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     extents->is_defined = 0;
  85
  86     cpml_pair_from_cairo(&p1, cpml_primitive_get_point(curve, 0));
  87     cpml_pair_from_cairo(&p2, cpml_primitive_get_point(curve, 1));
  88     cpml_pair_from_cairo(&p3, cpml_primitive_get_point(curve, 2));
  89     cpml_pair_from_cairo(&p4, cpml_primitive_get_point(curve, 3));
  90
  91     cpml_extents_pair_add(extents, &p1);
  92     cpml_extents_pair_add(extents, &p2);
  93     cpml_extents_pair_add(extents, &p3);
  94     cpml_extents_pair_add(extents, &p4);
  95 }
  96
  97 /**
  98  * cpml_curve_pair_at_time:
  99  * @curve: the #CpmlPrimitive curve data
 100  * @pair:  the destination pair
 101  * @t:     the "time" value
 102  *
 103  * Given the @curve Bézier cubic, finds the coordinates at time @t
 104  * (where 0 is the start and 1 is the end) and stores the result
 105  * in @pair. Keep in mind @t is not homogeneous, so 0.5 does not
 106  * necessarily means the mid point.
 107  *
 108  * The relation 0 < @t < 1 must be satisfied, as interpolating on
 109  * cubic curves is not allowed.
 110  **/
 111 void
 112 cpml_curve_pair_at_time(const CpmlPrimitive *curve, CpmlPair *pair, double t)
 113 {
 114     cairo_path_data_t *p1, *p2, *p3, *p4;
 115     double t_2, t_3, t1, t1_2, t1_3;
 116
 117     p1 = cpml_primitive_get_point(curve, 0);
 118     p2 = cpml_primitive_get_point(curve, 1);
 119     p3 = cpml_primitive_get_point(curve, 2);
 120     p4 = cpml_primitive_get_point(curve, 3);
 121
 122     t_2 = t * t;
 123     t_3 = t_2 * t;
 124     t1 = 1 - t;
 125     t1_2 = t1 * t1;
 126     t1_3 = t1_2 * t1;
 127
 128     pair->x = t1_3 * p1->point.x + 3 * t1_2 * t * p2->point.x
 129               + 3 * t1 * t_2 * p3->point.x + t_3 * p4->point.x;
 130     pair->y = t1_3 * p1->point.y + 3 * t1_2 * t * p2->point.y
 131               + 3 * t1 * t_2 * p3->point.y + t_3 * p4->point.y;
 132 }
 133
 134 /**
 135  * cpml_curve_pair_at:
 136  * @curve: the #CpmlPrimitive curve data
 137  * @pair:  the destination #CpmlPair
 138  * @pos:   the position value
 139  *
 140  * Given the @curve Bézier cubic, finds the coordinates at position
 141  * @pos (where 0 is the start and 1 is the end) and stores the result
 142  * in @pair. It is similar to cpml_curve_pair_at_time() but the @pos
 143  * value is evenly distribuited, that is 0.5 is exactly the mid point.
 144  * If you do not need this feature, use cpml_curve_pair_at_time()
 145  * as it is considerable faster.
 146  *
 147  * The relation 0 < @pos < 1 must be satisfied, as interpolating on
 148  * cubic curves is not allowed.
 149  *
 150  * <important>
 151  * <title>TODO</title>
 152  * <itemizedlist>
 153  * <listitem>To be implemented...</listitem>
 154  * </itemizedlist>
 155  * </important>
 156  **/
 157 void
 158 cpml_curve_pair_at(const CpmlPrimitive *curve, CpmlPair *pair, double pos)
 159 {
 160 }
 161
 162 /**
 163  * cpml_curve_vector_at_time:
 164  * @curve:  the #CpmlPrimitive curve data
 165  * @vector: the destination vector
 166  * @t:      the "time" value
 167  *
 168  * Given the @curve Bézier cubic, finds the slope at time @t
 169  * (where 0 is the start and 1 is the end) and stores the result
 170  * in @vector. Keep in mind @t is not homogeneous, so 0.5
 171  * does not necessarily means the mid point.
 172  *
 173  * @t must be inside the range 0 .. 1, as interpolating is not
 174  * allowed.
