cpml/cpml-curve.c

   1 /* CPML - Cairo Path Manipulation Library
   2  * Copyright (C) 2008, 2009  Nicola Fontana <ntd at entidi.it>
   3  *
   4  * This library is free software; you can redistribute it and/or
   5  * modify it under the terms of the GNU Lesser General Public
   6  * License as published by the Free Software Foundation; either
   7  * version 2 of the License, or (at your option) any later version.
   8  *
   9  * This library is distributed in the hope that it will be useful,
  10  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  11  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  12  * Lesser General Public License for more details.
  13  *
  14  * You should have received a copy of the GNU Lesser General Public
  15  * License along with this library; if not, write to the
  16  * Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
  17  * Boston, MA  02110-1301, USA.
  18  */
  19
  20 /**
  21  * SECTION:cpml-curve
  22  * @Section_Id:CpmlCurve
  23  * @title: CpmlCurve
  24  * @short_description: Bézier cubic curve primitive management
  25  *
  26  * The following functions manipulate %CAIRO_PATH_CURVE_TO #CpmlPrimitive.
  27  * No validation is made on the input so use the following methods
  28  * only when you are sure the <varname>primitive</varname> argument
  29  * is effectively a cubic Bézier curve.
  30  **/
  31
  32
  33 #include "cpml-curve.h"
  34 #include "cpml-pair.h"
  35
  36
  37 /**
  38  * cpml_curve_type_get_npoints:
  39  *
  40  * Returns the number of point needed to properly specify a curve primitive.
  41  *
  42  * Return value: 4
  43  **/
  44 int
  45 cpml_curve_type_get_npoints(void)
  46 {
  47     return 4;
  48 }
  49
  50 /**
  51  * cpml_curve_length:
  52  * @curve: the #CpmlPrimitive curve data
  53  *
  54  * Given the @curve primitive, returns the approximated length of
  55  * the Bézier curve.
  56  *
  57  * <important>
  58  * <title>TODO</title>
  59  * <itemizedlist>
  60  * <listitem>To be implemented...</listitem>
  61  * </itemizedlist>
  62  * </important>
  63  *
  64  * Return value: the requested length
  65  **/
  66 double
  67 cpml_curve_length(const CpmlPrimitive *curve)
  68 {
  69     return 0.;
  70 }
  71
  72 /**
  73  * cpml_curve_extents:
  74  * @curve: the #CpmlPrimitive curve data
  75  * @extents: where to store the extents
  76  *
  77  * Given a @curve primitive, returns its boundary box in @extents.
  78  **/
  79 void
  80 cpml_curve_extents(const CpmlPrimitive *curve, CpmlExtents *extents)
  81 {
  82     CpmlPair p1, p2, p3, p4;
  83
  84     cpml_pair_from_cairo(&p1, cpml_primitive_get_point(curve, 0));
  85     cpml_pair_from_cairo(&p2, cpml_primitive_get_point(curve, 1));
  86     cpml_pair_from_cairo(&p3, cpml_primitive_get_point(curve, 2));
  87     cpml_pair_from_cairo(&p4, cpml_primitive_get_point(curve, 3));
  88
  89     cpml_extents_pair_add(extents, &p1);
  90     cpml_extents_pair_add(extents, &p2);
  91     cpml_extents_pair_add(extents, &p3);
  92     cpml_extents_pair_add(extents, &p4);
  93 }
  94
  95 /**
  96  * cpml_curve_pair_at_time:
  97  * @curve: the #CpmlPrimitive curve data
  98  * @pair:  the destination pair
  99  * @t:     the "time" value
 100  *
 101  * Given the @curve Bézier cubic, finds the coordinates at time @t
 102  * (where 0 is the start and 1 is the end) and stores the result
 103  * in @pair. Keep in mind @t is not homogeneous, so 0.5 does not
 104  * necessarily means the mid point.
 105  *
 106  * The relation 0 < @t < 1 must be satisfied, as interpolating on
 107  * cubic curves is not allowed.
