src/2geom/elliptical-arc.cpp

   1 /*
   2  * SVG Elliptical Arc Class
   3  *
   4  * Copyright 2008  Marco Cecchetti <mrcekets at gmail.com>
   5  *
   6  * This library is free software; you can redistribute it and/or
   7  * modify it either under the terms of the GNU Lesser General Public
   8  * License version 2.1 as published by the Free Software Foundation
   9  * (the "LGPL") or, at your option, under the terms of the Mozilla
  10  * Public License Version 1.1 (the "MPL"). If you do not alter this
  11  * notice, a recipient may use your version of this file under either
  12  * the MPL or the LGPL.
  13  *
  14  * You should have received a copy of the LGPL along with this library
  15  * in the file COPYING-LGPL-2.1; if not, write to the Free Software
  16  * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
  17  * You should have received a copy of the MPL along with this library
  18  * in the file COPYING-MPL-1.1
  19  *
  20  * The contents of this file are subject to the Mozilla Public License
  21  * Version 1.1 (the "License"); you may not use this file except in
  22  * compliance with the License. You may obtain a copy of the License at
  23  * http://www.mozilla.org/MPL/
  24  *
  25  * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
  26  * OF ANY KIND, either express or implied. See the LGPL or the MPL for
  27  * the specific language governing rights and limitations.
  28  */
  29
  30
  31 #include "path.h"
  32 #include "angle.h"
  33
  34 #include "poly.h"
  35
  36 #include <cfloat>
  37
  38
  39
  40
  41 namespace Geom
  42 {
  43
  44
  45 Rect EllipticalArc::boundsExact() const
  46 {
  47         std::vector<double> extremes(4);
  48         double cosrot = std::cos(rotation_angle());
  49         double sinrot = std::sin(rotation_angle());
  50         extremes[0] = std::atan2( -ray(Y) * sinrot, ray(X) * cosrot );
  51         extremes[1] = extremes[0] + M_PI;
  52         if ( extremes[0] < 0 ) extremes[0] += 2*M_PI;
  53         extremes[2] = std::atan2( ray(Y) * cosrot, ray(X) * sinrot );
  54         extremes[3] = extremes[2] + M_PI;
  55         if ( extremes[2] < 0 ) extremes[2] += 2*M_PI;
  56
  57
  58         std::vector<double>arc_extremes(4);
  59         arc_extremes[0] = initialPoint()[X];
  60         arc_extremes[1] = finalPoint()[X];
  61         if ( arc_extremes[0] < arc_extremes[1] )
  62                 std::swap(arc_extremes[0], arc_extremes[1]);
  63         arc_extremes[2] = initialPoint()[Y];
  64         arc_extremes[3] = finalPoint()[Y];
  65         if ( arc_extremes[2] < arc_extremes[3] )
  66                 std::swap(arc_extremes[2], arc_extremes[3]);
  67
  68
  69         if ( start_angle() < end_angle() )
  70         {
  71                 if ( sweep_flag() )
  72                 {
  73                         for ( unsigned int i = 0; i < extremes.size(); ++i )
  74                         {
  75                                 if ( start_angle() < extremes[i] && extremes[i] < end_angle() )
  76                                 {
  77                                         arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
  78                                 }
  79                         }
  80                 }
  81                 else
  82                 {
  83                         for ( unsigned int i = 0; i < extremes.size(); ++i )
  84                         {
  85                                 if ( start_angle() > extremes[i] || extremes[i] > end_angle() )
  86                                 {
  87                                         arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
  88                                 }
  89                         }
  90                 }
  91         }
  92         else
  93         {
  94                 if ( sweep_flag() )
  95                 {
  96                         for ( unsigned int i = 0; i < extremes.size(); ++i )
  97                         {
  98                                 if ( start_angle() < extremes[i] || extremes[i] < end_angle() )
  99                                 {
 100                                         arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
 101                                 }
 102                         }
 103                 }
 104                 else
 105                 {
 106                         for ( unsigned int i = 0; i < extremes.size(); ++i )
 107                         {
 108                                 if ( start_angle() > extremes[i] && extremes[i] > end_angle() )
 109                                 {
 110                                         arc_extremes[i] = pointAtAngle(extremes[i])[i >> 1];
 111                                 }
 112                         }
 113                 }
 114         }
 115
 116         return Rect( Point(arc_extremes[1], arc_extremes[3]) ,
 117                              Point(arc_extremes[0], arc_extremes[2]) );
 118
 119 }
 120
 121
 122 std::vector<double>
 123 EllipticalArc::roots(double v, Dim2 d) const
 124 {
 125         if ( d > Y )
 126         {
 127                 THROW_RANGEERROR("dimention out of range");
 128         }
 129
 130         std::vector<double> sol;
 131         if ( are_near(ray(X), 0) && are_near(ray(Y), 0) )
 132         {
