cpml/cpml-arc.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 /**
  22  * SECTION:cpml-arc
  23  * @Section_Id:CpmlArc
  24  * @title: CpmlArc
  25  * @short_description: Manipulation of circular arcs
  26  *
  27  * The following functions manipulate #CAIRO_PATH_ARC_TO #CpmlPrimitive.
  28  * No validation is made on the input so use the following methods
  29  * only when you are sure the <varname>primitive</varname> argument
  30  * is effectively an arc-to.
  31  *
  32  * The arc primitive is defined by 3 points: the first one is the usual
  33  * implicit point got from the previous primitive, the second point is
  34  * an arbitrary intermediate point laying on the arc and the third point
  35  * is the end of the arc. These points identify univocally an arc:
  36  * furthermore, the intermediate point also gives the side of
  37  * the arc.
  38  *
  39  * As a special case, when the first point is coincident with the end
  40  * point the primitive is considered a circle with diameter defined by
  41  * the segment between the first and the intermediate point.
  42  *
  43  * <important>
  44  * <para>
  45  * An arc is not a native cairo primitive and should be treated specially.
  46  * </para>
  47  * </important>
  48  *
  49  * Using these CPML APIs you are free to use #CAIRO_PATH_ARC_TO whenever
  50  * you want but, if you are directly accessing the struct fields, you
  51  * are responsible of converting arcs to curves before passing them
  52  * to cairo. In other words, do not directly feed #CpmlPath struct to
  53  * cairo (throught cairo_append_path() for example) or at least do not
  54  * expect it will work.
  55  *
  56  * The conversion is provided by two APIs: cpml_arc_to_cairo() and
  57  * cpml_arc_to_curves(). The former directly renders to a cairo context
  58  * and is internally used by all the ..._to_cairo() functions when an
  59  * arc is met. The latter provided a more powerful (and more complex)
  60  * approach as it allows to specify the number of curves to use and do
  61  * not need a cairo context.
  62  **/
  63
  64
  65 #include "cpml-arc.h"
  66 #include "cpml-pair.h"
  67
  68 #include <stdlib.h>
  69 #include <math.h>
  70
  71
  72 /* Hardcoded max angle of the arc to be approximated by a Bézier curve:
  73  * this influence the arc quality (the default value is got from cairo) */
  74 #define ARC_MAX_ANGLE   M_PI_2
  75
  76
  77 static cairo_bool_t     get_center      (const CpmlPair *p,
  78                                          CpmlPair       *dest);
  79 static void             get_angles      (const CpmlPair *p,
  80                                          const CpmlPair *center,
  81                                          double         *start,
  82                                          double         *end);
  83 static void             arc_to_curve    (CpmlPrimitive  *curve,
  84                                          const CpmlPair *center,
  85                                          double          r,
  86                                          double          start,
  87                                          double          end);
  88
  89
  90 /**
  91  * cpml_arc_type_get_npoints:
  92  *
  93  * Returns the number of point needed to properly specify an arc primitive.
  94  *
  95  * Return value: 3
  96  **/
  97 int
  98 cpml_arc_type_get_npoints(void)
  99 {
 100     return 3;
 101 }
 102
 103 /**
 104  * cpml_arc_info:
 105  * @arc:    the #CpmlPrimitive arc data
 106  * @center: where to store the center coordinates (can be %NULL)
 107  * @r:      where to store the radius (can be %NULL)
 108  * @start:  where to store the starting angle (can be %NULL)
 109  * @end:    where to store the ending angle (can be %NULL)
 110  *
 111  * Given an @arc, this function calculates and returns its basic data.
 112  * Any pointer can be %NULL, in which case the requested info is not
 113  * returned. This function can fail (when the three points lay on a
 114  * straight line, for example) in which case 0 is returned and no
 115  * data can be considered valid.
 116  *
 117  * The radius @r can be 0 when the three points are coincidents: a
 118  * circle with radius 0 is considered a valid path.
 119  *
 120  * When the start and end angle are returned, together with their
 121  * values these angles implicitely gives another important information:
 122  * the arc direction.
 123  *
 124  * If @start < @end the arc must be rendered with increasing angle
 125  * value (clockwise direction using the ordinary cairo coordinate
 126  * system) while if @start > @end the arc must be rendered in reverse
 127  * order (that is counterclockwise in the cairo world). This is the
 128  * reason the angle values are returned in the range
 129  * { -M_PI < value < 3*M_PI } inclusive instead of the usual
 130  * { -M_PI < value < M_PI } range.