 175  **/
 176 void
 177 cpml_curve_vector_at_time(const CpmlPrimitive *curve,
 178                           CpmlVector *vector, double t)
 179 {
 180     cairo_path_data_t *p1, *p2, *p3, *p4;
 181     CpmlPair p21, p32, p43;
 182     double t1, t1_2, t_2;
 183
 184     p1 = cpml_primitive_get_point(curve, 0);
 185     p2 = cpml_primitive_get_point(curve, 1);
 186     p3 = cpml_primitive_get_point(curve, 2);
 187     p4 = cpml_primitive_get_point(curve, 3);
 188
 189     p21.x = p2->point.x - p1->point.x;
 190     p21.y = p2->point.y - p1->point.y;
 191     p32.x = p3->point.x - p2->point.x;
 192     p32.y = p3->point.y - p2->point.y;
 193     p43.x = p4->point.x - p3->point.x;
 194     p43.y = p4->point.y - p3->point.y;
 195
 196     t1 = 1 - t;
 197     t1_2 = t1 * t1;
 198     t_2 = t * t;
 199
 200     vector->x = 3 * t1_2 * p21.x + 6 * t1 * t * p32.x + 3 * t_2 * p43.x;
 201     vector->y = 3 * t1_2 * p21.y + 6 * t1 * t * p32.y + 3 * t_2 * p43.y;
 202 }
 203
 204 /**
 205  * cpml_curve_vector_at:
 206  * @curve:  the #CpmlPrimitive curve data
 207  * @vector: the destination vector
 208  * @pos:    the position value
 209  *
 210  * Given the @curve Bézier cubic, finds the slope at position @pos
 211  * (where 0 is the start and 1 is the end) and stores the result
 212  * in @vector. It is similar to cpml_curve_vector_at_time() but the
 213  * @pos value is evenly distribuited, that is 0.5 is exactly the
 214  * mid point. If you do not need this feature, use
 215  * cpml_curve_vector_at_time() as it is considerable faster.
 216  *
 217  * @pos must be inside the range 0 .. 1, as interpolating is not
 218  * allowed.
 219  *
 220  * <important>
 221  * <title>TODO</title>
 222  * <itemizedlist>
 223  * <listitem>To be implemented...</listitem>
 224  * </itemizedlist>
 225  * </important>
 226  **/
 227 void
 228 cpml_curve_vector_at(const CpmlPrimitive *curve,
 229                      CpmlVector *vector, double pos)
 230 {
 231 }
 232
 233 /**
 234  * cpml_curve_near_pos:
 235  * @curve: the #CpmlPrimitive curve data
 236  * @pair:  the coordinates of the subject point
 237  *
 238  * Returns the pos value of the point on @curve nearest to @pair.
 239  * The returned value is always between 0 and 1.
 240  *
 241  * <important>
 242  * <title>TODO</title>
 243  * <itemizedlist>
 244  * <listitem>To be implemented...</listitem>
 245  * </itemizedlist>
 246  * </important>
 247  *
 248  * Return value: the pos value, always between 0 and 1
 249  **/
 250 double
 251 cpml_curve_near_pos(const CpmlPrimitive *curve, const CpmlPair *pair)
 252 {
 253     /* TODO */
 254
 255     return 0;
 256 }
 257
 258 /**
 259  * cpml_curve_intersection:
 260  * @curve:  the first curve
 261  * @curve2: the second curve
 262  * @dest:   a vector of #CpmlPair
 263  * @max:    maximum number of intersections to return
 264  *          (that is, the size of @dest)
 265  *
 266  * Given two Bézier cubic curves (@curve and @curve2), gets their
 267  * intersection points and store the result in @dest. Because two
 268  * curves can have 4 intersections, @dest MUST be at least an array
 269  * of 4 #CpmlPair.
 270  *
 271  * If @max is 0, the function returns 0 immediately without any
 272  * further processing. If @curve and @curve2 are cohincident,
 273  * their intersections are not considered.