 133                 if ( center(d) == v )
 134                         sol.push_back(0);
 135                 return sol;
 136         }
 137
 138         const char* msg[2][2] =
 139         {
 140                 { "d == X; ray(X) == 0; "
 141                   "s = (v - center(X)) / ( -ray(Y) * std::sin(rotation_angle()) ); "
 142                   "s should be contained in [-1,1]",
 143                   "d == X; ray(Y) == 0; "
 144                   "s = (v - center(X)) / ( ray(X) * std::cos(rotation_angle()) ); "
 145                   "s should be contained in [-1,1]"
 146                 },
 147                 { "d == Y; ray(X) == 0; "
 148                   "s = (v - center(X)) / ( ray(Y) * std::cos(rotation_angle()) ); "
 149                   "s should be contained in [-1,1]",
 150                   "d == Y; ray(Y) == 0; "
 151                   "s = (v - center(X)) / ( ray(X) * std::sin(rotation_angle()) ); "
 152                   "s should be contained in [-1,1]"
 153                 },
 154         };
 155
 156         for ( unsigned int dim = 0; dim < 2; ++dim )
 157         {
 158                 if ( are_near(ray(dim), 0) )
 159                 {
 160
 161                         if ( initialPoint()[d] == v && finalPoint()[d] == v )
 162                         {
 163                                 THROW_INFINITESOLUTIONS(0);
 164                         }
 165                         if ( (initialPoint()[d] < finalPoint()[d])
 166                                  && (initialPoint()[d] > v || finalPoint()[d] < v) )
 167                         {
 168                                 return sol;
 169                         }
 170                         if ( (initialPoint()[d] > finalPoint()[d])
 171                                  && (finalPoint()[d] > v || initialPoint()[d] < v) )
 172                         {
 173                                 return sol;
 174                         }
 175                         double ray_prj;
 176                         switch(d)
 177                         {
 178                                 case X:
 179                                         switch(dim)
 180                                         {
 181                                                 case X: ray_prj = -ray(Y) * std::sin(rotation_angle());
 182                                                                 break;
 183                                                 case Y: ray_prj = ray(X) * std::cos(rotation_angle());
 184                                                                 break;
 185                                         }
 186                                         break;
 187                                 case Y:
 188                                         switch(dim)
 189                                         {
 190                                                 case X: ray_prj = ray(Y) * std::cos(rotation_angle());
 191                                                                 break;
 192                                                 case Y: ray_prj = ray(X) * std::sin(rotation_angle());
 193                                                                 break;
 194                                         }
 195                                         break;
 196                         }
 197
 198                         double s = (v - center(d)) / ray_prj;
 199                         if ( s < -1 || s > 1 )
 200                         {
 201                                 THROW_LOGICALERROR(msg[d][dim]);
 202                         }
 203                         switch(dim)
 204                         {
 205                                 case X:
 206                                         s = std::asin(s); // return a value in [-PI/2,PI/2]
 207                                         if ( logical_xor( sweep_flag(), are_near(start_angle(), M_PI/2) )  )
 208                                         {
 209                                                 if ( s < 0 ) s += 2*M_PI;
 210                                         }
 211                                         else
 212                                         {
 213                                                 s = M_PI - s;
 214                                                 if (!(s < 2*M_PI) ) s -= 2*M_PI;
 215                                         }
 216                                         break;
 217                                 case Y:
 218                                         s = std::acos(s); // return a value in [0,PI]
 219                                         if ( logical_xor( sweep_flag(), are_near(start_angle(), 0) ) )
 220                                         {
 221                                                 s = 2*M_PI - s;
 222                                                 if ( !(s < 2*M_PI) ) s -= 2*M_PI;
 223                                         }
 224                                         break;
 225                         }
 226
 227                         //std::cerr << "s = " << rad_to_deg(s);
 228                         s = map_to_01(s);
 229                         //std::cerr << " -> t: " << s << std::endl;
 230                         if ( !(s < 0 || s > 1) )
 231                                 sol.push_back(s);