 131  *
 132  * Return value: 1 if the function worked succesfully, 0 on errors
 133  **/
 134 cairo_bool_t
 135 cpml_arc_info(const CpmlPrimitive *arc, CpmlPair *center,
 136               double *r, double *start, double *end)
 137 {
 138     CpmlPair p[3], l_center;
 139
 140     cpml_pair_from_cairo(&p[0], arc->org);
 141     cpml_pair_from_cairo(&p[1], &arc->data[1]);
 142     cpml_pair_from_cairo(&p[2], &arc->data[2]);
 143
 144     if (!get_center(p, &l_center))
 145         return 0;
 146
 147     if (center)
 148         *center = l_center;
 149
 150     if (r != NULL)
 151         *r = cpml_pair_distance(&p[0], &l_center);
 152
 153     if (start != NULL || end != NULL) {
 154         double l_start, l_end;
 155
 156         get_angles(p, &l_center, &l_start, &l_end);
 157
 158         if (start != NULL)
 159             *start = l_start;
 160         if (end != NULL)
 161             *end = l_end;
 162     }
 163
 164     return 1;
 165 }
 166
 167 /**
 168  * cpml_arc_length:
 169  * @arc:  the #CpmlPrimitive arc data
 170  *
 171  * Given the @arc primitive, returns its length.
 172  *
 173  * Return value: the requested length or 0 on errors
 174  **/
 175 double
 176 cpml_arc_length(const CpmlPrimitive *arc)
 177 {
 178     double r, start, end, delta;
 179
 180     if (!cpml_arc_info(arc, NULL, &r, &start, &end) || start == end)
 181         return 0.;
 182
 183     delta = end - start;
 184     if (delta < 0)
 185         delta += M_PI*2;
 186
 187     return r*delta;
 188 }
 189
 190 /**
 191  * cpml_arc_extents:
 192  * @arc: the #CpmlPrimitive arc data
 193  * @extents: where to store the extents
 194  *
 195  * Given an @arc primitive, returns its boundary box in @extents.
 196  **/
 197 void
 198 cpml_arc_extents(const CpmlPrimitive *arc, CpmlExtents *extents)
 199 {
 200     double r, start, end;
 201     CpmlPair pair;
 202
 203     if (!cpml_arc_info(arc, NULL, &r, &start, &end))
 204         return;
 205
 206     /* Add the right quadrant point if needed */
 207     if ((start < 0 && end > 0) ||
 208         (end < M_PI * 2 && start > M_PI * 2)) {
 209         pair.x = r;
 210         pair.y = 0;
 211         cpml_extents_pair_add(extents, &pair);
 212     }
 213
 214     /* Add the bottom quadrant point if needed */
 215     if ((start < M_PI_2 && end > M_PI_2) ||
 216         (end < M_PI_2 * 5 && start > M_PI_2 * 5)) {
 217         pair.x = 0;
 218         pair.y = r;
 219         cpml_extents_pair_add(extents, &pair);
 220     }
 221
 222     /* Add the left quadrant point if needed */
 223     if (start < M_PI && end > M_PI) {
 224         pair.x = -r;
 225         pair.y = 0;
 226         cpml_extents_pair_add(extents, &pair);
 227     }
 228
 229     /* Add the top quadrant point if needed */
 230     if (start < M_PI_2 * 3 && end > M_PI) {
 231         pair.x = 0;
 232         pair.y = -r;
 233         cpml_extents_pair_add(extents, &pair);
 234     }
 235
 236     /* Add the start point */
 237     cpml_pair_from_cairo(&pair, cpml_primitive_get_point(arc, 0));
 238     cpml_extents_pair_add(extents, &pair);
 239
 240     /* Add the end point */
 241     cpml_pair_from_cairo(&pair, cpml_primitive_get_point(arc, -1));
 242     cpml_extents_pair_add(extents, &pair);
 243 }
 244
 245 /**
 246  * cpml_arc_pair_at:
 247  * @arc:  the #CpmlPrimitive arc data
 248  * @pair: the destination #CpmlPair
 249  * @pos:  the position value
 250  *
 251  * Given an @arc, finds the coordinates at position @pos (where 0 is
 252  * the start and 1 is the end) and stores the result in @pair.
 253  *
 254  * @pos can also be outside the 0..1 limit, as interpolating on an
 255  * arc is quite trivial.