 108  **/
 109 void
 110 cpml_curve_pair_at_time(const CpmlPrimitive *curve, CpmlPair *pair, double t)
 111 {
 112     cairo_path_data_t *p1, *p2, *p3, *p4;
 113     double t_2, t_3, t1, t1_2, t1_3;
 114
 115     p1 = cpml_primitive_get_point(curve, 0);
 116     p2 = cpml_primitive_get_point(curve, 1);
 117     p3 = cpml_primitive_get_point(curve, 2);
 118     p4 = cpml_primitive_get_point(curve, 3);
 119
 120     t_2 = t * t;
 121     t_3 = t_2 * t;
 122     t1 = 1 - t;
 123     t1_2 = t1 * t1;
 124     t1_3 = t1_2 * t1;
 125
 126     pair->x = t1_3 * p1->point.x + 3 * t1_2 * t * p2->point.x
 127               + 3 * t1 * t_2 * p3->point.x + t_3 * p4->point.x;
 128     pair->y = t1_3 * p1->point.y + 3 * t1_2 * t * p2->point.y
 129               + 3 * t1 * t_2 * p3->point.y + t_3 * p4->point.y;
 130 }
 131
 132 /**
 133  * cpml_curve_pair_at:
 134  * @curve: the #CpmlPrimitive curve data
 135  * @pair:  the destination #CpmlPair
 136  * @pos:   the position value
 137  *
 138  * Given the @curve Bézier cubic, finds the coordinates at position
 139  * @pos (where 0 is the start and 1 is the end) and stores the result
 140  * in @pair. It is similar to cpml_curve_pair_at_time() but the @pos
 141  * value is evenly distribuited, that is 0.5 is exactly the mid point.
 142  * If you do not need this feature, use cpml_curve_pair_at_time()
 143  * as it is considerable faster.
 144  *
 145  * The relation 0 < @pos < 1 must be satisfied, as interpolating on
 146  * cubic curves is not allowed.
 147  *
 148  * <important>
 149  * <title>TODO</title>
 150  * <itemizedlist>
 151  * <listitem>To be implemented...</listitem>
 152  * </itemizedlist>
 153  * </important>
 154  **/
 155 void
 156 cpml_curve_pair_at(const CpmlPrimitive *curve, CpmlPair *pair, double pos)
 157 {
 158 }
 159
 160 /**
 161  * cpml_curve_vector_at_time:
 162  * @curve:  the #CpmlPrimitive curve data
 163  * @vector: the destination vector
 164  * @t:      the "time" value
 165  *
 166  * Given the @curve Bézier cubic, finds the slope at time @t
 167  * (where 0 is the start and 1 is the end) and stores the result
 168  * in @vector. Keep in mind @t is not homogeneous, so 0.5
 169  * does not necessarily means the mid point.
 170  *
 171  * @t must be inside the range 0 .. 1, as interpolating is not
 172  * allowed.