 232                         return sol;
 233                 }
 234         }
 235
 236         double rotx, roty;
 237         switch(d)
 238         {
 239                 case X:
 240                         rotx = std::cos(rotation_angle());
 241                         roty = -std::sin(rotation_angle());
 242                         break;
 243                 case Y:
 244                         rotx = std::sin(rotation_angle());
 245                         roty = std::cos(rotation_angle());
 246                         break;
 247         }
 248         double rxrotx = ray(X) * rotx;
 249         double c_v = center(d) - v;
 250
 251         double a = -rxrotx + c_v;
 252         double b = ray(Y) * roty;
 253         double c = rxrotx + c_v;
 254         //std::cerr << "a = " << a << std::endl;
 255         //std::cerr << "b = " << b << std::endl;
 256         //std::cerr << "c = " << c << std::endl;
 257
 258         if ( are_near(a,0) )
 259         {
 260                 sol.push_back(M_PI);
 261                 if ( !are_near(b,0) )
 262                 {
 263                         double s = 2 * std::atan(-c/(2*b));
 264                         if ( s < 0 ) s += 2*M_PI;
 265                         sol.push_back(s);
 266                 }
 267         }
 268         else
 269         {
 270                 double delta = b * b - a * c;
 271                 //std::cerr << "delta = " << delta << std::endl;
 272                 if ( are_near(delta, 0) )
 273                 {
 274                         double s = 2 * std::atan(-b/a);
 275                         if ( s < 0 ) s += 2*M_PI;
 276                         sol.push_back(s);
 277                 }
 278                 else if ( delta > 0 )
 279                 {
 280                         double sq = std::sqrt(delta);
 281                         double s = 2 * std::atan( (-b - sq) / a );
 282                         if ( s < 0 ) s += 2*M_PI;
 283                         sol.push_back(s);
 284                         s = 2 * std::atan( (-b + sq) / a );
 285                         if ( s < 0 ) s += 2*M_PI;
 286                         sol.push_back(s);
 287                 }
 288         }
 289
 290         std::vector<double> arc_sol;
 291         for (unsigned int i = 0; i < sol.size(); ++i )
 292         {
 293                 //std::cerr << "s = " << rad_to_deg(sol[i]);
 294                 sol[i] = map_to_01(sol[i]);
 295                 //std::cerr << " -> t: " << sol[i] << std::endl;
 296                 if ( !(sol[i] < 0 || sol[i] > 1) )
 297                         arc_sol.push_back(sol[i]);
 298         }
 299         return arc_sol;
 300
 301
 302 //      return SBasisCurve(toSBasis()).roots(v, d);
 303 }
 304
 305 // D(E(t,C),t) = E(t+PI/2,O)
 306 Curve* EllipticalArc::derivative() const
 307 {
 308         EllipticalArc* result = new EllipticalArc(*this);
 309         result->m_center[X] = result->m_center[Y] = 0;
 310         result->m_start_angle += M_PI/2;
 311         if( !( result->m_start_angle < 2*M_PI ) )
 312         {
 313                 result->m_start_angle -= 2*M_PI;
 314         }
 315         result->m_end_angle += M_PI/2;
 316         if( !( result->m_end_angle < 2*M_PI ) )
 317         {
 318                 result->m_end_angle -= 2*M_PI;
 319         }
 320         result->m_initial_point = result->pointAtAngle( result->start_angle() );
 321         result->m_final_point = result->pointAtAngle( result->end_angle() );
 322         return result;
 323
 324 }
 325
 326 std::vector<Point>
 327 EllipticalArc::pointAndDerivatives(Coord t, unsigned int n) const
 328 {
 329         std::vector<Point> result;
 330         result.reserve(n);
 331         double angle = map_unit_interval_on_circular_arc(t, start_angle(),
 332                                                                  end_angle(), sweep_flag());
 333         EllipticalArc ea(*this);
 334         ea.m_center = Point(0,0);
 335         unsigned int m = std::min(n, 4u);
 336         for ( unsigned int i = 0; i < m; ++i )
 337         {
 338                 result.push_back( ea.pointAtAngle(angle) );
 339                 angle += M_PI/2;
 340                 if ( !(angle < 2*M_PI) ) angle -= 2*M_PI;
 341         }
 342         m = n / 4;
 343         for ( unsigned int i = 1; i < m; ++i )
 344         {
 345                 for ( unsigned int j = 0; j < 4; ++j )
 346                         result.push_back( result[j] );
 347         }
 348         m = n - 4 * m;
 349         for ( unsigned int i = 0; i < m; ++i )
 350         {
 351                 result.push_back( result[i] );
 352         }
 353         if ( !result.empty() ) // n != 0
 354                 result[0] = pointAtAngle(angle);
 355         return result;
 356 }
 357
 358 D2<SBasis> EllipticalArc::toSBasis() const
 359 {
 360     // the interval of parametrization has to be [0,1]
 361     Coord et = start_angle() + ( sweep_flag() ? sweep_angle() : -sweep_angle() );
 362     Linear param(start_angle(), et);
 363     Coord cos_rot_angle = std::cos(rotation_angle());
 364     Coord sin_rot_angle = std::sin(rotation_angle());
 365     // order = 4 seems to be enough to get a perfect looking elliptical arc
 366     // should it be choosen in function of the arc length anyway ?