 274  *
 275  * <important>
 276  * <title>TODO</title>
 277  * <itemizedlist>
 278  * <listitem>To be implemented...</listitem>
 279  * </itemizedlist>
 280  * </important>
 281  *
 282  * Return value: the number of intersections found (max 4)
 283  *               or 0 if the primitives do not intersect
 284  **/
 285 int
 286 cpml_curve_intersection(const CpmlPrimitive *curve,
 287                         const CpmlPrimitive *curve2,
 288                         CpmlPair *dest, int max)
 289 {
 290     return 0;
 291 }
 292
 293 /**
 294  * cpml_curve_intersection_with_arc:
 295  * @curve: a curve
 296  * @arc:   an arc
 297  * @dest:  a vector of #CpmlPair
 298  * @max:   maximum number of intersections to return
 299  *         (that is, the size of @dest)
 300  *
 301  * Given a Bézier cubic @curve and an @arc, gets their intersection
 302  * points and store the result in @dest. Because an arc and a cubic
 303  * curve can have up to 4 intersections, @dest MUST be at least an
 304  * array of 4 #CpmlPair.
 305  *
 306  * If @max is 0, the function returns 0 immediately without any
 307  * further processing.
 308  *
 309  * <important>
 310  * <title>TODO</title>
 311  * <itemizedlist>
 312  * <listitem>To be implemented...</listitem>
 313  * </itemizedlist>
 314  * </important>
 315  *
 316  * Return value: the number of intersections found (max 4)
 317  *               or 0 if the primitives do not intersect
 318  **/
 319 int
 320 cpml_curve_intersection_with_arc(const CpmlPrimitive *curve,
 321                                  const CpmlPrimitive *arc,
 322                                  CpmlPair *dest, int max)
 323 {
 324     return 0;
 325 }
 326
 327 /**
 328  * cpml_curve_intersection_with_line:
 329  * @curve: a curve
 330  * @line:  a line
 331  * @dest:  a vector of #CpmlPair
 332  * @max:   maximum number of intersections to return
 333  *         (that is, the size of @dest)
 334  *
 335  * Given a Bézier cubic @curve and a @line, gets their intersection
 336  * points and store the result in @dest. Because a line and a cubic
 337  * curve can have up to 4 intersections, @dest MUST be at least an
 338  * array of 4 #CpmlPair.
 339  *
 340  * If @max is 0, the function returns 0 immediately without any
 341  * further processing.
 342  *
 343  * <important>
 344  * <title>TODO</title>
 345  * <itemizedlist>
 346  * <listitem>To be implemented...</listitem>
 347  * </itemizedlist>
 348  * </important>
 349  *
 350  * Return value: the number of intersections found (max 4)
 351  *               or 0 if the primitives do not intersect
 352  **/
 353 int
 354 cpml_curve_intersection_with_line(const CpmlPrimitive *curve,
 355                                   const CpmlPrimitive *line,
 356                                   CpmlPair *dest, int max)
 357 {
 358     return 0;
 359 }
 360
 361 /**
 362  * cpml_curve_offset:
 363  * @curve:  the #CpmlPrimitive curve data
 364  * @offset: distance for the computed parallel curve
 365  *
 366  * Given a cubic Bézier primitive in @curve, this function finds
 367  * the approximated Bézier curve parallel to @curve at distance
 368  * @offset (an offset curve). The four points needed to build the
 369  * new curve are returned in the @curve struct.
 370  *
 371  * To solve the offset problem, a custom algorithm is used. First, the
 372  * resulting curve MUST have the same slope at the start and end point.
 373  * These constraints are not sufficient to resolve the system, so I let
 374  * the curve pass thought a given point (pm, known and got from the
 375  * original curve) at a given time (m, now hardcoded to 0.5).
 376  *
 377  * Firstly, I define some useful variables:
 378  *
 379  * v0 = unitvector(p[1]-p[0]) * offset;
 380  * v3 = unitvector(p[3]-p[2]) * offset;
 381  * p0 = p[0] + normal v0;
 382  * p3 = p[3] + normal v3.
 383  *
 384  * Now I want the curve to have the specified slopes at the start
 385  * and end point. Forcing the same slope at the start point means:
 386  *
 387  * p1 = p0 + k0 v0.
 388  *
 389  * where k0 is an arbitrary factor. Decomposing for x and y components:
 390  *
 391  * p1.x = p0.x + k0 v0.x;
 392  * p1.y = p0.y + k0 v0.y.
