/* 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-slope-private.h"
cairo_bool_t
_cairo_spline_init (cairo_spline_t *spline,
cairo_spline_add_point_func_t add_point_func,
void *closure,
const cairo_point_t *a, const cairo_point_t *b,
const cairo_point_t *c, const cairo_point_t *d)
{
spline->add_point_func = add_point_func;
spline->closure = closure;
spline->knots.a = *a;
spline->knots.b = *b;
spline->knots.c = *c;
spline->knots.d = *d;
if (a->x != b->x || a->y != b->y)
_cairo_slope_init (&spline->initial_slope, &spline->knots.a, &spline->knots.b);
else if (a->x != c->x || a->y != c->y)
_cairo_slope_init (&spline->initial_slope, &spline->knots.a, &spline->knots.c);
else if (a->x != d->x || a->y != d->y)
_cairo_slope_init (&spline->initial_slope, &spline->knots.a, &spline->knots.d);
else
return FALSE;
if (c->x != d->x || c->y != d->y)
_cairo_slope_init (&spline->final_slope, &spline->knots.c, &spline->knots.d);
else if (b->x != d->x || b->y != d->y)
_cairo_slope_init (&spline->final_slope, &spline->knots.b, &spline->knots.d);
else
_cairo_slope_init (&spline->final_slope, &spline->knots.a, &spline->knots.d);
return TRUE;
}
static cairo_status_t
_cairo_spline_add_point (cairo_spline_t *spline, cairo_point_t *point)
{
cairo_point_t *prev;
prev = &spline->last_point;
if (prev->x == point->x && prev->y == point->y)
return CAIRO_STATUS_SUCCESS;
spline->last_point = *point;
return spline->add_point_func (spline->closure, point);
}
static void
_lerp_half (const cairo_point_t *a, const cairo_point_t *b, cairo_point_t *result)
{
result->x = a->x + ((b->x - a->x) >> 1);
result->y = a->y + ((b->y - a->y) >> 1);
}
static void
_de_casteljau (cairo_spline_knots_t *s1, cairo_spline_knots_t *s2)
{
cairo_point_t ab, bc, cd;
cairo_point_t abbc, bccd;
cairo_point_t final;
_lerp_half (&s1->a, &s1->b, &ab);
_lerp_half (&s1->b, &s1->c, &bc);
_lerp_half (&s1->c, &s1->d, &cd);
_lerp_half (&ab, &bc, &abbc);
_lerp_half (&bc, &cd, &bccd);
_lerp_half (&abbc, &bccd, &final);
s2->a = final;
s2->b = bccd;
s2->c = cd;
s2->d = s1->d;
s1->b = ab;
s1->c = abbc;
s1->d = final;
}
/* Return an upper bound on the error (squared) that could result from
* approximating a spline as a line segment connecting the two endpoints. */
static double
_cairo_spline_error_squared (const cairo_spline_knots_t *knots)
{
double bdx, bdy, berr;
double cdx, cdy, cerr;
/* We are going to compute the distance (squared) between each of the the b
* and c control points and the segment a-b. The maximum of these two
* distances will be our approximation error. */
bdx = _cairo_fixed_to_double (knots->b.x - knots->a.x);
bdy = _cairo_fixed_to_double (knots->b.y - knots->a.y);
cdx = _cairo_fixed_to_double (knots->c.x - knots->a.x);
cdy = _cairo_fixed_to_double (knots->c.y - knots->a.y);
if (knots->a.x != knots->d.x || knots->a.y != knots->d.y) {
/* Intersection point (px):
* px = p1 + u(p2 - p1)
* (p - px) ∙ (p2 - p1) = 0
* Thus:
* u = ((p - p1) ∙ (p2 - p1)) / ∥p2 - p1∥²;
*/
double dx, dy, u, v;
dx = _cairo_fixed_to_double (knots->d.x - knots->a.x);
dy = _cairo_fixed_to_double (knots->d.y - knots->a.y);
v = dx * dx + dy * dy;
u = bdx * dx + bdy * dy;
if (u <= 0) {
/* bdx -= 0;
* bdy -= 0;
*/
} else if (u >= v) {
bdx -= dx;
bdy -= dy;
} else {
bdx -= u/v * dx;
bdy -= u/v * dy;
}
u = cdx * dx + cdy * dy;
if (u <= 0) {
/* cdx -= 0;
* cdy -= 0;
*/
} else if (u >= v) {
cdx -= dx;
cdy -= dy;
} else {
cdx -= u/v * dx;
cdy -= u/v * dy;
}
}
berr = bdx * bdx + bdy * bdy;
cerr = cdx * cdx + cdy * cdy;
if (berr > cerr)
return berr;
else
return cerr;
}
static cairo_status_t
_cairo_spline_decompose_into (cairo_spline_knots_t *s1, double tolerance_squared, cairo_spline_t *result)
{
cairo_spline_knots_t s2;
cairo_status_t status;
if (_cairo_spline_error_squared (s1) < tolerance_squared)
return _cairo_spline_add_point (result, &s1->a);
_de_casteljau (s1, &s2);
status = _cairo_spline_decompose_into (s1, tolerance_squared, result);
if (unlikely (status))
return status;
return _cairo_spline_decompose_into (&s2, tolerance_squared, result);
}
cairo_status_t
_cairo_spline_decompose (cairo_spline_t *spline, double tolerance)
{
cairo_spline_knots_t s1;
cairo_status_t status;
s1 = spline->knots;
spline->last_point = s1.a;
status = _cairo_spline_decompose_into (&s1, tolerance * tolerance, spline);
if (unlikely (status))
return status;
return _cairo_spline_add_point (spline, &spline->knots.d);
}
/* Note: this function is only good for computing bounds in device space. */
cairo_status_t
_cairo_spline_bound (cairo_spline_add_point_func_t add_point_func,
void *closure,
const cairo_point_t *p0, const cairo_point_t *p1,
const cairo_point_t *p2, const cairo_point_t *p3)
{
double x0, x1, x2, x3;
double y0, y1, y2, y3;
double a, b, c;
double t[4];
int t_num = 0, i;
cairo_status_t status;
x0 = _cairo_fixed_to_double (p0->x);
y0 = _cairo_fixed_to_double (p0->y);
x1 = _cairo_fixed_to_double (p1->x);
y1 = _cairo_fixed_to_double (p1->y);
x2 = _cairo_fixed_to_double (p2->x);
y2 = _cairo_fixed_to_double (p2->y);
x3 = _cairo_fixed_to_double (p3->x);
y3 = _cairo_fixed_to_double (p3->y);
/* The spline can be written as a polynomial of the four points:
*
* (1-t)³p0 + 3t(1-t)²p1 + 3t²(1-t)p2 + t³p3
*
* for 0≤t≤1. Now, the X and Y components of the spline follow the
* same polynomial but with x and y replaced for p. To find the
* bounds of the spline, we just need to find the X and Y bounds.
