forked from KolibriOS/kolibrios
309 lines
8.7 KiB
C
309 lines
8.7 KiB
C
/* cairo - a vector graphics library with display and print output
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*
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* Copyright © 2002 University of Southern California
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*
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* This library is free software; you can redistribute it and/or
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* modify it either under the terms of the GNU Lesser General Public
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* License version 2.1 as published by the Free Software Foundation
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* (the "LGPL") or, at your option, under the terms of the Mozilla
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* Public License Version 1.1 (the "MPL"). If you do not alter this
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* notice, a recipient may use your version of this file under either
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* the MPL or the LGPL.
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*
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* You should have received a copy of the LGPL along with this library
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* in the file COPYING-LGPL-2.1; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Suite 500, Boston, MA 02110-1335, USA
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* You should have received a copy of the MPL along with this library
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* in the file COPYING-MPL-1.1
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*
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* The contents of this file are subject to the Mozilla Public License
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* Version 1.1 (the "License"); you may not use this file except in
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* compliance with the License. You may obtain a copy of the License at
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* http://www.mozilla.org/MPL/
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*
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* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
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* OF ANY KIND, either express or implied. See the LGPL or the MPL for
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* the specific language governing rights and limitations.
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*
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* The Original Code is the cairo graphics library.
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*
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* The Initial Developer of the Original Code is University of Southern
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* California.
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*
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* Contributor(s):
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* Carl D. Worth <cworth@cworth.org>
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*/
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#include "cairoint.h"
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#include "cairo-arc-private.h"
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#define MAX_FULL_CIRCLES 65536
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/* Spline deviation from the circle in radius would be given by:
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error = sqrt (x**2 + y**2) - 1
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A simpler error function to work with is:
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e = x**2 + y**2 - 1
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From "Good approximation of circles by curvature-continuous Bezier
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curves", Tor Dokken and Morten Daehlen, Computer Aided Geometric
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Design 8 (1990) 22-41, we learn:
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abs (max(e)) = 4/27 * sin**6(angle/4) / cos**2(angle/4)
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and
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abs (error) =~ 1/2 * e
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Of course, this error value applies only for the particular spline
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approximation that is used in _cairo_gstate_arc_segment.
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*/
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static double
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_arc_error_normalized (double angle)
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{
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return 2.0/27.0 * pow (sin (angle / 4), 6) / pow (cos (angle / 4), 2);
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}
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static double
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_arc_max_angle_for_tolerance_normalized (double tolerance)
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{
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double angle, error;
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int i;
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/* Use table lookup to reduce search time in most cases. */
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struct {
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double angle;
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double error;
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} table[] = {
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{ M_PI / 1.0, 0.0185185185185185036127 },
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{ M_PI / 2.0, 0.000272567143730179811158 },
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{ M_PI / 3.0, 2.38647043651461047433e-05 },
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{ M_PI / 4.0, 4.2455377443222443279e-06 },
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{ M_PI / 5.0, 1.11281001494389081528e-06 },
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{ M_PI / 6.0, 3.72662000942734705475e-07 },
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{ M_PI / 7.0, 1.47783685574284411325e-07 },
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{ M_PI / 8.0, 6.63240432022601149057e-08 },
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{ M_PI / 9.0, 3.2715520137536980553e-08 },
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{ M_PI / 10.0, 1.73863223499021216974e-08 },
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{ M_PI / 11.0, 9.81410988043554039085e-09 },
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};
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int table_size = ARRAY_LENGTH (table);
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for (i = 0; i < table_size; i++)
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if (table[i].error < tolerance)
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return table[i].angle;
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++i;
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do {
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angle = M_PI / i++;
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error = _arc_error_normalized (angle);
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} while (error > tolerance);
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return angle;
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}
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static int
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_arc_segments_needed (double angle,
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double radius,
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cairo_matrix_t *ctm,
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double tolerance)
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{
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double major_axis, max_angle;
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/* the error is amplified by at most the length of the
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* major axis of the circle; see cairo-pen.c for a more detailed analysis
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* of this. */
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major_axis = _cairo_matrix_transformed_circle_major_axis (ctm, radius);
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max_angle = _arc_max_angle_for_tolerance_normalized (tolerance / major_axis);
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return ceil (fabs (angle) / max_angle);
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}
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/* We want to draw a single spline approximating a circular arc radius
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R from angle A to angle B. Since we want a symmetric spline that
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matches the endpoints of the arc in position and slope, we know
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that the spline control points must be:
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(R * cos(A), R * sin(A))
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(R * cos(A) - h * sin(A), R * sin(A) + h * cos (A))
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(R * cos(B) + h * sin(B), R * sin(B) - h * cos (B))
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(R * cos(B), R * sin(B))
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for some value of h.
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"Approximation of circular arcs by cubic polynomials", Michael
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Goldapp, Computer Aided Geometric Design 8 (1991) 227-238, provides
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various values of h along with error analysis for each.