 367     // or maybe a user settable parameter: toSBasis(unsigned int order) ?
 368     SBasis arc_x = ray(X) * cos(param,4);
 369     SBasis arc_y = ray(Y) * sin(param,4);
 370     D2<SBasis> arc;
 371     arc[0] = arc_x * cos_rot_angle - arc_y * sin_rot_angle + Linear(center(X),center(X));
 372     arc[1] = arc_x * sin_rot_angle + arc_y * cos_rot_angle + Linear(center(Y),center(Y));
 373     return arc;
 374 }
 375
 376
 377 bool EllipticalArc::containsAngle(Coord angle) const
 378 {
 379         if ( sweep_flag() )
 380                 if ( start_angle() < end_angle() )
 381                         return ( !( angle < start_angle() || angle > end_angle() ) );
 382                 else
 383                         return ( !( angle < start_angle() && angle > end_angle() ) );
 384         else
 385                 if ( start_angle() > end_angle() )
 386                         return ( !( angle > start_angle() || angle < end_angle() ) );
 387                 else
 388                         return ( !( angle > start_angle() && angle < end_angle() ) );
 389 }
 390
 391
 392 double EllipticalArc::valueAtAngle(Coord t, Dim2 d) const
 393 {
 394     double sin_rot_angle = std::sin(rotation_angle());
 395     double cos_rot_angle = std::cos(rotation_angle());
 396     if ( d == X )
 397     {
 398         return    ray(X) * cos_rot_angle * std::cos(t)
 399                 - ray(Y) * sin_rot_angle * std::sin(t)
 400                 + center(X);
 401     }
 402     else if ( d == Y )
 403     {
 404         return    ray(X) * sin_rot_angle * std::cos(t)
 405                 + ray(Y) * cos_rot_angle * std::sin(t)
 406                 + center(Y);
 407     }
 408     THROW_RANGEERROR("dimension parameter out of range");
 409 }
 410
 411
 412 Curve* EllipticalArc::portion(double f, double t) const
 413 {
 414         if (f < 0) f = 0;
 415         if (f > 1) f = 1;
 416         if (t < 0) t = 0;
 417         if (t > 1) t = 1;
 418         if ( are_near(f, t) )
 419         {
 420                 EllipticalArc* arc = new EllipticalArc();
 421                 arc->m_center = arc->m_initial_point = arc->m_final_point = pointAt(f);
 422                 arc->m_start_angle = arc->m_end_angle = m_start_angle;
 423                 arc->m_rot_angle = m_rot_angle;
 424                 arc->m_sweep = m_sweep;
 425                 arc->m_large_arc = m_large_arc;
 426         }
 427     EllipticalArc* arc = new EllipticalArc( *this );
 428     arc->m_initial_point = pointAt(f);
 429     arc->m_final_point = pointAt(t);
 430     double sa = sweep_flag() ? sweep_angle() : -sweep_angle();
 431     arc->m_start_angle = m_start_angle + sa * f;
 432     if ( !(arc->m_start_angle < 2*M_PI) )
 433         arc->m_start_angle -= 2*M_PI;
 434     if ( arc->m_start_angle < 0 )
 435         arc->m_start_angle += 2*M_PI;
 436     arc->m_end_angle = m_start_angle + sa * t;
 437     if ( !(arc->m_end_angle < 2*M_PI) )
 438         arc->m_end_angle -= 2*M_PI;
 439     if ( arc->m_end_angle < 0 )
 440         arc->m_end_angle += 2*M_PI;
 441     if ( f > t ) arc->m_sweep = !sweep_flag();
 442     if ( large_arc_flag() && (arc->sweep_angle() < M_PI) )
 443         arc->m_large_arc = false;
 444     return arc;
 445 }
 446
 447 // NOTE: doesn't work with 360 deg arcs
 448 void EllipticalArc::calculate_center_and_extreme_angles()
 449 {
 450     if ( are_near(initialPoint(), finalPoint()) )
 451     {
 452         if ( are_near(ray(X), 0) && are_near(ray(Y), 0) )
 453         {
 454                 m_start_angle = m_end_angle = 0;
 455                 m_center = initialPoint();
 456                 return;
 457         }
 458         else
 459         {
 460                 THROW_RANGEERROR("initial and final point are the same");
 461         }
 462     }
 463         if ( are_near(ray(X), 0) && are_near(ray(Y), 0) )
 464         { // but initialPoint != finalPoint
 465                 THROW_RANGEERROR(
 466                         "there is no ellipse that satisfies the given constraints: "
 467                         "ray(X) == 0 && ray(Y) == 0 but initialPoint != finalPoint"
 468                 );
 469         }
 470         if ( are_near(ray(Y), 0) )
 471         {
 472                 Point v = initialPoint() - finalPoint();
 473                 if ( are_near(L2sq(v), 4*ray(X)*ray(X)) )
 474                 {
 475                         double angle = std::atan2(v[Y], v[X]);
 476                         if (angle < 0) angle += 2*M_PI;
 477                         if ( are_near( angle, rotation_angle() ) )
