src/cairo-arc.c

   1 /* cairo - a vector graphics library with display and print output
   2  *
   3  * Copyright © 2002 University of Southern California
   4  *
   5  * This library is free software; you can redistribute it and/or
   6  * modify it either under the terms of the GNU Lesser General Public
   7  * License version 2.1 as published by the Free Software Foundation
   8  * (the "LGPL") or, at your option, under the terms of the Mozilla
   9  * Public License Version 1.1 (the "MPL"). If you do not alter this
  10  * notice, a recipient may use your version of this file under either
  11  * the MPL or the LGPL.
  12  *
  13  * You should have received a copy of the LGPL along with this library
  14  * in the file COPYING-LGPL-2.1; if not, write to the Free Software
  15  * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
  16  * You should have received a copy of the MPL along with this library
  17  * in the file COPYING-MPL-1.1
  18  *
  19  * The contents of this file are subject to the Mozilla Public License
  20  * Version 1.1 (the "License"); you may not use this file except in
  21  * compliance with the License. You may obtain a copy of the License at
  22  * http://www.mozilla.org/MPL/
  23  *
  24  * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
  25  * OF ANY KIND, either express or implied. See the LGPL or the MPL for
  26  * the specific language governing rights and limitations.
  27  *
  28  * The Original Code is the cairo graphics library.
  29  *
  30  * The Initial Developer of the Original Code is University of Southern
  31  * California.
  32  *
  33  * Contributor(s):
  34  *      Carl D. Worth <cworth@cworth.org>
  35  */
  36
  37 #include "cairoint.h"
  38
  39 #include "cairo-arc-private.h"
  40
  41 /* Spline deviation from the circle in radius would be given by:
  42
  43         error = sqrt (x**2 + y**2) - 1
  44
  45    A simpler error function to work with is:
  46
  47         e = x**2 + y**2 - 1
  48
  49    From "Good approximation of circles by curvature-continuous Bezier
  50    curves", Tor Dokken and Morten Daehlen, Computer Aided Geometric
  51    Design 8 (1990) 22-41, we learn:
  52
  53         abs (max(e)) = 4/27 * sin**6(angle/4) / cos**2(angle/4)
  54
  55    and
  56         abs (error) =~ 1/2 * e
  57
  58    Of course, this error value applies only for the particular spline
  59    approximation that is used in _cairo_gstate_arc_segment.
  60 */
  61 static double
  62 _arc_error_normalized (double angle)
  63 {
  64     return 2.0/27.0 * pow (sin (angle / 4), 6) / pow (cos (angle / 4), 2);
  65 }
  66
  67 static double
  68 _arc_max_angle_for_tolerance_normalized (double tolerance)
  69 {
  70     double angle, error;
  71     int i;
  72
  73     /* Use table lookup to reduce search time in most cases. */
  74     struct {
  75         double angle;
  76         double error;
  77     } table[] = {
  78         { M_PI / 1.0,   0.0185185185185185036127 },
  79         { M_PI / 2.0,   0.000272567143730179811158 },
  80         { M_PI / 3.0,   2.38647043651461047433e-05 },
  81         { M_PI / 4.0,   4.2455377443222443279e-06 },
  82         { M_PI / 5.0,   1.11281001494389081528e-06 },
  83         { M_PI / 6.0,   3.72662000942734705475e-07 },
  84         { M_PI / 7.0,   1.47783685574284411325e-07 },
  85         { M_PI / 8.0,   6.63240432022601149057e-08 },
  86         { M_PI / 9.0,   3.2715520137536980553e-08 },
  87         { M_PI / 10.0,  1.73863223499021216974e-08 },
  88         { M_PI / 11.0,  9.81410988043554039085e-09 },
  89     };
  90     int table_size = ARRAY_LENGTH (table);
  91
  92     for (i = 0; i < table_size; i++)
  93         if (table[i].error < tolerance)
  94             return table[i].angle;
  95
  96     ++i;
  97     do {
  98         angle = M_PI / i++;
  99         error = _arc_error_normalized (angle);
 100     } while (error > tolerance);
 101
 102     return angle;
 103 }
 104
 105 static int
 106 _arc_segments_needed (double          angle,
 107                       double          radius,
 108                       cairo_matrix_t *ctm,
 109                       double          tolerance)
 110 {
 111     double major_axis, max_angle;
 112
 113     /* the error is amplified by at most the length of the
 114      * major axis of the circle; see cairo-pen.c for a more detailed analysis
 115      * of this. */
 116     major_axis = _cairo_matrix_transformed_circle_major_axis (ctm, radius);
 117     max_angle = _arc_max_angle_for_tolerance_normalized (tolerance / major_axis);
 118
 119     return (int) ceil (angle / max_angle);
 120 }
 121
 122 /* We want to draw a single spline approximating a circular arc radius
 123    R from angle A to angle B. Since we want a symmetric spline that
 124    matches the endpoints of the arc in position and slope, we know
 125    that the spline control points must be:
 126
 127         (R * cos(A), R * sin(A))
 128         (R * cos(A) - h * sin(A), R * sin(A) + h * cos (A))
 129         (R * cos(B) + h * sin(B), R * sin(B) - h * cos (B))
 130         (R * cos(B), R * sin(B))
 131
 132    for some value of h.
 133
 134    "Approximation of circular arcs by cubic poynomials", Michael
 135    Goldapp, Computer Aided Geometric Design 8 (1991) 227-238, provides
 136    various values of h along with error analysis for each.