 393  *
 394  * Doing the same for the end point gives:
 395  *
 396  * p2.x = p3.x + k3 v3.x;
 397  * p2.y = p3.y + k3 v3.y.
 398  *
 399  * Now I interpolate the curve by forcing it to pass throught pm
 400  * when "time" is m, where 0 < m < 1. The cubic Bézier function is:
 401  *
 402  * C(t) = (1-t)³p0 + 3t(1-t)²p1 + 3t²(1-t)p2 + t³p3.
 403  *
 404  * and forcing t=m and C(t) = pm:
 405  *
 406  * pm = (1-m)³p0 + 3m(1-m)²p1 + 3m²(1-m)p2 + m³p3.
 407  *
 408  * (1-m) p1 + m p2 = (pm - (1-m)³p0 - m³p3) / (3m (1-m)).
 409  *
 410  * So the final system is:
 411  *
 412  * p1.x = p0.x + k0 v0.x;
 413  * p1.y = p0.y + k0 v0.y;
 414  * p2.x = p3.x + k3 v3.x;
 415  * p2.y = p3.y + k3 v3.y;
 416  * (1-m) p1.x + m p2.x = (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m));
 417  * (1-m) p1.y + m p2.y = (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)).
 418  *
 419  * Substituting and resolving for k0 and k3:
 420  *
 421  * (1-m) k0 v0.x + m k3 v3.x =
 422  *     (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m)) - (1-m) p0.x - m p3.x;
 423  * (1-m) k0 v0.y + m k3 v3.y =
 424  *     (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)) - (1-m) p0.y - m p3.y.
 425  *
 426  * (1-m) k0 v0.x + m k3 v3.x =
 427  *     (pm.x - (1-m)²(1+2m) p0.x - m²(3-2m) p3.x) / (3m (1-m));
 428  * (1-m) k0 v0.y + m k3 v3.y =
 429  *     (pm.y - (1-m)²(1+2m) p0.y - m²(3-2m) p3.y) / (3m (1-m)).
 430  *
 431  * Let:
 432  *
 433  * pk = (pm - (1-m)²(1+2m) p0 - m²(3-2m) p3) / (3m (1-m)).
 434  *
 435  * gives the following system:
 436  *
 437  * (1-m) k0 v0.x + m k3 v3.x = pk.x;
 438  * (1-m) k0 v0.y + m k3 v3.y = pk.y.
 439  *
 440  * Now I should avoid division by 0 troubles. If either v0.x and v3.x
 441  * are 0, the first equation will be inconsistent. More in general the
 442  * v0.x*v3.y = v3.x*v3.y condition should be avoided. This is the first
 443  * case to check, in which case an alternative approach is used. In the
 444  * other cases the above system can be used.
 445  *
 446  * If v0.x != 0 I can resolve for k0 and then find k3:
 447  *
 448  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 449  * (pk.x - m k3 v3.x) v0.y / v0.x + m k3 v3.y = pk.y.
 450  *
 451  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 452  * k3 m (v3.y - v3.x v0.y / v0.x) = pk.y - pk.x v0.y / v0.x.
 453  *
 454  * k3 = (pk.y - pk.x v0.y / v0.x) / (m (v3.y - v3.x v0.y / v0.x));
 455  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x).
 456  *
 457  * If v3.x != 0 I can resolve for k3 and then find k0:
 458  *
 459  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 460  * (1-m) k0 v0.y + (pk.x - (1-m) k0 v0.x) v3.y / v3.x = pk.y.
 461  *
 462  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 463  * k0 (1-m) (v0.y - k0 v0.x v3.y / v3.x) = pk.y - pk.x v3.y / v3.x.
 464  *
 465  * k0 = (pk.y - pk.x v3.y / v3.x) / ((1-m) (v0.y - v0.x v3.y / v3.x));
 466  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x).