 173  **/
 174 void
 175 cpml_curve_vector_at_time(const CpmlPrimitive *curve,
 176                           CpmlVector *vector, double t)
 177 {
 178     cairo_path_data_t *p1, *p2, *p3, *p4;
 179     CpmlPair p21, p32, p43;
 180     double t1, t1_2, t_2;
 181
 182     p1 = cpml_primitive_get_point(curve, 0);
 183     p2 = cpml_primitive_get_point(curve, 1);
 184     p3 = cpml_primitive_get_point(curve, 2);
 185     p4 = cpml_primitive_get_point(curve, 3);
 186
 187     p21.x = p2->point.x - p1->point.x;
 188     p21.y = p2->point.y - p1->point.y;
 189     p32.x = p3->point.x - p2->point.x;
 190     p32.y = p3->point.y - p2->point.y;
 191     p43.x = p4->point.x - p3->point.x;
 192     p43.y = p4->point.y - p3->point.y;
 193
 194     t1 = 1 - t;
 195     t1_2 = t1 * t1;
 196     t_2 = t * t;
 197
 198     vector->x = 3 * t1_2 * p21.x + 6 * t1 * t * p32.x + 3 * t_2 * p43.x;
 199     vector->y = 3 * t1_2 * p21.y + 6 * t1 * t * p32.y + 3 * t_2 * p43.y;
 200 }
 201
 202 /**
 203  * cpml_curve_vector_at:
 204  * @curve:  the #CpmlPrimitive curve data
 205  * @vector: the destination vector
 206  * @pos:    the position value
 207  *
 208  * Given the @curve Bézier cubic, finds the slope at position @pos
 209  * (where 0 is the start and 1 is the end) and stores the result
 210  * in @vector. It is similar to cpml_curve_vector_at_time() but the
 211  * @pos value is evenly distribuited, that is 0.5 is exactly the
 212  * mid point. If you do not need this feature, use
 213  * cpml_curve_vector_at_time() as it is considerable faster.
 214  *
 215  * @pos must be inside the range 0 .. 1, as interpolating is not
 216  * allowed.
 217  *
 218  * <important>
 219  * <title>TODO</title>
 220  * <itemizedlist>
 221  * <listitem>To be implemented...</listitem>
 222  * </itemizedlist>
 223  * </important>
 224  **/
 225 void
 226 cpml_curve_vector_at(const CpmlPrimitive *curve,
 227                      CpmlVector *vector, double pos)
 228 {
 229 }
 230
 231 /**
 232  * cpml_curve_near_pos:
 233  * @curve: the #CpmlPrimitive curve data
 234  * @pair:  the coordinates of the subject point
 235  *
 236  * Returns the pos value of the point on @curve nearest to @pair.
 237  * The returned value is always between 0 and 1.
 238  *
 239  * <important>
 240  * <title>TODO</title>
 241  * <itemizedlist>
 242  * <listitem>To be implemented...</listitem>
 243  * </itemizedlist>
 244  * </important>
 245  *
 246  * Return value: the pos value, always between 0 and 1
 247  **/
 248 double
 249 cpml_curve_near_pos(const CpmlPrimitive *curve, const CpmlPair *pair)
 250 {
 251     /* TODO */
 252
 253     return 0;
 254 }
 255
 256 /**
 257  * cpml_curve_intersection:
 258  * @curve:  the first curve
 259  * @curve2: the second curve
 260  * @dest:   a vector of #CpmlPair
 261  * @max:    maximum number of intersections to return
 262  *          (that is, the size of @dest)
 263  *
 264  * Given two Bézier cubic curves (@curve and @curve2), gets their
 265  * intersection points and store the result in @dest. Because two
 266  * curves can have 4 intersections, @dest MUST be at least an array
 267  * of 4 #CpmlPair.
 268  *
 269  * If @max is 0, the function returns 0 immediately without any
 270  * further processing. If @curve and @curve2 are cohincident,
 271  * their intersections are not considered.
 272  *
 273  * <important>
 274  * <title>TODO</title>
 275  * <itemizedlist>
 276  * <listitem>To be implemented...</listitem>
 277  * </itemizedlist>
 278  * </important>
 279  *
 280  * Return value: the number of intersections found (max 4)
 281  *               or 0 if the primitives do not intersect
 282  **/
 283 int
 284 cpml_curve_intersection(const CpmlPrimitive *curve,
 285                         const CpmlPrimitive *curve2,
 286                         CpmlPair *dest, int max)
 287 {
 288     return 0;
 289 }
 290
 291 /**
 292  * cpml_curve_intersection_with_arc:
 293  * @curve: a curve
 294  * @arc:   an arc
 295  * @dest:  a vector of #CpmlPair
 296  * @max:   maximum number of intersections to return
 297  *         (that is, the size of @dest)
 298  *
 299  * Given a Bézier cubic @curve and an @arc, gets their intersection
 300  * points and store the result in @dest. Because an arc and a cubic
 301  * curve can have up to 4 intersections, @dest MUST be at least an
 302  * array of 4 #CpmlPair.