 478                         {
 479                                 m_start_angle = 0;
 480                                 m_end_angle = M_PI;
 481                                 m_center = v/2 + finalPoint();
 482                                 return;
 483                         }
 484                         angle -= M_PI;
 485                         if ( angle < 0 ) angle += 2*M_PI;
 486                         if ( are_near( angle, rotation_angle() ) )
 487                         {
 488                                 m_start_angle = M_PI;
 489                                 m_end_angle = 0;
 490                                 m_center = v/2 + finalPoint();
 491                                 return;
 492                         }
 493                         THROW_RANGEERROR(
 494                                 "there is no ellipse that satisfies the given constraints: "
 495                                 "ray(Y) == 0 "
 496                                 "and slope(initialPoint - finalPoint) != rotation_angle "
 497                                 "and != rotation_angle + PI"
 498                         );
 499                 }
 500                 if ( L2sq(v) > 4*ray(X)*ray(X) )
 501                 {
 502                         THROW_RANGEERROR(
 503                                 "there is no ellipse that satisfies the given constraints: "
 504                                 "ray(Y) == 0 and distance(initialPoint, finalPoint) > 2*ray(X)"
 505                         );
 506                 }
 507                 else
 508                 {
 509                         THROW_RANGEERROR(
 510                                 "there is infinite ellipses that satisfy the given constraints: "
 511                                 "ray(Y) == 0  and distance(initialPoint, finalPoint) < 2*ray(X)"
 512                         );
 513                 }
 514
 515         }
 516
 517         if ( are_near(ray(X), 0) )
 518         {
 519                 Point v = initialPoint() - finalPoint();
 520                 if ( are_near(L2sq(v), 4*ray(Y)*ray(Y)) )
 521                 {
 522                         double angle = std::atan2(v[Y], v[X]);
 523                         if (angle < 0) angle += 2*M_PI;
 524                         double rot_angle = rotation_angle() + M_PI/2;
 525                         if ( !(rot_angle < 2*M_PI) ) rot_angle -= 2*M_PI;
 526                         if ( are_near( angle, rot_angle ) )
 527                         {
 528                                 m_start_angle = M_PI/2;
 529                                 m_end_angle = 3*M_PI/2;
 530                                 m_center = v/2 + finalPoint();
 531                                 return;
 532                         }
 533                         angle -= M_PI;
 534                         if ( angle < 0 ) angle += 2*M_PI;
 535                         if ( are_near( angle, rot_angle ) )
 536                         {
 537                                 m_start_angle = 3*M_PI/2;
 538                                 m_end_angle = M_PI/2;
 539                                 m_center = v/2 + finalPoint();
 540                                 return;
 541                         }
 542                         THROW_RANGEERROR(
 543                                 "there is no ellipse that satisfies the given constraints: "
 544                                 "ray(X) == 0 "
 545                                 "and slope(initialPoint - finalPoint) != rotation_angle + PI/2 "
 546                                 "and != rotation_angle + (3/2)*PI"
 547                         );
 548                 }
 549                 if ( L2sq(v) > 4*ray(Y)*ray(Y) )
 550                 {
 551                         THROW_RANGEERROR(
 552                                 "there is no ellipse that satisfies the given constraints: "
 553                                 "ray(X) == 0 and distance(initialPoint, finalPoint) > 2*ray(Y)"
 554                         );
 555                 }
 556                 else
 557                 {
 558                         THROW_RANGEERROR(
 559                                 "there is infinite ellipses that satisfy the given constraints: "
 560                                 "ray(X) == 0  and distance(initialPoint, finalPoint) < 2*ray(Y)"
 561                         );
 562                 }
 563
 564         }
 565
 566     double sin_rot_angle = std::sin(rotation_angle());
