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  * <important>
  32  * <title>TODO</title>
  33  * <itemizedlist>
  34  * <listitem>the get_length() method must be implemented;</listitem>
  35  * <listitem>actually the put_extents() method is implemented by computing
  36  *           the bounding box of the control polygon and this will likely
  37  *           include some empty space: there is room for improvements;</listitem>
  38  * </itemizedlist>
  39  * </important>
  40  **/
  41
  42
  43 #include "cpml-internal.h"
  44 #include "cpml-extents.h"
  45 #include "cpml-segment.h"
  46 #include "cpml-primitive.h"
  47 #include "cpml-primitive-private.h"
  48 #include "cpml-curve.h"
  49
  50
  51 static void             put_extents     (const CpmlPrimitive    *curve,
  52                                          CpmlExtents            *extents);
  53
  54
  55 const _CpmlPrimitiveClass *
  56 _cpml_curve_get_class(void)
  57 {
  58     static _CpmlPrimitiveClass *p_class = NULL;
  59
  60     if (p_class == NULL) {
  61         static _CpmlPrimitiveClass class_data = {
  62             "curve", 4,
  63             NULL,
  64             put_extents,
  65             NULL,
  66             NULL,
  67             NULL,
  68             NULL,
  69             NULL,
  70             NULL
  71         };
  72         p_class = &class_data;
  73     }
  74
  75     return p_class;
  76 }
  77
  78
  79 /**
  80  * cpml_curve_put_pair_at_time:
  81  * @curve: the #CpmlPrimitive curve data
  82  * @t:     the "time" value
  83  * @pair:  the destination pair
  84  *
  85  * Given the @curve Bézier cubic, finds the coordinates at time @t
  86  * (where 0 is the start and 1 is the end) and stores the result
  87  * in @pair. Keep in mind @t is not homogeneous, so 0.5 does not
  88  * necessarily means the mid point.
  89  *
  90  * The relation 0 < @t < 1 must be satisfied, as interpolating on
  91  * cubic curves is not allowed.
  92  **/
  93 void
  94 cpml_curve_put_pair_at_time(const CpmlPrimitive *curve, double t,
  95                             CpmlPair *pair)
  96 {
  97     cairo_path_data_t *p1, *p2, *p3, *p4;
  98     double t_2, t_3, t1, t1_2, t1_3;
  99
 100     p1 = cpml_primitive_get_point(curve, 0);
 101     p2 = cpml_primitive_get_point(curve, 1);
 102     p3 = cpml_primitive_get_point(curve, 2);
 103     p4 = cpml_primitive_get_point(curve, 3);
 104
 105     t_2 = t * t;
 106     t_3 = t_2 * t;
 107     t1 = 1 - t;
 108     t1_2 = t1 * t1;
 109     t1_3 = t1_2 * t1;
 110
 111     pair->x = t1_3 * p1->point.x + 3 * t1_2 * t * p2->point.x
 112               + 3 * t1 * t_2 * p3->point.x + t_3 * p4->point.x;
 113     pair->y = t1_3 * p1->point.y + 3 * t1_2 * t * p2->point.y
 114               + 3 * t1 * t_2 * p3->point.y + t_3 * p4->point.y;
 115 }
 116
 117 /**
 118  * cpml_curve_put_pair_at:
 119  * @curve: the #CpmlPrimitive curve data
 120  * @pos:   the position value
 121  * @pair:  the destination #CpmlPair
 122  *
 123  * Given the @curve Bézier cubic, finds the coordinates at position
 124  * @pos (where 0 is the start and 1 is the end) and stores the result
 125  * in @pair. It is similar to cpml_curve_put_pair_at_time() but the @pos
 126  * value is evenly distribuited, that is 0.5 is exactly the mid point.
 127  * If you do not need this feature, use cpml_curve_put_pair_at_time()
 128  * as it is considerable faster.
 129  *
 130  * The relation 0 < @pos < 1 must be satisfied, as interpolating on
 131  * cubic curves is not allowed.
 132  *
 133  * <important>
 134  * <title>TODO</title>
 135  * <itemizedlist>
 136  * <listitem>To be implemented...</listitem>
 137  * </itemizedlist>
 138  * </important>
 139  **/
 140 void
 141 cpml_curve_put_pair_at(const CpmlPrimitive *curve, double pos, CpmlPair *pair)
 142 {
 143 }
 144
 145 /**
 146  * cpml_curve_put_vector_at_time:
 147  * @curve:  the #CpmlPrimitive curve data
 148  * @t:      the "time" value
 149  * @vector: the destination vector
 150  *
 151  * Given the @curve Bézier cubic, finds the slope at time @t
 152  * (where 0 is the start and 1 is the end) and stores the result
 153  * in @vector. Keep in mind @t is not homogeneous, so 0.5
 154  * does not necessarily means the mid point.
