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gtk/gsk/gskcurve.c
Matthias Clasen ad9fb1e101 Improve quad and conic decomposition 
If the control point is equal to either
start or end, just emit a line. This improves
the rendering of rounded rectangles with such
corners.
2023-08-27 16:57:03 -04:00

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/*
* Copyright © 2020 Benjamin Otte
*
* This library is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* This library is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this library. If not, see <http://www.gnu.org/licenses/>.
*
* Authors: Benjamin Otte <otte@gnome.org>
*/
#include "config.h"
#include "gskcurveprivate.h"
#include "gskboundingboxprivate.h"
/* GskCurve collects all the functionality we need for Bézier segments */
#define MIN_PROGRESS (1/1024.f)
typedef struct _GskCurveClass GskCurveClass;
struct _GskCurveClass
{
void (* init) (GskCurve *curve,
gskpathop op);
void (* init_foreach) (GskCurve *curve,
GskPathOperation op,
const graphene_point_t *pts,
gsize n_pts,
float weight);
void (* print) (const GskCurve *curve,
GString *string);
gskpathop (* pathop) (const GskCurve *curve);
const graphene_point_t * (* get_start_point) (const GskCurve *curve);
const graphene_point_t * (* get_end_point) (const GskCurve *curve);
void (* get_start_tangent) (const GskCurve *curve,
graphene_vec2_t *tangent);
void (* get_end_tangent) (const GskCurve *curve,
graphene_vec2_t *tangent);
void (* get_point) (const GskCurve *curve,
float t,
graphene_point_t *pos);
void (* get_tangent) (const GskCurve *curve,
float t,
graphene_vec2_t *tangent);
void (* reverse) (const GskCurve *curve,
GskCurve *reverse);
float (* get_curvature) (const GskCurve *curve,
float t);
void (* split) (const GskCurve *curve,
float progress,
GskCurve *result1,
GskCurve *result2);
void (* segment) (const GskCurve *curve,
float start,
float end,
GskCurve *segment);
gboolean (* decompose) (const GskCurve *curve,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data);
gboolean (* decompose_curve) (const GskCurve *curve,
GskPathForeachFlags flags,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data);
void (* get_bounds) (const GskCurve *curve,
GskBoundingBox *bounds);
void (* get_tight_bounds) (const GskCurve *curve,
GskBoundingBox *bounds);
void (* get_derivative_at) (const GskCurve *curve,
float t,
graphene_point_t *value);
int (* get_crossing) (const GskCurve *curve,
const graphene_point_t *point);
float (* get_length_to) (const GskCurve *curve,
float t);
};
/* {{{ Utilities */
#define RAD_TO_DEG(r) ((r)*180.f/M_PI)
#define DEG_TO_RAD(d) ((d)*M_PI/180.f)
static void
get_tangent (const graphene_point_t *p0,
const graphene_point_t *p1,
graphene_vec2_t *t)
{
graphene_vec2_init (t, p1->x - p0->x, p1->y - p0->y);
graphene_vec2_normalize (t, t);
}
static int
line_get_crossing (const graphene_point_t *p,
const graphene_point_t *p1,
const graphene_point_t *p2)
{
if (p1->y <= p->y)
{
if (p2->y > p->y)
{
if ((p2->x - p1->x) * (p->y - p1->y) - (p->x - p1->x) * (p2->y - p1->y) > 0)
return 1;
}
}
else if (p2->y <= p->y)
{
if ((p2->x - p1->x) * (p->y - p1->y) - (p->x - p1->x) * (p2->y - p1->y) < 0)
return -1;
}
return 0;
}
static int
get_crossing_by_bisection (const GskCurve *curve,
const graphene_point_t *point)
{
GskBoundingBox bounds;
GskCurve c1, c2;
gsk_curve_get_bounds (curve, &bounds);
if (bounds.max.y < point->y || bounds.min.y > point->y || bounds.max.x < point->x)
return 0;
if (bounds.min.x > point->x)
return line_get_crossing (point, gsk_curve_get_start_point (curve), gsk_curve_get_end_point (curve));
if (graphene_point_distance (&bounds.min, &bounds.max, NULL, NULL) < 0.001)
return line_get_crossing (point, gsk_curve_get_start_point (curve), gsk_curve_get_end_point (curve));
gsk_curve_split (curve, 0.5, &c1, &c2);
return gsk_curve_get_crossing (&c1, point) + gsk_curve_get_crossing (&c2, point);
}
/* Replace a line by an equivalent quad,
* and a quad by an equivalent cubic.