 256  **/
 257 void
 258 cpml_arc_pair_at(const CpmlPrimitive *arc, CpmlPair *pair, double pos)
 259 {
 260     if (pos == 0.) {
 261         cpml_pair_from_cairo(pair, arc->org);
 262     } else if (pos == 1.) {
 263         cpml_pair_from_cairo(pair, &arc->data[2]);
 264     } else {
 265         CpmlPair center;
 266         double r, start, end, angle;
 267
 268         if (!cpml_arc_info(arc, &center, &r, &start, &end))
 269             return;
 270
 271         angle = (end-start)*pos + start;
 272         cpml_vector_from_angle(pair, angle);
 273         cpml_vector_set_length(pair, r);
 274         cpml_pair_add(pair, &center);
 275     }
 276 }
 277
 278 /**
 279  * cpml_arc_vector_at:
 280  * @arc:    the #CpmlPrimitive arc data
 281  * @vector: the destination vector
 282  * @pos:    the position value
 283  *
 284  * Given an @arc, finds the slope at position @pos (where 0 is
 285  * the start and 1 is the end) and stores the result in @vector.
 286  *
 287  * @pos can also be outside the 0..1 limit, as interpolating on an
 288  * arc is quite trivial.
 289  **/
 290 void
 291 cpml_arc_vector_at(const CpmlPrimitive *arc, CpmlVector *vector, double pos)
 292 {
 293     double start, end, angle;
 294
 295     if (!cpml_arc_info(arc, NULL, NULL, &start, &end))
 296         return;
 297
 298     angle = (end-start)*pos + start;
 299     cpml_vector_from_angle(vector, angle);
 300     cpml_vector_normal(vector);
 301
 302     if (start > end)
 303         cpml_pair_negate(vector);
 304 }
 305
 306 /**
 307  * cpml_arc_near_pos:
 308  * @arc: the #CpmlPrimitive arc data
 309  * @pair: the coordinates of the subject point
 310  *
 311  * Returns the pos value of the point on @arc nearest to @pair.
 312  * The returned value is always between 0 and 1.
 313  *
 314  * <important>
 315  * <title>TODO</title>
 316  * <itemizedlist>
 317  * <listitem>To be implemented...</listitem>
 318  * </itemizedlist>
 319  * </important>
 320  *
 321  * Return value: the pos value, always between 0 and 1
 322  **/
 323 double
 324 cpml_arc_near_pos(const CpmlPrimitive *arc, const CpmlPair *pair)
 325 {
 326     /* TODO */
 327
 328     return 0;
 329 }
 330
 331 /**
 332  * cpml_arc_intersection:
 333  * @arc:  the first arc
 334  * @arc2: the second arc
 335  * @dest: a vector of #CpmlPair
 336  * @max:  maximum number of intersections to return
 337  *        (that is, the size of @dest)
 338  *
 339  * Given two arcs (@arc and @arc2), gets their intersection points
 340  * and store the result in @dest. Keep in mind two arcs can have
 341  * up to 2 intersections.
 342  *
 343  * If @max is 0, the function returns 0 immediately without any
 344  * further processing. If @arc and @arc2 are cohincident (same
 345  * center and same radius), their intersections are not considered.
 346  *
 347  * <important>
 348  * <title>TODO</title>
 349  * <itemizedlist>
 350  * <listitem>To be implemented...</listitem>
 351  * </itemizedlist>
 352  * </important>
 353  *
 354  * Return value: the number of intersections found (max 2)
 355  *               or 0 if the primitives do not intersect
 356  **/
 357 int
 358 cpml_arc_intersection(const CpmlPrimitive *arc, const CpmlPrimitive *arc2,
 359                       CpmlPair *dest, int max)
 360 {
 361     return 0;
 362 }
 363
 364 /**
 365  * cpml_arc_intersection_with_line:
 366  * @arc:  an arc
 367  * @line: a line
 368  * @dest: a vector of #CpmlPair
 369  * @max:  maximum number of intersections to return
 370  *        (that is, the size of @dest)
 371  *
 372  * Given an @arc and a @line, gets their intersection points
 373  * and store the result in @dest. Keep in mind an arc and a
 374  * line can have up to 2 intersections.
 375  *
 376  * If @max is 0, the function returns 0 immediately without any
 377  * further processing.
