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