*/
static void
gsk_curve_elevate (const GskCurve *curve,
GskCurve *elevated)
{
if (curve->op == GSK_PATH_LINE)
{
graphene_point_t p[3];
p[0] = curve->line.points[0];
graphene_point_interpolate (&curve->line.points[0],
&curve->line.points[1],
0.5,
&p[1]);
p[2] = curve->line.points[1];
gsk_curve_init (elevated, gsk_pathop_encode (GSK_PATH_QUAD, p));
}
else if (curve->op == GSK_PATH_QUAD)
{
graphene_point_t p[4];
p[0] = curve->quad.points[0];
graphene_point_interpolate (&curve->quad.points[0],
&curve->quad.points[1],
2/3.,
&p[1]);
graphene_point_interpolate (&curve->quad.points[2],
&curve->quad.points[1],
2/3.,
&p[2]);
p[3] = curve->quad.points[2];
gsk_curve_init (elevated, gsk_pathop_encode (GSK_PATH_CUBIC, p));
}
else
g_assert_not_reached ();
}
/* Compute arclength by using Gauss quadrature on
*
* \int_0^z \sqrt{ (dx/dt)^2 + (dy/dt)^2 } dt
*/
#include "gskcurve-ct-values.c"
static float
get_length_by_approximation (const GskCurve *curve,
float t)
{
double z = t / 2;
double sum = 0;
graphene_point_t d;
for (unsigned int i = 0; i < G_N_ELEMENTS (T); i++)
{
gsk_curve_get_derivative_at (curve, z * T[i] + z, &d);
sum += C[i] * sqrt (d.x * d.x + d.y * d.y);
}
return z * sum;
}
/* }}} */
/* {{{ Line */
static void
gsk_line_curve_init_from_points (GskLineCurve *self,
GskPathOperation op,
const graphene_point_t *start,
const graphene_point_t *end)
{
self->op = op;
self->points[0] = *start;
self->points[1] = *end;
}
static void
gsk_line_curve_init (GskCurve *curve,
gskpathop op)
{
GskLineCurve *self = &curve->line;
const graphene_point_t *pts = gsk_pathop_points (op);
gsk_line_curve_init_from_points (self, gsk_pathop_op (op), &pts[0], &pts[1]);
}
static void
gsk_line_curve_init_foreach (GskCurve *curve,
GskPathOperation op,
const graphene_point_t *pts,
gsize n_pts,
float weight)
{
GskLineCurve *self = &curve->line;
g_assert (n_pts == 2);
gsk_line_curve_init_from_points (self, op, &pts[0], &pts[1]);
}
static void
gsk_line_curve_print (const GskCurve *curve,
GString *string)
{
g_string_append_printf (string,
"M %g %g L %g %g",
curve->line.points[0].x, curve->line.points[0].y,
curve->line.points[1].x, curve->line.points[1].y);
}
static gskpathop
gsk_line_curve_pathop (const GskCurve *curve)
{
const GskLineCurve *self = &curve->line;
return gsk_pathop_encode (self->op, self->points);
}
static const graphene_point_t *
gsk_line_curve_get_start_point (const GskCurve *curve)
{
const GskLineCurve *self = &curve->line;
return &self->points[0];
}
static const graphene_point_t *
gsk_line_curve_get_end_point (const GskCurve *curve)
{
const GskLineCurve *self = &curve->line;
return &self->points[1];
}
static void
gsk_line_curve_get_start_end_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
const GskLineCurve *self = &curve->line;
get_tangent (&self->points[0], &self->points[1], tangent);
}
static void
gsk_line_curve_get_point (const GskCurve *curve,
float t,
graphene_point_t *pos)
{
const GskLineCurve *self = &curve->line;
graphene_point_interpolate (&self->points[0], &self->points[1], t, pos);
}
static void
gsk_line_curve_get_tangent (const GskCurve *curve,
float t,
graphene_vec2_t *tangent)
{
const GskLineCurve *self = &curve->line;
get_tangent (&self->points[0], &self->points[1], tangent);
}
static float
gsk_line_curve_get_curvature (const GskCurve *curve,
float t)
{
return 0;
}
static void
gsk_line_curve_reverse (const GskCurve *curve,
GskCurve *reverse)
{
const GskLineCurve *self = &curve->line;
reverse->op = GSK_PATH_LINE;
reverse->line.points[0] = self->points[1];