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From that paper, a very practical value of h is:
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h = 4/3 * R * tan(angle/4)
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This value does not give the spline with minimal error, but it does
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provide a very good approximation, (6th-order convergence), and the
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error expression is quite simple, (see the comment for
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_arc_error_normalized).
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*/
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static void
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_cairo_arc_segment (cairo_t *cr,
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double xc,
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double yc,
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double radius,
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double angle_A,
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double angle_B)
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{
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double r_sin_A, r_cos_A;
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double r_sin_B, r_cos_B;
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double h;
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r_sin_A = radius * sin (angle_A);
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r_cos_A = radius * cos (angle_A);
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r_sin_B = radius * sin (angle_B);
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r_cos_B = radius * cos (angle_B);
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h = 4.0/3.0 * tan ((angle_B - angle_A) / 4.0);
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cairo_curve_to (cr,
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xc + r_cos_A - h * r_sin_A,
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yc + r_sin_A + h * r_cos_A,
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xc + r_cos_B + h * r_sin_B,
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yc + r_sin_B - h * r_cos_B,
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xc + r_cos_B,
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yc + r_sin_B);
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}
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static void
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_cairo_arc_in_direction (cairo_t *cr,
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double xc,
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double yc,
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double radius,
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double angle_min,
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double angle_max,
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cairo_direction_t dir)
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{
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if (cairo_status (cr))
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return;
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assert (angle_max >= angle_min);
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if (angle_max - angle_min > 2 * M_PI * MAX_FULL_CIRCLES) {
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angle_max = fmod (angle_max - angle_min, 2 * M_PI);
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angle_min = fmod (angle_min, 2 * M_PI);
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angle_max += angle_min + 2 * M_PI * MAX_FULL_CIRCLES;
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}
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/* Recurse if drawing arc larger than pi */
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if (angle_max - angle_min > M_PI) {
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double angle_mid = angle_min + (angle_max - angle_min) / 2.0;
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if (dir == CAIRO_DIRECTION_FORWARD) {
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_cairo_arc_in_direction (cr, xc, yc, radius,
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angle_min, angle_mid,
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dir);
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_cairo_arc_in_direction (cr, xc, yc, radius,
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angle_mid, angle_max,
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dir);
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} else {
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_cairo_arc_in_direction (cr, xc, yc, radius,
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angle_mid, angle_max,
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dir);
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_cairo_arc_in_direction (cr, xc, yc, radius,
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angle_min, angle_mid,
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dir);
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}
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} else if (angle_max != angle_min) {
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cairo_matrix_t ctm;
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int i, segments;
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double step;
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cairo_get_matrix (cr, &ctm);
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segments = _arc_segments_needed (angle_max - angle_min,
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radius, &ctm,
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cairo_get_tolerance (cr));
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step = (angle_max - angle_min) / segments;
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segments -= 1;
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if (dir == CAIRO_DIRECTION_REVERSE) {
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double t;
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t = angle_min;
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angle_min = angle_max;
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angle_max = t;
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step = -step;
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}
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for (i = 0; i < segments; i++, angle_min += step) {
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_cairo_arc_segment (cr, xc, yc, radius,
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angle_min, angle_min + step);
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}
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_cairo_arc_segment (cr, xc, yc, radius,
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angle_min, angle_max);
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} else {
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cairo_line_to (cr,
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xc + radius * cos (angle_min),
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yc + radius * sin (angle_min));
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}
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}
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/**
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* _cairo_arc_path:
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* @cr: a cairo context
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* @xc: X position of the center of the arc
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* @yc: Y position of the center of the arc
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* @radius: the radius of the arc
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* @angle1: the start angle, in radians
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* @angle2: the end angle, in radians
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*
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* Compute a path for the given arc and append it onto the current
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* path within @cr. The arc will be accurate within the current
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* tolerance and given the current transformation.
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**/
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void
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_cairo_arc_path (cairo_t *cr,
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double xc,
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double yc,
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double radius,
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double angle1,
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double angle2)
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{
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_cairo_arc_in_direction (cr, xc, yc,
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radius,
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angle1, angle2,
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CAIRO_DIRECTION_FORWARD);
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}
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/**
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* _cairo_arc_path_negative:
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* @xc: X position of the center of the arc
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* @yc: Y position of the center of the arc
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* @radius: the radius of the arc
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* @angle1: the start angle, in radians
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* @angle2: the end angle, in radians
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* @ctm: the current transformation matrix
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* @tolerance: the current tolerance value
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* @path: the path onto which the arc will be appended
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*
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* Compute a path for the given arc (defined in the negative
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* direction) and append it onto the current path within @cr. The arc
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* will be accurate within the current tolerance and given the current
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* transformation.
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**/
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void
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_cairo_arc_path_negative (cairo_t *cr,
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double xc,
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double yc,
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double radius,
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double angle1,
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double angle2)
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{
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_cairo_arc_in_direction (cr, xc, yc,
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radius,
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angle2, angle1,
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CAIRO_DIRECTION_REVERSE);
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}
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