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