reverse->line.points[1] = self->points[0];
}
static void
gsk_line_curve_split (const GskCurve *curve,
float progress,
GskCurve *start,
GskCurve *end)
{
const GskLineCurve *self = &curve->line;
graphene_point_t point;
graphene_point_interpolate (&self->points[0], &self->points[1], progress, &point);
if (start)
gsk_line_curve_init_from_points (&start->line, GSK_PATH_LINE, &self->points[0], &point);
if (end)
gsk_line_curve_init_from_points (&end->line, GSK_PATH_LINE, &point, &self->points[1]);
}
static void
gsk_line_curve_segment (const GskCurve *curve,
float start,
float end,
GskCurve *segment)
{
const GskLineCurve *self = &curve->line;
const graphene_point_t *pts = self->points;
graphene_point_t p0, p1;
graphene_point_interpolate (&pts[0], &pts[1], start, &p0);
graphene_point_interpolate (&pts[0], &pts[1], end, &p1);
gsk_line_curve_init_from_points (&segment->line, GSK_PATH_LINE, &p0, &p1);
}
static gboolean
gsk_line_curve_decompose (const GskCurve *curve,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
const GskLineCurve *self = &curve->line;
const graphene_point_t *pts = self->points;
return add_line_func (&pts[0], &pts[1], 0.f, 1.f, GSK_CURVE_LINE_REASON_STRAIGHT, user_data);
}
static gboolean
gsk_line_curve_decompose_curve (const GskCurve *curve,
GskPathForeachFlags flags,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data)
{
const GskLineCurve *self = &curve->line;
return add_curve_func (GSK_PATH_LINE, self->points, 2, 0.f, user_data);
}
static void
gsk_line_curve_get_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
const GskLineCurve *self = &curve->line;
const graphene_point_t *pts = self->points;
gsk_bounding_box_init (bounds, &pts[0], &pts[1]);
}
static void
gsk_line_curve_get_derivative_at (const GskCurve *curve,
float t,
graphene_point_t *value)
{
const GskLineCurve *self = &curve->line;
const graphene_point_t *pts = self->points;
value->x = pts[1].x - pts[0].x;
value->y = pts[1].y - pts[0].y;
}
static int
gsk_line_curve_get_crossing (const GskCurve *curve,
const graphene_point_t *point)
{
const GskLineCurve *self = &curve->line;
const graphene_point_t *pts = self->points;
return line_get_crossing (point, &pts[0], &pts[1]);
}
static float
gsk_line_curve_get_length_to (const GskCurve *curve,
float t)
{
const GskLineCurve *self = &curve->line;
const graphene_point_t *pts = self->points;
return t * graphene_point_distance (&pts[0], &pts[1], NULL, NULL);
}
static const GskCurveClass GSK_LINE_CURVE_CLASS = {
gsk_line_curve_init,
gsk_line_curve_init_foreach,
gsk_line_curve_print,
gsk_line_curve_pathop,
gsk_line_curve_get_start_point,
gsk_line_curve_get_end_point,
gsk_line_curve_get_start_end_tangent,
gsk_line_curve_get_start_end_tangent,
gsk_line_curve_get_point,
gsk_line_curve_get_tangent,
gsk_line_curve_reverse,
gsk_line_curve_get_curvature,
gsk_line_curve_split,
gsk_line_curve_segment,
gsk_line_curve_decompose,
gsk_line_curve_decompose_curve,
gsk_line_curve_get_bounds,
gsk_line_curve_get_bounds,
gsk_line_curve_get_derivative_at,
gsk_line_curve_get_crossing,
gsk_line_curve_get_length_to,
};
/* }}} */
/* {{{ Quadratic */
static void
gsk_quad_curve_ensure_coefficients (const GskQuadCurve *curve)
{
GskQuadCurve *self = (GskQuadCurve *) curve;
const graphene_point_t *pts = self->points;
if (self->has_coefficients)
return;
self->coeffs[2] = pts[0];
self->coeffs[1] = GRAPHENE_POINT_INIT (2 * (pts[1].x - pts[0].x),
2 * (pts[1].y - pts[0].y));
self->coeffs[0] = GRAPHENE_POINT_INIT (pts[2].x - 2 * pts[1].x + pts[0].x,