 155  *
 156  * @t must be inside the range 0 .. 1, as interpolating is not
 157  * allowed.
 158  **/
 159 void
 160 cpml_curve_put_vector_at_time(const CpmlPrimitive *curve,
 161                               double t, CpmlVector *vector)
 162 {
 163     cairo_path_data_t *p1, *p2, *p3, *p4;
 164     CpmlPair p21, p32, p43;
 165     double t1, t1_2, t_2;
 166
 167     p1 = cpml_primitive_get_point(curve, 0);
 168     p2 = cpml_primitive_get_point(curve, 1);
 169     p3 = cpml_primitive_get_point(curve, 2);
 170     p4 = cpml_primitive_get_point(curve, 3);
 171
 172     p21.x = p2->point.x - p1->point.x;
 173     p21.y = p2->point.y - p1->point.y;
 174     p32.x = p3->point.x - p2->point.x;
 175     p32.y = p3->point.y - p2->point.y;
 176     p43.x = p4->point.x - p3->point.x;
 177     p43.y = p4->point.y - p3->point.y;
 178
 179     t1 = 1 - t;
 180     t1_2 = t1 * t1;
 181     t_2 = t * t;
 182
 183     vector->x = 3 * t1_2 * p21.x + 6 * t1 * t * p32.x + 3 * t_2 * p43.x;
 184     vector->y = 3 * t1_2 * p21.y + 6 * t1 * t * p32.y + 3 * t_2 * p43.y;
 185 }
 186
 187 /**
 188  * cpml_curve_put_vector_at:
 189  * @curve:  the #CpmlPrimitive curve data
 190  * @pos:    the position value
 191  * @vector: the destination vector
 192  *
 193  * Given the @curve Bézier cubic, finds the slope at position @pos
 194  * (where 0 is the start and 1 is the end) and stores the result
 195  * in @vector. It is similar to cpml_curve_put_vector_at_time()
 196  * but the @pos value is evenly distribuited, that is 0.5 is
 197  * exactly the mid point. If you do not need this feature, use
 198  * cpml_curve_put_vector_at_time() as it is considerable faster.
 199  *
 200  * @pos must be inside the range 0 .. 1, as interpolating is not
 201  * allowed.
 202  *
 203  * <important>
 204  * <title>TODO</title>
 205  * <itemizedlist>
 206  * <listitem>To be implemented...</listitem>
 207  * </itemizedlist>
 208  * </important>
 209  **/
 210 void
 211 cpml_curve_put_vector_at(const CpmlPrimitive *curve, double pos,
 212                          CpmlVector *vector)
 213 {
 214 }
 215
 216 /**
 217  * cpml_curve_get_closest_pos:
 218  * @curve: the #CpmlPrimitive curve data
 219  * @pair:  the coordinates of the subject point
 220  *
 221  * Returns the pos value of the point on @curve nearest to @pair.
 222  * The returned value is always between 0 and 1.
 223  *
 224  * <important>
 225  * <title>TODO</title>
 226  * <itemizedlist>
 227  * <listitem>To be implemented...</listitem>
 228  * </itemizedlist>
 229  * </important>
 230  *
 231  * Returns: the pos value, always between 0 and 1
 232  **/
 233 double
 234 cpml_curve_get_closest_pos(const CpmlPrimitive *curve, const CpmlPair *pair)
 235 {
 236     /* TODO */
 237
 238     return 0;
 239 }
 240
 241 /**
 242  * cpml_curve_put_intersections:
 243  * @curve:  the first curve
 244  * @curve2: the second curve
 245  * @max:    maximum number of intersections to return
 246  *          (that is, the size of @dest)
 247  * @dest:   a vector of #CpmlPair
 248  *
 249  * Given two Bézier cubic curves (@curve and @curve2), gets their
 250  * intersection points and store the result in @dest. Because two
 251  * curves can have 4 intersections, @dest MUST be at least an array
 252  * of 4 #CpmlPair.
 253  *
 254  * If @max is 0, the function returns 0 immediately without any
 255  * further processing. If @curve and @curve2 are cohincident,
 256  * their intersections are not considered.