 567     double cos_rot_angle = std::cos(rotation_angle());
 568
 569     Point sp = sweep_flag() ? initialPoint() : finalPoint();
 570     Point ep = sweep_flag() ? finalPoint() : initialPoint();
 571
 572     Matrix m( ray(X) * cos_rot_angle, ray(X) * sin_rot_angle,
 573              -ray(Y) * sin_rot_angle, ray(Y) * cos_rot_angle,
 574               0,                      0 );
 575     Matrix im = m.inverse();
 576     Point sol = (ep - sp) * im;
 577     double half_sum_angle = std::atan2(-sol[X], sol[Y]);
 578     double half_diff_angle;
 579     if ( are_near(std::fabs(half_sum_angle), M_PI/2) )
 580     {
 581         double anti_sgn_hsa = (half_sum_angle > 0) ? -1 : 1;
 582         double arg = anti_sgn_hsa * sol[X] / 2;
 583         // if |arg| is a little bit > 1 acos returns nan
 584         if ( are_near(arg, 1) )
 585             half_diff_angle = 0;
 586         else if ( are_near(arg, -1) )
 587             half_diff_angle = M_PI;
 588         else
 589         {
 590                 if ( !(-1 < arg && arg < 1) )
 591                         THROW_RANGEERROR(
 592                                 "there is no ellipse that satisfies the given constraints"
 593                         );
 594             // assert( -1 < arg && arg < 1 );
 595             // if it fails
 596             // => there is no ellipse that satisfies the given constraints
 597             half_diff_angle = std::acos( arg );
 598         }
 599
 600         half_diff_angle = M_PI/2 - half_diff_angle;
 601     }
 602     else
 603     {
 604         double  arg = sol[Y] / ( 2 * std::cos(half_sum_angle) );
 605         // if |arg| is a little bit > 1 asin returns nan
 606         if ( are_near(arg, 1) )
 607             half_diff_angle = M_PI/2;
 608         else if ( are_near(arg, -1) )
 609             half_diff_angle = -M_PI/2;
 610         else
 611         {
 612                 if ( !(-1 < arg && arg < 1) )
 613                         THROW_RANGEERROR(
 614                                 "there is no ellipse that satisfies the given constraints"
 615                         );
 616             // assert( -1 < arg && arg < 1 );
 617             // if it fails
 618             // => there is no ellipse that satisfies the given constraints
 619             half_diff_angle = std::asin( arg );
 620         }
 621     }
 622
 623     if (   ( m_large_arc && half_diff_angle > 0 )
 624         || (!m_large_arc && half_diff_angle < 0 ) )
 625     {
 626         half_diff_angle = -half_diff_angle;
 627     }
 628     if ( half_sum_angle < 0 ) half_sum_angle += 2*M_PI;
 629     if ( half_diff_angle < 0 ) half_diff_angle += M_PI;
 630
 631     m_start_angle = half_sum_angle - half_diff_angle;
 632     m_end_angle =  half_sum_angle + half_diff_angle;
 633     // 0 <= m_start_angle, m_end_angle < 2PI
 634     if ( m_start_angle < 0 ) m_start_angle += 2*M_PI;
 635     if( !(m_end_angle < 2*M_PI) ) m_end_angle -= 2*M_PI;
 636     sol[0] = std::cos(m_start_angle);
 637     sol[1] = std::sin(m_start_angle);
 638     m_center = sp - sol * m;
 639     if ( !sweep_flag() )
 640     {
 641         double angle = m_start_angle;
 642         m_start_angle = m_end_angle;
 643         m_end_angle = angle;
 644     }
 645 }
 646
 647 Coord EllipticalArc::map_to_02PI(Coord t) const
 648 {
 649     if ( sweep_flag() )
 650     {
 651         Coord angle = start_angle() + sweep_angle() * t;
 652         if ( !(angle < 2*M_PI) )
 653             angle -= 2*M_PI;
 654         return angle;
 655     }
 656     else
 657     {
 658         Coord angle = start_angle() - sweep_angle() * t;
 659         if ( angle < 0 ) angle += 2*M_PI;
 660         return angle;
 661     }
 662 }
 663
 664 Coord EllipticalArc::map_to_01(Coord angle) const
 665 {
 666         return map_circular_arc_on_unit_interval(angle, start_angle(),
 667                                                          end_angle(), sweep_flag());
 668 }
 669
 670
 671 std::vector<double> EllipticalArc::
 672 allNearestPoints( Point const& p, double from, double to ) const
 673 {
 674         if ( from > to ) std::swap(from, to);
 675         if ( from < 0 || to > 1 )
 676         {
 677                 THROW_RANGEERROR("[from,to] interval out of range");
 678         }
 679         std::vector<double> result;
 680         if ( ( are_near(ray(X), 0) && are_near(ray(Y), 0) )  || are_near(from, to) )
 681         {
 682                 result.push_back(from);
 683                 return result;
 684         }
