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_intersection:
 207  * @curve:  the first curve
 208  * @curve2: the second curve
 209  * @dest:   a vector of at least 4 #CpmlPair
 210  *
 211  * Given two Bézier cubic curves (@curve and @curve2), gets their
 212  * intersection points and store the result in @dest. Because two
 213  * curves can have 4 intersections, @dest MUST be at least an array
 214  * of 4 #CpmlPair.
 215  *
 216  * <important>
 217  * <title>TODO</title>
 218  * <itemizedlist>
 219  * <listitem>To be implemented...</listitem>
 220  * </itemizedlist>
 221  * </important>
 222  *
 223  * Return value: the number of intersections (max 4)
 224  **/
 225 int
 226 cpml_curve_intersection(const CpmlPrimitive *curve,
 227                         const CpmlPrimitive *curve2, CpmlPair *dest)
 228 {
 229     return 0;
 230 }
 231
 232 /**
 233  * cpml_curve_intersection_with_arc:
 234  * @curve: a curve
 235  * @arc:   an arc
 236  * @dest:  a vector of at least 4 #CpmlPair
 237  *
 238  * Given a Bézier cubic @curve and an @arc, gets their intersection
 239  * points and store the result in @dest. Because an arc and a cubic
 240  * curve can have up to 4 intersections, @dest MUST be at least an
 241  * array of 4 #CpmlPair.
 242  *
 243  * <important>
 244  * <title>TODO</title>
 245  * <itemizedlist>
 246  * <listitem>To be implemented...</listitem>
 247  * </itemizedlist>
 248  * </important>
 249  *
 250  * Return value: the number of intersections (max 4)
 251  **/
 252 int
 253 cpml_curve_intersection_with_arc(const CpmlPrimitive *curve,
 254                                  const CpmlPrimitive *arc, CpmlPair *dest)
 255 {
 256     return 0;
 257 }
 258
 259 /**
 260  * cpml_curve_intersection_with_line:
 261  * @curve: a curve
 262  * @line:  a line
 263  * @dest:  a vector of at least 4 #CpmlPair
 264  *
 265  * Given a Bézier cubic @curve and a @line, gets their intersection
 266  * points and store the result in @dest. Because a line and a cubic
 267  * curve can have up to 4 intersections, @dest MUST be at least an
 268  * array of 4 #CpmlPair.
 269  *
 270  * <important>
 271  * <title>TODO</title>
 272  * <itemizedlist>
 273  * <listitem>To be implemented...</listitem>
 274  * </itemizedlist>
 275  * </important>
 276  *
 277  * Return value: the number of intersections (max 4)
 278  **/
 279 int
 280 cpml_curve_intersection_with_line(const CpmlPrimitive *curve,
 281                                   const CpmlPrimitive *line, CpmlPair *dest)
 282 {
 283     return 0;
 284 }
 285
 286 /**
 287  * cpml_curve_offset:
 288  * @curve:  the #CpmlPrimitive curve data
 289  * @offset: distance for the computed parallel curve
 290  *
 291  * Given a cubic Bézier primitive in @curve, this function finds
 292  * the approximated Bézier curve parallel to @curve at distance
 293  * @offset (an offset curve). The four points needed to build the
 294  * new curve are returned in the @curve struct.
 295  *
 296  * To solve the offset problem, a custom algorithm is used. First, the
 297  * resulting curve MUST have the same slope at the start and end point.
 298  * These constraints are not sufficient to resolve the system, so I let
 299  * the curve pass thought a given point (pm, known and got from the
 300  * original curve) at a given time (m, now hardcoded to 0.5).
 301  *
 302  * Firstly, I define some useful variables:
 303  *
 304  * v0 = unitvector(p[1]-p[0]) * offset;
 305  * v3 = unitvector(p[3]-p[2]) * offset;
 306  * p0 = p[0] + normal v0;
 307  * p3 = p[3] + normal v3.
 308  *
 309  * Now I want the curve to have the specified slopes at the start
 310  * and end point. Forcing the same slope at the start point means:
 311  *
 312  * p1 = p0 + k0 v0.
 313  *
 314  * where k0 is an arbitrary factor. Decomposing for x and y components:
 315  *
 316  * p1.x = p0.x + k0 v0.x;
 317  * p1.y = p0.y + k0 v0.y.
