.phi) * sin(C) * (y = 1. / sin(D));
  			y *= D * sin(lp.phi);
  		} else
  			x = y = 0.;
  		if (Q->mode) { /* Winkel Tripel */
  			x = (x + lp.lam * Q->cosphi1) * 0.5;
  			y = (y + lp.phi) * 0.5;
  		}
  	/* if too far from given values of x,y, repeat with better approximation of phi,lam */
  	} while (((fabs(xy.x-x) > EPSILON) || (fabs(xy.y-y) > EPSILON)) && (round++ < MAXROUND));
  
  	if (iter == MAXITER && round == MAXROUND) fprintf(stderr, "Warning: Accuracy of 1e-12 not reached. Last increments: dlat=%e and dlon=%e\n", dp, dl);
  
  	return lp;
  }
  
  
  
  static void *freeup_new (PJ *P) {                        /* Destructor */
      if (0==P)
          return 0;
      if (0==P->opaque)
          return pj_dealloc (P);
  
      pj_dealloc (P->opaque);
      return pj_dealloc(P);
  }
  
  static void freeup (PJ *P) {
      freeup_new (P);
      return;
  }
  
  static PJ *setup(PJ *P) {
  	P->inv = s_inverse;
  	P->fwd = s_forward;
  	P->es = 0.;
  	return P;
  }
  
  
  PJ *PROJECTION(aitoff) {
      struct pj_opaque *Q = pj_calloc (1, sizeof (struct pj_opaque));
      if (0==Q)
          return freeup_new (P);
      P->opaque = Q;
  
  	Q->mode = 0;
      return setup(P);
  }
  
  
  PJ *PROJECTION(wintri) {
      struct pj_opaque *Q = pj_calloc (1, sizeof (struct pj_opaque));
      if (0==Q)
          return freeup_new (P);
      P->opaque = Q;
  
  	Q->mode = 1;
  	if (pj_param(P->ctx, P->params, "tlat_1").i) {
  		if ((Q->cosphi1 = cos(pj_param(P->ctx, P->params, "rlat_1").f)) == 0.)
  			E_ERROR(-22)
      }
  	else /* 50d28' or acos(2/pi) */
  		Q->cosphi1 = 0.636619772367581343;
      return setup(P);
  }
  
  
  #ifdef PJ_OMIT_SELFTEST
  int pj_aitoff_selftest (void) {return 0;}
  #else
  
  int pj_aitoff_selftest (void) {
      double tolerance_lp = 1e-10;
      double tolerance_xy = 1e-7;
  
      char s_args[] = {"+proj=aitoff   +a=6400000    +lat_1=0 +lat_2=2"};
  
      LP fwd_in[] = {
          { 2, 1},
          { 2,-1},
          {-2, 1},
          {-2,-1}
      };
  
  
      XY s_fwd_expect[] = {
          {223379.45881169615,  111706.74288385305},
          {223379.45881169615,  -111706.74288385305},
          {-223379.45881169615,  111706.74288385305},
          {-223379.45881169615,  -111706.74288385305},
      };
  
      XY inv_in[] = {
          { 200, 100},
          { 200,-100},
          {-200, 100},
          {-200,-100}
      };
  
  
      LP s_inv_expect[] = {
          {0.0017904931100388164,  0.00089524655491012516},
          {0.0017904931100388164,  -0.00089524655491012516},
          {-0.0017904931100388164,  0.00089524655491012516},
          {-0.0017904931100388164,  -0.00089524655491012516},
      };
  
      return pj_generic_selftest (0, s_args, tolerance_xy, tolerance_lp, 4, 4, fwd_in, 0, s_fwd_expect, inv_in, 0, s_inv_expect);
  }
  
  
  #endif
  
  
  
  #ifdef PJ_OMIT_SELFTEST
  int pj_wintri_selftest (void) {return 0;}
  #else
  
  int pj_wintri_selftest (void) {
      double tolerance_lp = 1e-10;
      double tolerance_xy = 1e-7;
  
      char s_args[] = {"+proj=wintri   +a=6400000    +lat_1=0 +lat_2=2"};
  
      LP fwd_in[] = {
          { 2, 1},
          { 2,-1},
          {-2, 1},
          {-2,-1}
      };
  
      XY s_fwd_expect[] = {
          {223390.80153348515,  111703.90750574505},
          {223390.80153348515,  -111703.90750574505},
          {-223390.80153348515,  111703.90750574505},
          {-223390.80153348515,  -111703.90750574505},
      };
  
      XY inv_in[] = {
          { 200, 100},
          { 200,-100},
          {-200, 100},
          {-200,-100}
      };
  
      LP s_inv_expect[] = {
          {0.0017904931099113196,  0.00089524655490101819},
          {0.0017904931099113196,  -0.00089524655490101819},
          {-0.0017904931099113196,  0.00089524655490101819},
          {-0.0017904931099113196,  -0.00089524655490101819},
      };
  
      return pj_generic_selftest (0, s_args, tolerance_xy, tolerance_lp, 4, 4, fwd_in, 0, s_fwd_expect, inv_in, 0, s_inv_expect);
  }
  
  
  #endif
  