 257  *
 258  * <important>
 259  * <title>TODO</title>
 260  * <itemizedlist>
 261  * <listitem>To be implemented...</listitem>
 262  * </itemizedlist>
 263  * </important>
 264  *
 265  * Returns: the number of intersections found (max 4)
 266  *               or 0 if the primitives do not intersect
 267  **/
 268 int
 269 cpml_curve_put_intersections(const CpmlPrimitive *curve,
 270                              const CpmlPrimitive *curve2,
 271                              int max, CpmlPair *dest)
 272 {
 273     return 0;
 274 }
 275
 276 /**
 277  * cpml_curve_put_intersections_with_arc:
 278  * @curve: a curve
 279  * @arc:   an arc
 280  * @max:   maximum number of intersections to return
 281  *         (that is, the size of @dest)
 282  * @dest:  a vector of #CpmlPair
 283  *
 284  * Given a Bézier cubic @curve and an @arc, gets their intersection
 285  * points and store the result in @dest. Because an arc and a cubic
 286  * curve can have up to 4 intersections, @dest MUST be at least an
 287  * array of 4 #CpmlPair.
 288  *
 289  * If @max is 0, the function returns 0 immediately without any
 290  * further processing.
 291  *
 292  * <important>
 293  * <title>TODO</title>
 294  * <itemizedlist>
 295  * <listitem>To be implemented...</listitem>
 296  * </itemizedlist>
 297  * </important>
 298  *
 299  * Returns: the number of intersections found (max 4)
 300  *          or 0 if the primitives do not intersect
 301  **/
 302 int
 303 cpml_curve_put_intersections_with_arc(const CpmlPrimitive *curve,
 304                                       const CpmlPrimitive *arc,
 305                                       int max, CpmlPair *dest)
 306 {
 307     return 0;
 308 }
 309
 310 /**
 311  * cpml_curve_put_intersections_with_line:
 312  * @curve: a curve
 313  * @line:  a line
 314  * @max:   maximum number of intersections to return
 315  *         (that is, the size of @dest)
 316  * @dest:  a vector of #CpmlPair
 317  *
 318  * Given a Bézier cubic @curve and a @line, gets their intersection
 319  * points and store the result in @dest. Because a line and a cubic
 320  * curve can have up to 4 intersections, @dest MUST be at least an
 321  * array of 4 #CpmlPair.
 322  *
 323  * If @max is 0, the function returns 0 immediately without any
 324  * further processing.
 325  *
 326  * <important>
 327  * <title>TODO</title>
 328  * <itemizedlist>
 329  * <listitem>To be implemented...</listitem>
 330  * </itemizedlist>
 331  * </important>
 332  *
 333  * Returns: the number of intersections found (max 4)
 334  *          or 0 if the primitives do not intersect
 335  **/
 336 int
 337 cpml_curve_put_intersections_with_line(const CpmlPrimitive *curve,
 338                                        const CpmlPrimitive *line,
 339                                        int max, CpmlPair *dest)
 340 {
 341     return 0;
 342 }
 343
 344 /**
 345  * cpml_curve_offset:
 346  * @curve:  the #CpmlPrimitive curve data
 347  * @offset: distance for the computed parallel curve
 348  *
 349  * Given a cubic Bézier primitive in @curve, this function finds
 350  * the approximated Bézier curve parallel to @curve at distance
 351  * @offset (an offset curve). The four points needed to build the
 352  * new curve are returned in the @curve struct.
 353  *
 354  * To solve the offset problem, a custom algorithm is used. First, the
 355  * resulting curve MUST have the same slope at the start and end point.
 356  * These constraints are not sufficient to resolve the system, so I let
 357  * the curve pass thought a given point (pm, known and got from the
 358  * original curve) at a given time (m, now hardcoded to 0.5).
 359  *
 360  * Firstly, I define some useful variables:
 361  *
 362  * v0 = unitvector(p[1]-p[0]) * offset;
 363  * v3 = unitvector(p[3]-p[2]) * offset;
 364  * p0 = p[0] + normal v0;
 365  * p3 = p[3] + normal v3.
 366  *
 367  * Now I want the curve to have the specified slopes at the start
 368  * and end point. Forcing the same slope at the start point means:
 369  *
 370  * p1 = p0 + k0 v0.
 371  *
 372  * where k0 is an arbitrary factor. Decomposing for x and y components:
 373  *
 374  * p1.x = p0.x + k0 v0.x;
 375  * p1.y = p0.y + k0 v0.y.
 376  *
 377  * Doing the same for the end point gives:
 378  *
 379  * p2.x = p3.x + k3 v3.x;
 380  * p2.y = p3.y + k3 v3.y.