 318  *
 319  * Doing the same for the end point gives:
 320  *
 321  * p2.x = p3.x + k3 v3.x;
 322  * p2.y = p3.y + k3 v3.y.
 323  *
 324  * Now I interpolate the curve by forcing it to pass throught pm
 325  * when "time" is m, where 0 < m < 1. The cubic Bézier function is:
 326  *
 327  * C(t) = (1-t)³p0 + 3t(1-t)²p1 + 3t²(1-t)p2 + t³p3.
 328  *
 329  * and forcing t=m and C(t) = pm:
 330  *
 331  * pm = (1-m)³p0 + 3m(1-m)²p1 + 3m²(1-m)p2 + m³p3.
 332  *
 333  * (1-m) p1 + m p2 = (pm - (1-m)³p0 - m³p3) / (3m (1-m)).
 334  *
 335  * So the final system is:
 336  *
 337  * p1.x = p0.x + k0 v0.x;
 338  * p1.y = p0.y + k0 v0.y;
 339  * p2.x = p3.x + k3 v3.x;
 340  * p2.y = p3.y + k3 v3.y;
 341  * (1-m) p1.x + m p2.x = (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m));
 342  * (1-m) p1.y + m p2.y = (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)).
 343  *
 344  * Substituting and resolving for k0 and k3:
 345  *
 346  * (1-m) k0 v0.x + m k3 v3.x =
 347  *     (pm.x - (1-m)³p0.x - m³p3.x) / (3m (1-m)) - (1-m) p0.x - m p3.x;
 348  * (1-m) k0 v0.y + m k3 v3.y =
 349  *     (pm.y - (1-m)³p0.y - m³p3.y) / (3m (1-m)) - (1-m) p0.y - m p3.y.
 350  *
 351  * (1-m) k0 v0.x + m k3 v3.x =
 352  *     (pm.x - (1-m)²(1+2m) p0.x - m²(3-2m) p3.x) / (3m (1-m));
 353  * (1-m) k0 v0.y + m k3 v3.y =
 354  *     (pm.y - (1-m)²(1+2m) p0.y - m²(3-2m) p3.y) / (3m (1-m)).
 355  *
 356  * Let:
 357  *
 358  * pk = (pm - (1-m)²(1+2m) p0 - m²(3-2m) p3) / (3m (1-m)).
 359  *
 360  * gives the following system:
 361  *
 362  * (1-m) k0 v0.x + m k3 v3.x = pk.x;
 363  * (1-m) k0 v0.y + m k3 v3.y = pk.y.
 364  *
 365  * Now I should avoid division by 0 troubles. If either v0.x and v3.x
 366  * are 0, the first equation will be inconsistent. More in general the
 367  * v0.x*v3.y = v3.x*v3.y condition should be avoided. This is the first
 368  * case to check, in which case an alternative approach is used. In the
 369  * other cases the above system can be used.
 370  *
 371  * If v0.x != 0 I can resolve for k0 and then find k3:
 372  *
 373  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 374  * (pk.x - m k3 v3.x) v0.y / v0.x + m k3 v3.y = pk.y.
 375  *
 376  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x);
 377  * k3 m (v3.y - v3.x v0.y / v0.x) = pk.y - pk.x v0.y / v0.x.
 378  *
 379  * k3 = (pk.y - pk.x v0.y / v0.x) / (m (v3.y - v3.x v0.y / v0.x));
 380  * k0 = (pk.x - m k3 v3.x) / ((1-m) v0.x).
 381  *
 382  * If v3.x != 0 I can resolve for k3 and then find k0:
 383  *
 384  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 385  * (1-m) k0 v0.y + (pk.x - (1-m) k0 v0.x) v3.y / v3.x = pk.y.
 386  *
 387  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x);
 388  * k0 (1-m) (v0.y - k0 v0.x v3.y / v3.x) = pk.y - pk.x v3.y / v3.x.
 389  *
 390  * k0 = (pk.y - pk.x v3.y / v3.x) / ((1-m) (v0.y - v0.x v3.y / v3.x));
 391  * k3 = (pk.x - (1-m) k0 v0.x) / (m v3.x).