 137
 138    From that paper, a very practical value of h is:
 139
 140         h = 4/3 * tan(angle/4)
 141
 142    This value does not give the spline with minimal error, but it does
 143    provide a very good approximation, (6th-order convergence), and the
 144    error expression is quite simple, (see the comment for
 145    _arc_error_normalized).
 146 */
 147 static void
 148 _cairo_arc_segment (cairo_t *cr,
 149                     double   xc,
 150                     double   yc,
 151                     double   radius,
 152                     double   angle_A,
 153                     double   angle_B)
 154 {
 155     double r_sin_A, r_cos_A;
 156     double r_sin_B, r_cos_B;
 157     double h;
 158
 159     r_sin_A = radius * sin (angle_A);
 160     r_cos_A = radius * cos (angle_A);
 161     r_sin_B = radius * sin (angle_B);
 162     r_cos_B = radius * cos (angle_B);
 163
 164     h = 4.0/3.0 * tan ((angle_B - angle_A) / 4.0);
 165
 166     cairo_curve_to (cr,
 167                     xc + r_cos_A - h * r_sin_A,
 168                     yc + r_sin_A + h * r_cos_A,
 169                     xc + r_cos_B + h * r_sin_B,
 170                     yc + r_sin_B - h * r_cos_B,
 171                     xc + r_cos_B,
 172                     yc + r_sin_B);
 173 }
 174
 175 static void
 176 _cairo_arc_in_direction (cairo_t          *cr,
 177                          double            xc,
 178                          double            yc,
 179                          double            radius,
 180                          double            angle_min,
 181                          double            angle_max,
 182                          cairo_direction_t dir)
 183 {
 184     if (cairo_status (cr))
 185         return;
 186
 187     while (angle_max - angle_min > 4 * M_PI)
 188         angle_max -= 2 * M_PI;
 189
 190     /* Recurse if drawing arc larger than pi */
 191     if (angle_max - angle_min > M_PI) {
 192         double angle_mid = angle_min + (angle_max - angle_min) / 2.0;
 193         if (dir == CAIRO_DIRECTION_FORWARD) {
 194             _cairo_arc_in_direction (cr, xc, yc, radius,
 195                                      angle_min, angle_mid,
 196                                      dir);
 197
 198             _cairo_arc_in_direction (cr, xc, yc, radius,
 199                                      angle_mid, angle_max,
 200                                      dir);
 201         } else {
 202             _cairo_arc_in_direction (cr, xc, yc, radius,
 203                                      angle_mid, angle_max,
 204                                      dir);
 205
 206             _cairo_arc_in_direction (cr, xc, yc, radius,
 207                                      angle_min, angle_mid,
 208                                      dir);
 209         }
 210     } else if (angle_max != angle_min) {
 211         cairo_matrix_t ctm;
 212         int i, segments;
 213         double angle, angle_step;
 214
 215         cairo_get_matrix (cr, &ctm);
 216         segments = _arc_segments_needed (angle_max - angle_min,
 217                                          radius, &ctm,
 218                                          cairo_get_tolerance (cr));
 219         angle_step = (angle_max - angle_min) / (double) segments;
 220
 221         if (dir == CAIRO_DIRECTION_FORWARD) {
 222             angle = angle_min;
 223         } else {
 224             angle = angle_max;
 225             angle_step = - angle_step;
 226         }
 227
 228         for (i = 0; i < segments; i++, angle += angle_step) {
 229             _cairo_arc_segment (cr, xc, yc,
 230                                 radius,
 231                                 angle,
 232                                 angle + angle_step);
 233         }
 234     }
 235 }
 236
 237 /**
 238  * _cairo_arc_path
 239  * @cr: a cairo context
 240  * @xc: X position of the center of the arc
 241  * @yc: Y position of the center of the arc
 242  * @radius: the radius of the arc
 243  * @angle1: the start angle, in radians
 244  * @angle2: the end angle, in radians
 245  *
 246  * Compute a path for the given arc and append it onto the current
 247  * path within @cr. The arc will be accurate within the current
 248  * tolerance and given the current transformation.
 249  **/
 250 void
 251 _cairo_arc_path (cairo_t *cr,
 252                  double   xc,
 253                  double   yc,
 254                  double   radius,
 255                  double   angle1,
 256                  double   angle2)
 257 {
 258     _cairo_arc_in_direction (cr, xc, yc,
 259                              radius,
 260                              angle1, angle2,
 261                              CAIRO_DIRECTION_FORWARD);
 262 }
 263
 264 /**
 265  * _cairo_arc_path_negative:
 266  * @xc: X position of the center of the arc
 267  * @yc: Y position of the center of the arc
 268  * @radius: the radius of the arc
 269  * @angle1: the start angle, in radians
 270  * @angle2: the end angle, in radians
 271  * @ctm: the current transformation matrix
 272  * @tolerance: the current tolerance value
 273  * @path: the path onto which the arc will be appended
 274  *
 275  * Compute a path for the given arc (defined in the negative
 276  * direction) and append it onto the current path within @cr. The arc
 277  * will be accurate within the current tolerance and given the current
 278  * transformation.
 279  **/
 280 void
 281 _cairo_arc_path_negative (cairo_t *cr,
 282                           double   xc,
 283                           double   yc,
 284                           double   radius,
 285                           double   angle1,
 286                           double   angle2)
 287 {
 288     _cairo_arc_in_direction (cr, xc, yc,
 289                              radius,
 290                              angle2, angle1,
 291                              CAIRO_DIRECTION_REVERSE);
 292 }