 303  *
 304  * If @max is 0, the function returns 0 immediately without any
 305  * further processing.
 306  *
 307  * <important>
 308  * <title>TODO</title>
 309  * <itemizedlist>
 310  * <listitem>To be implemented...</listitem>
 311  * </itemizedlist>
 312  * </important>
 313  *
 314  * Return value: the number of intersections found (max 4)
 315  *               or 0 if the primitives do not intersect
 316  **/
 317 int
 318 cpml_curve_intersection_with_arc(const CpmlPrimitive *curve,
 319                                  const CpmlPrimitive *arc,
 320                                  CpmlPair *dest, int max)
 321 {
 322     return 0;
 323 }
 324
 325 /**
 326  * cpml_curve_intersection_with_line:
 327  * @curve: a curve
 328  * @line:  a line
 329  * @dest:  a vector of #CpmlPair
 330  * @max:   maximum number of intersections to return
 331  *         (that is, the size of @dest)
 332  *
 333  * Given a Bézier cubic @curve and a @line, gets their intersection
 334  * points and store the result in @dest. Because a line and a cubic
 335  * curve can have up to 4 intersections, @dest MUST be at least an
 336  * array of 4 #CpmlPair.
 337  *
 338  * If @max is 0, the function returns 0 immediately without any
 339  * further processing.
 340  *
 341  * <important>
 342  * <title>TODO</title>
 343  * <itemizedlist>
 344  * <listitem>To be implemented...</listitem>
 345  * </itemizedlist>
 346  * </important>
 347  *
 348  * Return value: the number of intersections found (max 4)
 349  *               or 0 if the primitives do not intersect
 350  **/
 351 int
 352 cpml_curve_intersection_with_line(const CpmlPrimitive *curve,
 353                                   const CpmlPrimitive *line,
 354                                   CpmlPair *dest, int max)
 355 {
 356     return 0;
 357 }
 358
 359 /**
 360  * cpml_curve_offset:
 361  * @curve:  the #CpmlPrimitive curve data
 362  * @offset: distance for the computed parallel curve
 363  *
 364  * Given a cubic Bézier primitive in @curve, this function finds
 365  * the approximated Bézier curve parallel to @curve at distance
 366  * @offset (an offset curve). The four points needed to build the
 367  * new curve are returned in the @curve struct.
 368  *
 369  * To solve the offset problem, a custom algorithm is used. First, the
 370  * resulting curve MUST have the same slope at the start and end point.
 371  * These constraints are not sufficient to resolve the system, so I let
 372  * the curve pass thought a given point (pm, known and got from the
 373  * original curve) at a given time (m, now hardcoded to 0.5).
 374  *
 375  * Firstly, I define some useful variables:
 376  *
 377  * v0 = unitvector(p[1]-p[0]) * offset;
 378  * v3 = unitvector(p[3]-p[2]) * offset;
 379  * p0 = p[0] + normal v0;
 380  * p3 = p[3] + normal v3.
 381  *
 382  * Now I want the curve to have the specified slopes at the start
 383  * and end point. Forcing the same slope at the start point means:
 384  *
 385  * p1 = p0 + k0 v0.
 386  *
 387  * where k0 is an arbitrary factor. Decomposing for x and y components:
 388  *
 389  * p1.x = p0.x + k0 v0.x;
 390  * p1.y = p0.y + k0 v0.y.
 391  *
 392  * Doing the same for the end point gives:
 393  *
 394  * p2.x = p3.x + k3 v3.x;
 395  * p2.y = p3.y + k3 v3.y.
 396  *
 397  * Now I interpolate the curve by forcing it to pass throught pm
 398  * when "time" is m, where 0 < m < 1. The cubic Bézier function is:
 399  *
 400  * C(t) = (1-t)³p0 + 3t(1-t)²p1 + 3t²(1-t)p2 + t³p3.