 467  *
 468  * <important>
 469  * <title>TODO</title>
 470  * <itemizedlist>
 471  * <listitem>By default, interpolation of the new curve is made by offseting
 472  *           the mid point: use a better candidate.</listitem>
 473  * <listitem>When the equations are inconsistent, the alternative approach
 474  *           performs very bad if <varname>v0</varname> and
 475  *           <varname>v3</varname> are opposite or staggered.</listitem>
 476  * </itemizedlist>
 477  * </important>
 478  **/
 479 void
 480 cpml_curve_offset(CpmlPrimitive *curve, double offset)
 481 {
 482     double m, mm;
 483     CpmlVector v0, v3, vm, vtmp;
 484     CpmlPair p0, p1, p2, p3, pm;
 485
 486     m = 0.5;
 487     mm = 1-m;
 488
 489     /* Firstly, convert the curve points from cairo format to cpml format
 490      * and store them (temporary) in p0..p3 */
 491     cpml_pair_from_cairo(&p0, curve->org);
 492     cpml_pair_from_cairo(&p1, &curve->data[1]);
 493     cpml_pair_from_cairo(&p2, &curve->data[2]);
 494     cpml_pair_from_cairo(&p3, &curve->data[3]);
 495
 496     /* v0 = p1-p0 */
 497     cpml_pair_sub(cpml_pair_copy(&v0, &p1), &p0);
 498
 499     /* v3 = p3-p2 */
 500     cpml_pair_sub(cpml_pair_copy(&v3, &p3), &p2);
 501
 502     /* pm = point in C(m) offseted the requested @offset distance */
 503     cpml_curve_vector_at_time(curve, &vm, m);
 504     cpml_vector_set_length(&vm, offset);
 505     cpml_vector_normal(&vm);
 506     cpml_curve_pair_at_time(curve, &pm, m);
 507     cpml_pair_add(&pm, &vm);
 508
 509     /* p0 = p0 + normal of v0 of @offset magnitude (exact value) */
 510     cpml_vector_set_length(cpml_pair_copy(&vtmp, &v0), offset);
 511     cpml_vector_normal(&vtmp);
 512     cpml_pair_add(&p0, &vtmp);
 513
 514     /* p3 = p3 + normal of v3 of @offset magnitude, as done for p0 */
 515     cpml_vector_set_length(cpml_pair_copy(&vtmp, &v3), offset);
 516     cpml_vector_normal(&vtmp);
 517     cpml_pair_add(&p3, &vtmp);
 518
 519     if (v0.x*v3.y == v3.x*v0.y) {
 520         /* Inconsistent equations: use the alternative approach */
 521         p1.x = p0.x + v0.x + vm.x * 4/3;
 522         p1.y = p0.y + v0.y + vm.y * 4/3;
 523         p2.x = p3.x - v3.x + vm.x * 4/3;
 524         p2.y = p3.y - v3.y + vm.y * 4/3;
 525     } else {
 526         CpmlPair pk;
 527         double k0, k3;
 528
 529         pk.x = (pm.x - mm*mm*(1+m+m)*p0.x - m*m*(1+mm+mm)*p3.x) / (3*m*(1-m));
 530         pk.y = (pm.y - mm*mm*(1+m+m)*p0.y - m*m*(1+mm+mm)*p3.y) / (3*m*(1-m));
 531
 532         if (v0.x != 0) {
 533             k3 = (pk.y - pk.x*v0.y / v0.x) / (m*(v3.y - v3.x*v0.y / v0.x));
 534             k0 = (pk.x - m*k3*v3.x) / (mm*v0.x);
 535         } else {
 536             k0 = (pk.y - pk.x*v3.y / v3.x) / (mm*(v0.y - v0.x*v3.y / v3.x));
 537             k3 = (pk.x - mm*k0*v0.x) / (m*v3.x);
 538         }
 539
 540         p1.x = p0.x + k0*v0.x;
 541         p1.y = p0.y + k0*v0.y;
 542         p2.x = p3.x + k3*v3.x;
 543         p2.y = p3.y + k3*v3.y;
 544     }
 545
 546     /* Return the new curve in the original array */
 547     cpml_pair_to_cairo(&p0, curve->org);
 548     cpml_pair_to_cairo(&p1, &curve->data[1]);
 549     cpml_pair_to_cairo(&p2, &curve->data[2]);
 550     cpml_pair_to_cairo(&p3, &curve->data[3]);
 551 }