 392  *
 393  * <important>
 394  * <title>TODO</title>
 395  * <itemizedlist>
 396  * <listitem>By default, interpolation of the new curve is made by offseting
 397  *           the mid point: use a better candidate.</listitem>
 398  * <listitem>When the equations are inconsistent, the alternative approach
 399  *           performs very bad if <varname>v0</varname> and
 400  *           <varname>v3</varname> are opposite or staggered.</listitem>
 401  * </itemizedlist>
 402  * </important>
 403  **/
 404 void
 405 cpml_curve_offset(CpmlPrimitive *curve, double offset)
 406 {
 407     double m, mm;
 408     CpmlVector v0, v3, vm, vtmp;
 409     CpmlPair p0, p1, p2, p3, pm;
 410
 411     m = 0.5;
 412     mm = 1-m;
 413
 414     /* Firstly, convert the curve points from cairo format to cpml format
 415      * and store them (temporary) in p0..p3 */
 416     cpml_pair_from_cairo(&p0, curve->org);
 417     cpml_pair_from_cairo(&p1, &curve->data[1]);
 418     cpml_pair_from_cairo(&p2, &curve->data[2]);
 419     cpml_pair_from_cairo(&p3, &curve->data[3]);
 420
 421     /* v0 = p1-p0 */
 422     cpml_pair_sub(cpml_pair_copy(&v0, &p1), &p0);
 423
 424     /* v3 = p3-p2 */
 425     cpml_pair_sub(cpml_pair_copy(&v3, &p3), &p2);
 426
 427     /* pm = point in C(m) offseted the requested @offset distance */
 428     cpml_curve_vector_at_time(curve, &vm, m);
 429     cpml_vector_set_length(&vm, offset);
 430     cpml_vector_normal(&vm);
 431     cpml_curve_pair_at_time(curve, &pm, m);
 432     cpml_pair_add(&pm, &vm);
 433
 434     /* p0 = p0 + normal of v0 of @offset magnitude (exact value) */
 435     cpml_vector_set_length(cpml_pair_copy(&vtmp, &v0), offset);
 436     cpml_vector_normal(&vtmp);
 437     cpml_pair_add(&p0, &vtmp);
 438
 439     /* p3 = p3 + normal of v3 of @offset magnitude, as done for p0 */
 440     cpml_vector_set_length(cpml_pair_copy(&vtmp, &v3), offset);
 441     cpml_vector_normal(&vtmp);
 442     cpml_pair_add(&p3, &vtmp);
 443
 444     if (v0.x*v3.y == v3.x*v0.y) {
 445         /* Inconsistent equations: use the alternative approach */
 446         p1.x = p0.x + v0.x + vm.x * 4/3;
 447         p1.y = p0.y + v0.y + vm.y * 4/3;
 448         p2.x = p3.x - v3.x + vm.x * 4/3;
 449         p2.y = p3.y - v3.y + vm.y * 4/3;
 450     } else {
 451         CpmlPair pk;
 452         double k0, k3;
 453
 454         pk.x = (pm.x - mm*mm*(1+m+m)*p0.x - m*m*(1+mm+mm)*p3.x) / (3*m*(1-m));
 455         pk.y = (pm.y - mm*mm*(1+m+m)*p0.y - m*m*(1+mm+mm)*p3.y) / (3*m*(1-m));
 456
 457         if (v0.x != 0) {
 458             k3 = (pk.y - pk.x*v0.y / v0.x) / (m*(v3.y - v3.x*v0.y / v0.x));
 459             k0 = (pk.x - m*k3*v3.x) / (mm*v0.x);
 460         } else {
 461             k0 = (pk.y - pk.x*v3.y / v3.x) / (mm*(v0.y - v0.x*v3.y / v3.x));
 462             k3 = (pk.x - mm*k0*v0.x) / (m*v3.x);
 463         }
 464
 465         p1.x = p0.x + k0*v0.x;
 466         p1.y = p0.y + k0*v0.y;
 467         p2.x = p3.x + k3*v3.x;
 468         p2.y = p3.y + k3*v3.y;
 469     }
 470
 471     /* Return the new curve in the original array */
 472     cpml_pair_to_cairo(&p0, curve->org);
 473     cpml_pair_to_cairo(&p1, &curve->data[1]);
 474     cpml_pair_to_cairo(&p2, &curve->data[2]);
 475     cpml_pair_to_cairo(&p3, &curve->data[3]);
 476 }