 401  *
 402  * and forcing t=m and C(t) = pm:
 403  *
 404  * pm = (1-m)³p0 + 3m(1-m)²p1 + 3m²(1-m)p2 + m³p3.
 405  *
 406  * (1-m) p1 + m p2 = (pm - (1-m)³p0 - m³p3) / (3m (1-m)).
 407  *
 408  * So the final system is:
 409  *
 410  * p1.x = p0.x + k0 v0.x;
 411  * p1.y = p0.y + k0 v0.y;
 412  * p2.x = p3.x + k3 v3.x;
 413  * p2.y = p3.y + k3 v3.y;
 414  * (1-m) p1.x + m p2.x = (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m));
 415  * (1-m) p1.y + m p2.y = (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)).
 416  *
 417  * Substituting and resolving for k0 and k3:
 418  *
 419  * (1-m) k0 v0.x + m k3 v3.x =
 420  *     (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m)) - (1-m) p0.x - m p3.x;
 421  * (1-m) k0 v0.y + m k3 v3.y =
 422  *     (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)) - (1-m) p0.y - m p3.y.
 423  *
 424  * (1-m) k0 v0.x + m k3 v3.x =
 425  *     (pm.x - (1-m)²(1+2m) p0.x - m²(3-2m) p3.x) / (3m (1-m));
 426  * (1-m) k0 v0.y + m k3 v3.y =
 427  *     (pm.y - (1-m)²(1+2m) p0.y - m²(3-2m) p3.y) / (3m (1-m)).
 428  *
 429  * Let:
 430  *
 431  * pk = (pm - (1-m)²(1+2m) p0 - m²(3-2m) p3) / (3m (1-m)).
 432  *
 433  * gives the following system:
 434  *
 435  * (1-m) k0 v0.x + m k3 v3.x = pk.x;
 436  * (1-m) k0 v0.y + m k3 v3.y = pk.y.
 437  *
 438  * Now I should avoid division by 0 troubles. If either v0.x and v3.x
 439  * are 0, the first equation will be inconsistent. More in general the
 440  * v0.x*v3.y = v3.x*v3.y condition should be avoided. This is the first
 441  * case to check, in which case an alternative approach is used. In the
 442  * other cases the above system can be used.
 443  *
 444  * If v0.x != 0 I can resolve for k0 and then find k3:
 445  *
 446  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 447  * (pk.x - m k3 v3.x) v0.y / v0.x + m k3 v3.y = pk.y.
 448  *
 449  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 450  * k3 m (v3.y - v3.x v0.y / v0.x) = pk.y - pk.x v0.y / v0.x.
 451  *
 452  * k3 = (pk.y - pk.x v0.y / v0.x) / (m (v3.y - v3.x v0.y / v0.x));
 453  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x).
 454  *
 455  * If v3.x != 0 I can resolve for k3 and then find k0:
 456  *
 457  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 458  * (1-m) k0 v0.y + (pk.x - (1-m) k0 v0.x) v3.y / v3.x = pk.y.
 459  *
 460  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 461  * k0 (1-m) (v0.y - k0 v0.x v3.y / v3.x) = pk.y - pk.x v3.y / v3.x.
 462  *
 463  * k0 = (pk.y - pk.x v3.y / v3.x) / ((1-m) (v0.y - v0.x v3.y / v3.x));
 464  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x).
