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Diffstat (limited to 'src/cairo-arc.c')
| -rw-r--r-- | src/cairo-arc.c | 296 |
1 files changed, 296 insertions, 0 deletions
diff --git a/src/cairo-arc.c b/src/cairo-arc.c new file mode 100644 index 000000000..2e6c6895e --- /dev/null +++ b/src/cairo-arc.c @@ -0,0 +1,296 @@ +/* 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 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 <math.h> + +#include "cairo-arc-private.h" + +/* 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 = (sizeof (table) / sizeof (table[0])); + + 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 l1, l2, lmax; + double max_angle; + + _cairo_matrix_compute_eigen_values (ctm, &l1, &l2); + + l1 = fabs (l1); + l2 = fabs (l2); + if (l1 > l2) + lmax = l1; + else + lmax = l2; + + max_angle = _arc_max_angle_for_tolerance_normalized (tolerance / (radius * lmax)); + + return (int) ceil (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 poynomials", 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 * 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) +{ + while (angle_max - angle_min > 4 * M_PI) + angle_max -= 2 * M_PI; + + /* 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; + /* XXX: Something tells me this block could be condensed. */ + 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 { + cairo_matrix_t ctm; + int i, segments; + double angle, angle_step; + + cairo_get_matrix (cr, &ctm); + segments = _arc_segments_needed (angle_max - angle_min, + radius, &ctm, + cairo_get_tolerance (cr)); + angle_step = (angle_max - angle_min) / (double) segments; + + if (dir == CAIRO_DIRECTION_FORWARD) { + angle = angle_min; + } else { + angle = angle_max; + angle_step = - angle_step; + } + + for (i = 0; i < segments; i++, angle += angle_step) { + _cairo_arc_segment (cr, xc, yc, + radius, + angle, + angle + angle_step); + } + } +} + +/** + * _cairo_arc_path_negative: + * @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 th earc 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) +{ + return _cairo_arc_in_direction (cr, xc, yc, + radius, + angle2, angle1, + CAIRO_DIRECTION_REVERSE); +} |