pts[2].y - 2 * pts[1].y + pts[0].y);
self->has_coefficients = TRUE;
}
static void
gsk_quad_curve_init_from_points (GskQuadCurve *self,
const graphene_point_t pts[3])
{
self->op = GSK_PATH_QUAD;
self->has_coefficients = FALSE;
memcpy (self->points, pts, sizeof (graphene_point_t) * 3);
}
static void
gsk_quad_curve_init (GskCurve *curve,
gskpathop op)
{
GskQuadCurve *self = &curve->quad;
gsk_quad_curve_init_from_points (self, gsk_pathop_points (op));
}
static void
gsk_quad_curve_init_foreach (GskCurve *curve,
GskPathOperation op,
const graphene_point_t *pts,
gsize n_pts,
float weight)
{
GskQuadCurve *self = &curve->quad;
g_assert (n_pts == 3);
gsk_quad_curve_init_from_points (self, pts);
}
static void
gsk_quad_curve_print (const GskCurve *curve,
GString *string)
{
g_string_append_printf (string,
"M %g %g Q %g %g %g %g",
curve->quad.points[0].x, curve->quad.points[0].y,
curve->quad.points[1].x, curve->cubic.points[1].y,
curve->quad.points[2].x, curve->cubic.points[2].y);
}
static gskpathop
gsk_quad_curve_pathop (const GskCurve *curve)
{
const GskQuadCurve *self = &curve->quad;
return gsk_pathop_encode (self->op, self->points);
}
static const graphene_point_t *
gsk_quad_curve_get_start_point (const GskCurve *curve)
{
const GskQuadCurve *self = &curve->quad;
return &self->points[0];
}
static const graphene_point_t *
gsk_quad_curve_get_end_point (const GskCurve *curve)
{
const GskQuadCurve *self = &curve->quad;
return &self->points[2];
}
static void
gsk_quad_curve_get_start_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
const GskQuadCurve *self = &curve->quad;
get_tangent (&self->points[0], &self->points[1], tangent);
}
static void
gsk_quad_curve_get_end_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
const GskQuadCurve *self = &curve->quad;
get_tangent (&self->points[1], &self->points[2], tangent);
}
static void
gsk_quad_curve_get_point (const GskCurve *curve,
float t,
graphene_point_t *pos)
{
GskQuadCurve *self = (GskQuadCurve *) &curve->quad;
const graphene_point_t *c = self->coeffs;
gsk_quad_curve_ensure_coefficients (self);
*pos = GRAPHENE_POINT_INIT ((c[0].x * t + c[1].x) * t + c[2].x,
(c[0].y * t + c[1].y) * t + c[2].y);
}
static void
gsk_quad_curve_get_tangent (const GskCurve *curve,
float t,
graphene_vec2_t *tangent)
{
GskQuadCurve *self = (GskQuadCurve *) &curve->quad;
const graphene_point_t *c = self->coeffs;
gsk_quad_curve_ensure_coefficients (self);
graphene_vec2_init (tangent,
2.0f * c[0].x * t + c[1].x,
2.0f * c[0].y * t + c[1].y);
graphene_vec2_normalize (tangent, tangent);
}
static float gsk_cubic_curve_get_curvature (const GskCurve *curve,
float t);
static float
gsk_quad_curve_get_curvature (const GskCurve *curve,
float t)
{
return gsk_cubic_curve_get_curvature (curve, t);
}
static void
gsk_quad_curve_reverse (const GskCurve *curve,
GskCurve *reverse)
{
const GskCubicCurve *self = &curve->cubic;
reverse->op = GSK_PATH_QUAD;
reverse->cubic.points[0] = self->points[2];
reverse->cubic.points[1] = self->points[1];
reverse->cubic.points[2] = self->points[0];
reverse->cubic.has_coefficients = FALSE;
}
static void
gsk_quad_curve_split (const GskCurve *curve,
float progress,
GskCurve *start,
GskCurve *end)
{
GskQuadCurve *self = (GskQuadCurve *) &curve->quad;
const graphene_point_t *pts = self->points;
graphene_point_t ab, bc;
graphene_point_t final;
graphene_point_interpolate (&pts[0], &pts[1], progress, &ab);
graphene_point_interpolate (&pts[1], &pts[2], progress, &bc);
graphene_point_interpolate (&ab, &bc, progress, &final);