 465  *
 466  * <important>
 467  * <title>TODO</title>
 468  * <itemizedlist>
 469  * <listitem>By default, interpolation of the new curve is made by offseting
 470  *           the mid point: use a better candidate.</listitem>
 471  * <listitem>When the equations are inconsistent, the alternative approach
 472  *           performs very bad if <varname>v0</varname> and
 473  *           <varname>v3</varname> are opposite or staggered.</listitem>
 474  * </itemizedlist>
 475  * </important>
 476  **/
 477 void
 478 cpml_curve_offset(CpmlPrimitive *curve, double offset)
 479 {
 480     double m, mm;
 481     CpmlVector v0, v3, vm, vtmp;
 482     CpmlPair p0, p1, p2, p3, pm;
 483
 484     m = 0.5;
 485     mm = 1-m;
 486
 487     /* Firstly, convert the curve points from cairo format to cpml format
 488      * and store them (temporary) in p0..p3 */
 489     cpml_pair_from_cairo(&p0, curve->org);
 490     cpml_pair_from_cairo(&p1, &curve->data[1]);
 491     cpml_pair_from_cairo(&p2, &curve->data[2]);
 492     cpml_pair_from_cairo(&p3, &curve->data[3]);
 493
 494     /* v0 = p1-p0 */
 495     cpml_pair_sub(cpml_pair_copy(&v0, &p1), &p0);
 496
 497     /* v3 = p3-p2 */
 498     cpml_pair_sub(cpml_pair_copy(&v3, &p3), &p2);
 499
 500     /* pm = point in C(m) offseted the requested @offset distance */
 501     cpml_curve_vector_at_time(curve, &vm, m);
 502     cpml_vector_set_length(&vm, offset);
 503     cpml_vector_normal(&vm);
 504     cpml_curve_pair_at_time(curve, &pm, m);
 505     cpml_pair_add(&pm, &vm);
 506
 507     /* p0 = p0 + normal of v0 of @offset magnitude (exact value) */
 508     cpml_vector_set_length(cpml_pair_copy(&vtmp, &v0), offset);
 509     cpml_vector_normal(&vtmp);
 510     cpml_pair_add(&p0, &vtmp);
 511
 512     /* p3 = p3 + normal of v3 of @offset magnitude, as done for p0 */
 513     cpml_vector_set_length(cpml_pair_copy(&vtmp, &v3), offset);
 514     cpml_vector_normal(&vtmp);
 515     cpml_pair_add(&p3, &vtmp);
 516
 517     if (v0.x*v3.y == v3.x*v0.y) {
 518         /* Inconsistent equations: use the alternative approach */
 519         p1.x = p0.x + v0.x + vm.x * 4/3;
 520         p1.y = p0.y + v0.y + vm.y * 4/3;
 521         p2.x = p3.x - v3.x + vm.x * 4/3;
 522         p2.y = p3.y - v3.y + vm.y * 4/3;
 523     } else {
 524         CpmlPair pk;
 525         double k0, k3;
 526
 527         pk.x = (pm.x - mm*mm*(1+m+m)*p0.x - m*m*(1+mm+mm)*p3.x) / (3*m*(1-m));
 528         pk.y = (pm.y - mm*mm*(1+m+m)*p0.y - m*m*(1+mm+mm)*p3.y) / (3*m*(1-m));
 529
 530         if (v0.x != 0) {
 531             k3 = (pk.y - pk.x*v0.y / v0.x) / (m*(v3.y - v3.x*v0.y / v0.x));
 532             k0 = (pk.x - m*k3*v3.x) / (mm*v0.x);
 533         } else {
 534             k0 = (pk.y - pk.x*v3.y / v3.x) / (mm*(v0.y - v0.x*v3.y / v3.x));
 535             k3 = (pk.x - mm*k0*v0.x) / (m*v3.x);
 536         }
 537
 538         p1.x = p0.x + k0*v0.x;
 539         p1.y = p0.y + k0*v0.y;
 540         p2.x = p3.x + k3*v3.x;
 541         p2.y = p3.y + k3*v3.y;
 542     }
 543
 544     /* Return the new curve in the original array */
 545     cpml_pair_to_cairo(&p0, curve->org);
 546     cpml_pair_to_cairo(&p1, &curve->data[1]);
 547     cpml_pair_to_cairo(&p2, &curve->data[2]);
 548     cpml_pair_to_cairo(&p3, &curve->data[3]);
 549 }