* To find the bound, we take the derivative and equal it to zero,
* and solve to find the t's that give the extreme points.
*
* Here is the derivative of the curve, sorted on t:
*
* 3t²(-p0+3p1-3p2+p3) + 2t(3p0-6p1+3p2) -3p0+3p1
*
* Let:
*
* a = -p0+3p1-3p2+p3
* b = p0-2p1+p2
* c = -p0+p1
*
* Gives:
*
* a.t² + 2b.t + c = 0
*
* With:
*
* delta = b*b - a*c
*
* the extreme points are at -c/2b if a is zero, at (-b±√delta)/a if
* delta is positive, and at -b/a if delta is zero.
*/
#define ADD(t0) \
{ \
double _t0 = (t0); \
if (0 < _t0 && _t0 < 1) \
t[t_num++] = _t0; \
}
#define FIND_EXTREMES(a,b,c) \
{ \
if (a == 0) { \
if (b != 0) \
ADD (-c / (2*b)); \
} else { \
double b2 = b * b; \
double delta = b2 - a * c; \
if (delta > 0) { \
cairo_bool_t feasible; \
double _2ab = 2 * a * b; \
/* We are only interested in solutions t that satisfy 0<t<1 \
* here. We do some checks to avoid sqrt if the solutions \
* are not in that range. The checks can be derived from: \
* \
* 0 < (-b±√delta)/a < 1 \
*/ \
if (_2ab >= 0) \
feasible = delta > b2 && delta < a*a + b2 + _2ab; \
else if (-b / a >= 1) \
feasible = delta < b2 && delta > a*a + b2 + _2ab; \
else \
feasible = delta < b2 || delta < a*a + b2 + _2ab; \
\
if (unlikely (feasible)) { \
double sqrt_delta = sqrt (delta); \
ADD ((-b - sqrt_delta) / a); \
ADD ((-b + sqrt_delta) / a); \
} \
} else if (delta == 0) { \
ADD (-b / a); \
} \
} \
}
/* Find X extremes */
a = -x0 + 3*x1 - 3*x2 + x3;
b = x0 - 2*x1 + x2;
c = -x0 + x1;
FIND_EXTREMES (a, b, c);
/* Find Y extremes */
a = -y0 + 3*y1 - 3*y2 + y3;
b = y0 - 2*y1 + y2;
c = -y0 + y1;
FIND_EXTREMES (a, b, c);
status = add_point_func (closure, p0);
if (unlikely (status))
return status;
for (i = 0; i < t_num; i++) {
cairo_point_t p;
double x, y;
double t_1_0, t_0_1;
double t_2_0, t_0_2;
double t_3_0, t_2_1_3, t_1_2_3, t_0_3;
t_1_0 = t[i]; /* t */
t_0_1 = 1 - t_1_0; /* (1 - t) */
t_2_0 = t_1_0 * t_1_0; /* t * t */
t_0_2 = t_0_1 * t_0_1; /* (1 - t) * (1 - t) */
t_3_0 = t_2_0 * t_1_0; /* t * t * t */
t_2_1_3 = t_2_0 * t_0_1 * 3; /* t * t * (1 - t) * 3 */
t_1_2_3 = t_1_0 * t_0_2 * 3; /* t * (1 - t) * (1 - t) * 3 */
t_0_3 = t_0_1 * t_0_2; /* (1 - t) * (1 - t) * (1 - t) */
/* Bezier polynomial */
x = x0 * t_0_3
+ x1 * t_1_2_3
+ x2 * t_2_1_3
+ x3 * t_3_0;
y = y0 * t_0_3
+ y1 * t_1_2_3
+ y2 * t_2_1_3
+ y3 * t_3_0;
p.x = _cairo_fixed_from_double (x);
p.y = _cairo_fixed_from_double (y);
status = add_point_func (closure, &p);
if (unlikely (status))
return status;
}
return add_point_func (closure, p3);
}