 381  *
 382  * Now I interpolate the curve by forcing it to pass throught pm
 383  * when "time" is m, where 0 < m < 1. The cubic Bézier function is:
 384  *
 385  * C(t) = (1-t)³p0 + 3t(1-t)²p1 + 3t²(1-t)p2 + t³p3.
 386  *
 387  * and forcing t=m and C(t) = pm:
 388  *
 389  * pm = (1-m)³p0 + 3m(1-m)²p1 + 3m²(1-m)p2 + m³p3.
 390  *
 391  * (1-m) p1 + m p2 = (pm - (1-m)³p0 - m³p3) / (3m (1-m)).
 392  *
 393  * So the final system is:
 394  *
 395  * p1.x = p0.x + k0 v0.x;
 396  * p1.y = p0.y + k0 v0.y;
 397  * p2.x = p3.x + k3 v3.x;
 398  * p2.y = p3.y + k3 v3.y;
 399  * (1-m) p1.x + m p2.x = (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m));
 400  * (1-m) p1.y + m p2.y = (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)).
 401  *
 402  * Substituting and resolving for k0 and k3:
 403  *
 404  * (1-m) k0 v0.x + m k3 v3.x =
 405  *     (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m)) - (1-m) p0.x - m p3.x;
 406  * (1-m) k0 v0.y + m k3 v3.y =
 407  *     (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)) - (1-m) p0.y - m p3.y.
 408  *
 409  * (1-m) k0 v0.x + m k3 v3.x =
 410  *     (pm.x - (1-m)²(1+2m) p0.x - m²(3-2m) p3.x) / (3m (1-m));
 411  * (1-m) k0 v0.y + m k3 v3.y =
 412  *     (pm.y - (1-m)²(1+2m) p0.y - m²(3-2m) p3.y) / (3m (1-m)).
 413  *
 414  * Let:
 415  *
 416  * pk = (pm - (1-m)²(1+2m) p0 - m²(3-2m) p3) / (3m (1-m)).
 417  *
 418  * gives the following system:
 419  *
 420  * (1-m) k0 v0.x + m k3 v3.x = pk.x;
 421  * (1-m) k0 v0.y + m k3 v3.y = pk.y.
 422  *
 423  * Now I should avoid division by 0 troubles. If either v0.x and v3.x
 424  * are 0, the first equation will be inconsistent. More in general the
 425  * v0.x*v3.y = v3.x*v3.y condition should be avoided. This is the first
 426  * case to check, in which case an alternative approach is used. In the
 427  * other cases the above system can be used.
 428  *
 429  * If v0.x != 0 I can resolve for k0 and then find k3:
 430  *
 431  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 432  * (pk.x - m k3 v3.x) v0.y / v0.x + m k3 v3.y = pk.y.
 433  *
 434  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 435  * k3 m (v3.y - v3.x v0.y / v0.x) = pk.y - pk.x v0.y / v0.x.
 436  *
 437  * k3 = (pk.y - pk.x v0.y / v0.x) / (m (v3.y - v3.x v0.y / v0.x));
 438  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x).
 439  *
 440  * If v3.x != 0 I can resolve for k3 and then find k0:
 441  *
 442  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 443  * (1-m) k0 v0.y + (pk.x - (1-m) k0 v0.x) v3.y / v3.x = pk.y.
 444  *
 445  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 446  * k0 (1-m) (v0.y - k0 v0.x v3.y / v3.x) = pk.y - pk.x v3.y / v3.x.
 447  *
 448  * k0 = (pk.y - pk.x v3.y / v3.x) / ((1-m) (v0.y - v0.x v3.y / v3.x));
 449  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x).
