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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
*
* 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);
}