 685         else if ( are_near(ray(X), 0) || are_near(ray(Y), 0) )
 686         {
 687                 LineSegment seg(pointAt(from), pointAt(to));
 688                 Point np = seg.pointAt( seg.nearestPoint(p) );
 689                 if ( are_near(ray(Y), 0) )
 690                 {
 691                         if ( are_near(rotation_angle(), M_PI/2)
 692                                  || are_near(rotation_angle(), 3*M_PI/2) )
 693                         {
 694                                 result = roots(np[Y], Y);
 695                         }
 696                         else
 697                         {
 698                                 result = roots(np[X], X);
 699                         }
 700                 }
 701                 else
 702                 {
 703                         if ( are_near(rotation_angle(), M_PI/2)
 704                                  || are_near(rotation_angle(), 3*M_PI/2) )
 705                         {
 706                                 result = roots(np[X], X);
 707                         }
 708                         else
 709                         {
 710                                 result = roots(np[Y], Y);
 711                         }
 712                 }
 713                 return result;
 714         }
 715         else if ( are_near(ray(X), ray(Y)) )
 716         {
 717                 Point r = p - center();
 718                 if ( are_near(r, Point(0,0)) )
 719                 {
 720                         THROW_INFINITESOLUTIONS(0);
 721                 }
 722                 // TODO: implement case r != 0
 723 //              Point np = ray(X) * unit_vector(r);
 724 //              std::vector<double> solX = roots(np[X],X);
 725 //              std::vector<double> solY = roots(np[Y],Y);
 726 //              double t;
 727 //              if ( are_near(solX[0], solY[0]) || are_near(solX[0], solY[1]))
 728 //              {
 729 //                      t = solX[0];
 730 //              }
 731 //              else
 732 //              {
 733 //                      t = solX[1];
 734 //              }
 735 //              if ( !(t < from || t > to) )
 736 //              {
 737 //                      result.push_back(t);
 738 //              }
 739 //              else
 740 //              {
 741 //
 742 //              }
 743         }
 744
 745         // solve the equation <D(E(t),t)|E(t)-p> == 0
 746         // that provides min and max distance points
 747         // on the ellipse E wrt the point p
 748         // after the substitutions:
 749         // cos(t) = (1 - s^2) / (1 + s^2)
 750         // sin(t) = 2t / (1 + s^2)
 751         // where s = tan(t/2)
 752         // we get a 4th degree equation in s
 753         /*
 754          *      ry s^4 ((-cy + py) Cos[Phi] + (cx - px) Sin[Phi]) +
 755          *      ry ((cy - py) Cos[Phi] + (-cx + px) Sin[Phi]) +
 756          *      2 s^3 (rx^2 - ry^2 + (-cx + px) rx Cos[Phi] + (-cy + py) rx Sin[Phi]) +
 757          *      2 s (-rx^2 + ry^2 + (-cx + px) rx Cos[Phi] + (-cy + py) rx Sin[Phi])
 758          */
 759
 760         Point p_c = p - center();
 761         double rx2_ry2 = (ray(X) - ray(Y)) * (ray(X) + ray(Y));
 762         double cosrot = std::cos( rotation_angle() );
 763         double sinrot = std::sin( rotation_angle() );
 764         double expr1 = ray(X) * (p_c[X] * cosrot + p_c[Y] * sinrot);
 765         Poly coeff;
 766         coeff.resize(5);
 767         coeff[4] = ray(Y) * ( p_c[Y] * cosrot - p_c[X] * sinrot );
 768         coeff[3] = 2 * ( rx2_ry2 + expr1 );
 769         coeff[2] = 0;
 770         coeff[1] = 2 * ( -rx2_ry2 + expr1 );
 771         coeff[0] = -coeff[4];
 772
 773 //      for ( unsigned int i = 0; i < 5; ++i )
 774 //              std::cerr << "c[" << i << "] = " << coeff[i] << std::endl;
 775
 776         std::vector<double> real_sol;
 777         // gsl_poly_complex_solve raises an error
 778         // if the leading coefficient is zero
 779         if ( are_near(coeff[4], 0) )
 780         {
 781                 real_sol.push_back(0);
 782                 if ( !are_near(coeff[3], 0) )
 783                 {
 784                         double sq = -coeff[1] / coeff[3];
 785                         if ( sq > 0 )
 786                         {
 787                                 double s = std::sqrt(sq);
 788                                 real_sol.push_back(s);
 789                                 real_sol.push_back(-s);
 790                         }
 791                 }
 792         }
