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  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., 51 Franklin Street, Suite 500, Boston, MA 02110-1335, 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 ceil (fabs (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.     } else {
  235.         cairo_line_to (cr,
  236.                        xc + radius * cos (angle_min),
  237.                        yc + radius * sin (angle_min));
  238.     }
  239. }
  240.  
  241. /**
  242.  * _cairo_arc_path
  243.  * @cr: a cairo context
  244.  * @xc: X position of the center of the arc
  245.  * @yc: Y position of the center of the arc
  246.  * @radius: the radius of the arc
  247.  * @angle1: the start angle, in radians
  248.  * @angle2: the end angle, in radians
  249.  *
  250.  * Compute a path for the given arc and append it onto the current
  251.  * path within @cr. The arc will be accurate within the current
  252.  * tolerance and given the current transformation.
  253.  **/
  254. void
  255. _cairo_arc_path (cairo_t *cr,
  256.                  double   xc,
  257.                  double   yc,
  258.                  double   radius,
  259.                  double   angle1,
  260.                  double   angle2)
  261. {
  262.     _cairo_arc_in_direction (cr, xc, yc,
  263.                              radius,
  264.                              angle1, angle2,
  265.                              CAIRO_DIRECTION_FORWARD);
  266. }
  267.  
  268. /**
  269.  * _cairo_arc_path_negative:
  270.  * @xc: X position of the center of the arc
  271.  * @yc: Y position of the center of the arc
  272.  * @radius: the radius of the arc
  273.  * @angle1: the start angle, in radians
  274.  * @angle2: the end angle, in radians
  275.  * @ctm: the current transformation matrix
  276.  * @tolerance: the current tolerance value
  277.  * @path: the path onto which the arc will be appended
  278.  *
  279.  * Compute a path for the given arc (defined in the negative
  280.  * direction) and append it onto the current path within @cr. The arc
  281.  * will be accurate within the current tolerance and given the current
  282.  * transformation.
  283.  **/
  284. void
  285. _cairo_arc_path_negative (cairo_t *cr,
  286.                           double   xc,
  287.                           double   yc,
  288.                           double   radius,
  289.                           double   angle1,
  290.                           double   angle2)
  291. {
  292.     _cairo_arc_in_direction (cr, xc, yc,
  293.                              radius,
  294.                              angle2, angle1,
  295.                              CAIRO_DIRECTION_REVERSE);
  296. }
  297.