 378  *
 379  * <important>
 380  * <title>TODO</title>
 381  * <itemizedlist>
 382  * <listitem>To be implemented...</listitem>
 383  * </itemizedlist>
 384  * </important>
 385  *
 386  * Return value: the number of intersections found (max 2)
 387  *               or 0 if the primitives do not intersect
 388  **/
 389 int
 390 cpml_arc_intersection_with_line(const CpmlPrimitive *arc,
 391                                 const CpmlPrimitive *line,
 392                                 CpmlPair *dest, int max)
 393 {
 394     return 0;
 395 }
 396
 397 /**
 398  * cpml_arc_offset:
 399  * @arc:    the #CpmlPrimitive arc data
 400  * @offset: distance for the computed parallel arc
 401  *
 402  * Given an @arc, this function computes the parallel arc at
 403  * distance @offset. The three points needed to build the
 404  * new arc are returned in the @arc data (substituting the
 405  * previous ones.
 406  **/
 407 void
 408 cpml_arc_offset(CpmlPrimitive *arc, double offset)
 409 {
 410     CpmlPair p[3], center;
 411     double r;
 412
 413     cpml_pair_from_cairo(&p[0], arc->org);
 414     cpml_pair_from_cairo(&p[1], &arc->data[1]);
 415     cpml_pair_from_cairo(&p[2], &arc->data[2]);
 416
 417     if (!get_center(p, &center))
 418         return;
 419
 420     r = cpml_pair_distance(&p[0], &center) + offset;
 421
 422     /* Offset the three points by calculating their vector from the center,
 423      * setting the new radius as length and readding the center */
 424     cpml_pair_sub(&p[0], &center);
 425     cpml_pair_sub(&p[1], &center);
 426     cpml_pair_sub(&p[2], &center);
 427
 428     cpml_vector_set_length(&p[0], r);
 429     cpml_vector_set_length(&p[1], r);
 430     cpml_vector_set_length(&p[2], r);
 431
 432     cpml_pair_add(&p[0], &center);
 433     cpml_pair_add(&p[1], &center);
 434     cpml_pair_add(&p[2], &center);
 435
 436     cpml_pair_to_cairo(&p[0], arc->org);
 437     cpml_pair_to_cairo(&p[1], &arc->data[1]);
 438     cpml_pair_to_cairo(&p[2], &arc->data[2]);
 439 }
 440
 441 /**
 442  * cpml_arc_to_cairo:
 443  * @arc: the #CpmlPrimitive arc data
 444  * @cr:  the destination cairo context
 445  *
 446  * Renders @arc to the @cr cairo context. As cairo does not support
 447  * arcs natively, it is approximated using one or more Bézier curves.
 448  *
 449  * The number of curves used is dependent from the angle of the arc.
 450  * Anyway, this function uses internally the hardcoded %M_PI_2 value
 451  * as threshold value. This means the maximum arc approximated by a
 452  * single curve will be a quarter of a circle and, consequently, a
 453  * whole circle will be approximated by 4 Bézier curves.
 454  **/
 455 void
 456 cpml_arc_to_cairo(const CpmlPrimitive *arc, cairo_t *cr)
 457 {
 458     CpmlPair center;
 459     double r, start, end;
 460     int n_curves;
 461     double step, angle;
 462     CpmlPrimitive curve;
 463     cairo_path_data_t data[4];
 464
 465     if (!cpml_arc_info(arc, &center, &r, &start, &end))
 466         return;
 467
 468     n_curves = ceil(fabs(end-start) / ARC_MAX_ANGLE);
 469     step = (end-start) / (double) n_curves;
 470     curve.data = data;
 471
 472     for (angle = start; n_curves--; angle += step) {
 473         arc_to_curve(&curve, &center, r, angle, angle+step);
 474         cairo_curve_to(cr,
 475                        curve.data[1].point.x, curve.data[1].point.y,
 476                        curve.data[2].point.x, curve.data[2].point.y,
 477                        curve.data[3].point.x, curve.data[3].point.y);
 478     }
 479 }
 480
 481 /**
 482  * cpml_arc_to_curves:
 483  * @arc:      the #CpmlPrimitive arc data
 484  * @segment:  the destination #CpmlSegment
 485  * @n_curves: number of Bézier to use
 486  *
 487  * Converts @arc to a serie of @n_curves Bézier curves and puts them
 488  * inside @segment. Obviously, @segment must have enough space to
 489  * contain at least @n_curves curves.
 490  *
 491  * This function works in a similar way as cpml_arc_to_cairo() but
 492  * has two important differences: it does not need a cairo context
 493  * and the number of curves to be generated is explicitely defined.
 494  * The latter difference allows a more specific error control from
 495  * the application: in the file src/cairo-arc.c, found in the cairo
 496  * tarball (at least in cairo-1.9.1), there is a table showing the
 497  * magnitude of error of this curve approximation algorithm.