 793         else
 794         {
 795                 real_sol = solve_reals(coeff);
 796         }
 797 //      else
 798 //      {
 799 //              double sol[8];
 800 //              gsl_poly_complex_workspace * w = gsl_poly_complex_workspace_alloc(5);
 801 //              gsl_poly_complex_solve(coeff, 5, w, sol );
 802 //              gsl_poly_complex_workspace_free(w);
 803 //
 804 //              for ( unsigned int i = 0; i < 4; ++i )
 805 //              {
 806 //                      if ( sol[2*i+1] == 0 ) real_sol.push_back(sol[2*i]);
 807 //              }
 808 //      }
 809
 810         for ( unsigned int i = 0; i < real_sol.size(); ++i )
 811         {
 812                 real_sol[i] = 2 * std::atan(real_sol[i]);
 813                 if ( real_sol[i] < 0 ) real_sol[i] += 2*M_PI;
 814         }
 815         // when s -> Infinity then <D(E)|E-p> -> 0 iff coeff[4] == 0
 816         // so we add M_PI to the solutions being lim arctan(s) = PI when s->Infinity
 817         if ( (real_sol.size() % 2) != 0 )
 818         {
 819                 real_sol.push_back(M_PI);
 820         }
 821
 822         double mindistsq1 = std::numeric_limits<double>::max();
 823         double mindistsq2 = std::numeric_limits<double>::max();
 824         double dsq;
 825         unsigned int mi1, mi2;
 826         for ( unsigned int i = 0; i < real_sol.size(); ++i )
 827         {
 828                 dsq = distanceSq(p, pointAtAngle(real_sol[i]));
 829                 if ( mindistsq1 > dsq )
 830                 {
 831                         mindistsq2 = mindistsq1;
 832                         mi2 = mi1;
 833                         mindistsq1 = dsq;
 834                         mi1 = i;
 835                 }
 836                 else if ( mindistsq2 > dsq )
 837                 {
 838                         mindistsq2 = dsq;
 839                         mi2 = i;
 840                 }
 841         }
 842
 843         double t = map_to_01( real_sol[mi1] );
 844         if ( !(t < from || t > to) )
 845         {
 846                 result.push_back(t);
 847         }
 848
 849         bool second_sol = false;
 850         t = map_to_01( real_sol[mi2] );
 851         if ( real_sol.size() == 4 && !(t < from || t > to) )
 852         {
 853         if ( result.empty() || are_near(mindistsq1, mindistsq2) )
 854         {
 855                 result.push_back(t);
 856                 second_sol = true;
 857         }
 858         }
 859
 860         // we need to test extreme points too
 861         double dsq1 = distanceSq(p, pointAt(from));
 862         double dsq2 = distanceSq(p, pointAt(to));
 863         if ( second_sol )
 864         {
 865                 if ( mindistsq2 > dsq1 )
 866                 {
 867                         result.clear();
 868                         result.push_back(from);
 869                         mindistsq2 = dsq1;
 870                 }
 871                 else if ( are_near(mindistsq2, dsq) )
 872                 {
 873                         result.push_back(from);
 874                 }
 875                 if ( mindistsq2 > dsq2 )
 876                 {
 877                         result.clear();
 878                         result.push_back(to);
 879                 }
 880                 else if ( are_near(mindistsq2, dsq2) )
 881                 {
 882                         result.push_back(to);
 883                 }
 884
 885         }
 886         else
 887         {
 888                 if ( result.empty() )
 889                 {
 890                         if ( are_near(dsq1, dsq2) )
 891                         {
 892                                 result.push_back(from);
 893                                 result.push_back(to);
 894                         }
 895                         else if ( dsq2 > dsq1 )
 896                         {
 897                                 result.push_back(from);
 898                         }
 899                         else
 900                         {
 901                                 result.push_back(to);
 902                         }
 903                 }
 904         }
 905
 906         return result;
 907 }
 908
 909
 910 } // end namespace Geom
 911
 912
 913 /*
 914   Local Variables:
 915   mode:c++
 916   c-file-style:"stroustrup"
 917   c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
 918   indent-tabs-mode:nil
 919   fill-column:99
 920   End:
 921 */
 922 // vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :
 923
 924