 450  *
 451  * <important>
 452  * <title>TODO</title>
 453  * <itemizedlist>
 454  * <listitem>By default, interpolation of the new curve is made by offseting
 455  *           the mid point: use a better candidate.</listitem>
 456  * <listitem>When the equations are inconsistent, the alternative approach
 457  *           performs very bad if <varname>v0</varname> and
 458  *           <varname>v3</varname> are opposite or staggered.</listitem>
 459  * </itemizedlist>
 460  * </important>
 461  **/
 462 void
 463 cpml_curve_offset(CpmlPrimitive *curve, double offset)
 464 {
 465     double m, mm;
 466     CpmlVector v0, v3, vm, vtmp;
 467     CpmlPair p0, p1, p2, p3, pm;
 468
 469     m = 0.5;
 470     mm = 1-m;
 471
 472     /* Firstly, convert the curve points from cairo format to cpml format
 473      * and store them (temporary) in p0..p3 */
 474     cpml_pair_from_cairo(&p0, curve->org);
 475     cpml_pair_from_cairo(&p1, &curve->data[1]);
 476     cpml_pair_from_cairo(&p2, &curve->data[2]);
 477     cpml_pair_from_cairo(&p3, &curve->data[3]);
 478
 479     /* v0 = p1-p0 */
 480     cpml_pair_copy(&v0, &p1);
 481     cpml_pair_sub(&v0, &p0);
 482
 483     /* v3 = p3-p2 */
 484     cpml_pair_copy(&v3, &p3);
 485     cpml_pair_sub(&v3, &p2);
 486
 487     /* pm = point in C(m) offseted the requested @offset distance */
 488     cpml_curve_put_vector_at_time(curve, m, &vm);
 489     cpml_vector_set_length(&vm, offset);
 490     cpml_vector_normal(&vm);
 491     cpml_curve_put_pair_at_time(curve, m, &pm);
 492     cpml_pair_add(&pm, &vm);
 493
 494     /* p0 = p0 + normal of v0 of @offset magnitude (exact value) */
 495     cpml_pair_copy(&vtmp, &v0);
 496     cpml_vector_set_length(&vtmp, offset);
 497     cpml_vector_normal(&vtmp);
 498     cpml_pair_add(&p0, &vtmp);
 499
 500     /* p3 = p3 + normal of v3 of @offset magnitude, as done for p0 */
 501     cpml_pair_copy(&vtmp, &v3);
 502     cpml_vector_set_length(&vtmp, offset);
 503     cpml_vector_normal(&vtmp);
 504     cpml_pair_add(&p3, &vtmp);
 505
 506     if (v0.x*v3.y == v3.x*v0.y) {
 507         /* Inconsistent equations: use the alternative approach */
 508         p1.x = p0.x + v0.x + vm.x * 4/3;
 509         p1.y = p0.y + v0.y + vm.y * 4/3;
 510         p2.x = p3.x - v3.x + vm.x * 4/3;
 511         p2.y = p3.y - v3.y + vm.y * 4/3;
 512     } else {
 513         CpmlPair pk;
 514         double k0, k3;
 515
 516         pk.x = (pm.x - mm*mm*(1+m+m)*p0.x - m*m*(1+mm+mm)*p3.x) / (3*m*(1-m));
 517         pk.y = (pm.y - mm*mm*(1+m+m)*p0.y - m*m*(1+mm+mm)*p3.y) / (3*m*(1-m));
 518
 519         if (v0.x != 0) {
 520             k3 = (pk.y - pk.x*v0.y / v0.x) / (m*(v3.y - v3.x*v0.y / v0.x));
 521             k0 = (pk.x - m*k3*v3.x) / (mm*v0.x);
 522         } else {
 523             k0 = (pk.y - pk.x*v3.y / v3.x) / (mm*(v0.y - v0.x*v3.y / v3.x));
 524             k3 = (pk.x - mm*k0*v0.x) / (m*v3.x);
 525         }
 526
 527         p1.x = p0.x + k0*v0.x;
 528         p1.y = p0.y + k0*v0.y;
 529         p2.x = p3.x + k3*v3.x;
 530         p2.y = p3.y + k3*v3.y;
 531     }
 532
 533     /* Return the new curve in the original array */
 534     cpml_pair_to_cairo(&p0, curve->org);
 535     cpml_pair_to_cairo(&p1, &curve->data[1]);
 536     cpml_pair_to_cairo(&p2, &curve->data[2]);
 537     cpml_pair_to_cairo(&p3, &curve->data[3]);
 538 }
 539
 540
 541 static void
 542 put_extents(const CpmlPrimitive *curve, CpmlExtents *extents)
 543 {
 544     CpmlPair p1, p2, p3, p4;
 545
 546     extents->is_defined = 0;
 547
 548     cpml_pair_from_cairo(&p1, cpml_primitive_get_point(curve, 0));
 549     cpml_pair_from_cairo(&p2, cpml_primitive_get_point(curve, 1));
 550     cpml_pair_from_cairo(&p3, cpml_primitive_get_point(curve, 2));
 551     cpml_pair_from_cairo(&p4, cpml_primitive_get_point(curve, 3));
 552
 553     cpml_extents_pair_add(extents, &p1);
 554     cpml_extents_pair_add(extents, &p2);
 555     cpml_extents_pair_add(extents, &p3);
 556     cpml_extents_pair_add(extents, &p4);
 557 }