 498  **/
 499 void
 500 cpml_arc_to_curves(const CpmlPrimitive *arc, CpmlSegment *segment,
 501                    int n_curves)
 502 {
 503     CpmlPair center;
 504     double r, start, end;
 505     double step, angle;
 506     CpmlPrimitive curve;
 507
 508     if (!cpml_arc_info(arc, &center, &r, &start, &end))
 509         return;
 510
 511     step = (end-start) / (double) n_curves;
 512     segment->num_data = n_curves*4;
 513     curve.segment = segment;
 514     curve.data = segment->data;
 515
 516     for (angle = start; n_curves--; angle += step) {
 517         arc_to_curve(&curve, &center, r, angle, angle+step);
 518         curve.data += 4;
 519     }
 520 }
 521
 522
 523 static cairo_bool_t
 524 get_center(const CpmlPair *p, CpmlPair *dest)
 525 {
 526     CpmlPair b, c;
 527     double d, b2, c2;
 528
 529     /* When p[0] == p[2], p[0]..p[1] is considered the diameter of a circle */
 530     if (p[0].x == p[2].x && p[0].y == p[2].y) {
 531         dest->x = (p[0].x + p[1].x) / 2;
 532         dest->y = (p[0].y + p[1].y) / 2;
 533         return 1;
 534     }
 535
 536     /* Translate the 3 points of -p0, to simplify the formula */
 537     cpml_pair_sub(cpml_pair_copy(&b, &p[1]), &p[0]);
 538     cpml_pair_sub(cpml_pair_copy(&c, &p[2]), &p[0]);
 539
 540     /* Check for division by 0, that is the case where the 3 given points
 541      * are laying on a straight line and there is no fitting circle */
 542     d = (b.x*c.y - b.y*c.x) * 2;
 543     if (d == 0.)
 544         return 0;
 545
 546     b2 = b.x*b.x + b.y*b.y;
 547     c2 = c.x*c.x + c.y*c.y;
 548
 549     dest->x = (c.y*b2 - b.y*c2) / d + p[0].x;
 550     dest->y = (b.x*c2 - c.x*b2) / d + p[0].y;
 551
 552     return 1;
 553 }
 554
 555 static void
 556 get_angles(const CpmlPair *p, const CpmlPair *center,
 557            double *start, double *end)
 558 {
 559     CpmlVector vector;
 560     double mid;
 561
 562     /* Calculate the starting angle */
 563     cpml_pair_sub(cpml_pair_copy(&vector, &p[0]), center);
 564     *start = cpml_vector_angle(&vector);
 565
 566     if (p[0].x == p[2].x && p[0].y == p[2].y) {
 567         /* When p[0] and p[2] are cohincidents, p[0]..p[1] is the diameter
 568          * of a circle: return by convention start=start end=start+2PI */
 569         *end = *start + M_PI*2;
 570     } else {
 571         /* Calculate the mid and end angle */
 572         cpml_pair_sub(cpml_pair_copy(&vector, &p[1]), center);
 573         mid = cpml_vector_angle(&vector);
 574         cpml_pair_sub(cpml_pair_copy(&vector, &p[2]), center);
 575         *end = cpml_vector_angle(&vector);
 576
 577         if (*end > *start) {
 578             if (mid > *end || mid < *start)
 579                 *start += M_PI*2;
 580         } else {
 581             if (mid < *end || mid > *start)
 582                 *end += M_PI*2;
 583         }
 584     }
 585 }
 586
 587 static void
 588 arc_to_curve(CpmlPrimitive *curve, const CpmlPair *center,
 589              double r, double start, double end)
 590 {
 591     double r_sin1, r_cos1;
 592     double r_sin2, r_cos2;
 593     double h;
 594
 595     r_sin1 = r*sin(start);
 596     r_cos1 = r*cos(start);
 597     r_sin2 = r*sin(end);
 598     r_cos2 = r*cos(end);
 599
 600     h = 4./3. * tan((end-start) / 4.);
 601
 602     curve->data[0].header.type = CAIRO_PATH_CURVE_TO;
 603     curve->data[0].header.length = 4;
 604     curve->data[1].point.x = center->x + r_cos1 - h*r_sin1;
 605     curve->data[1].point.y = center->y + r_sin1 + h*r_cos1;
 606     curve->data[2].point.x = center->x + r_cos2 + h*r_sin2;
 607     curve->data[2].point.y = center->y + r_sin2 - h*r_cos2;
 608     curve->data[3].point.x = center->x + r_cos2;
 609     curve->data[3].point.y = center->y + r_sin2;
 610 }