if (start)
gsk_quad_curve_init_from_points (&start->quad, (graphene_point_t[3]) { pts[0], ab, final });
if (end)
gsk_quad_curve_init_from_points (&end->quad, (graphene_point_t[3]) { final, bc, pts[2] });
}
static void
gsk_quad_curve_segment (const GskCurve *curve,
float start,
float end,
GskCurve *segment)
{
GskCurve tmp;
gsk_quad_curve_split (curve, start, NULL, &tmp);
gsk_quad_curve_split (&tmp, (end - start) / (1.0f - start), segment, NULL);
}
/* taken from Skia, including the very descriptive name */
static gboolean
gsk_quad_curve_too_curvy (const GskQuadCurve *self,
float tolerance)
{
const graphene_point_t *pts = self->points;
float dx, dy;
dx = fabs (pts[1].x / 2 - (pts[0].x + pts[2].x) / 4);
dy = fabs (pts[1].y / 2 - (pts[0].y + pts[2].y) / 4);
return MAX (dx, dy) > tolerance;
}
static gboolean
gsk_quad_curve_decompose_step (const GskCurve *curve,
float start_progress,
float end_progress,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
const GskQuadCurve *self = &curve->quad;
GskCurve left, right;
float mid_progress;
if (!gsk_quad_curve_too_curvy (self, tolerance))
return add_line_func (&self->points[0], &self->points[2], start_progress, end_progress, GSK_CURVE_LINE_REASON_STRAIGHT, user_data);
if (end_progress - start_progress <= MIN_PROGRESS)
return add_line_func (&self->points[0], &self->points[2], start_progress, end_progress, GSK_CURVE_LINE_REASON_SHORT, user_data);
gsk_quad_curve_split ((const GskCurve *) self, 0.5, &left, &right);
mid_progress = (start_progress + end_progress) / 2;
return gsk_quad_curve_decompose_step (&left, start_progress, mid_progress, tolerance, add_line_func, user_data)
&& gsk_quad_curve_decompose_step (&right, mid_progress, end_progress, tolerance, add_line_func, user_data);
}
static gboolean
gsk_quad_curve_decompose (const GskCurve *curve,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
return gsk_quad_curve_decompose_step (curve, 0.0, 1.0, tolerance, add_line_func, user_data);
}
typedef struct
{
GskCurveAddCurveFunc add_curve;
gpointer user_data;
} AddLineData;
static gboolean
gsk_curve_add_line_cb (const graphene_point_t *from,
const graphene_point_t *to,
float from_progress,
float to_progress,
GskCurveLineReason reason,
gpointer user_data)
{
AddLineData *data = user_data;
graphene_point_t p[2] = { *from, *to };
return data->add_curve (GSK_PATH_LINE, p, 2, 0.f, data->user_data);
}
static gboolean
gsk_quad_curve_decompose_curve (const GskCurve *curve,
GskPathForeachFlags flags,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data)
{
const GskQuadCurve *self = &curve->quad;
if (flags & GSK_PATH_FOREACH_ALLOW_QUAD)
return add_curve_func (GSK_PATH_QUAD, self->points, 3, 0.f, user_data);
else if (graphene_point_equal (&curve->conic.points[0], &curve->conic.points[1]) ||
graphene_point_equal (&curve->conic.points[1], &curve->conic.points[2]))
{
if (!graphene_point_equal (&curve->conic.points[0], &curve->conic.points[2]))
return add_curve_func (GSK_PATH_LINE,
(graphene_point_t[2]) {
curve->conic.points[0],
curve->conic.points[2],
},
2, 0.f, user_data);
else
return TRUE;
}
else if (flags & GSK_PATH_FOREACH_ALLOW_CUBIC)
{
GskCurve c;
gsk_curve_elevate (curve, &c);
return add_curve_func (GSK_PATH_CUBIC, c.cubic.points, 4, 0.f, user_data);
}
else
{
return gsk_quad_curve_decompose (curve,
tolerance,
gsk_curve_add_line_cb,
&(AddLineData) { add_curve_func, user_data });
}
}
static void
gsk_quad_curve_get_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
const GskQuadCurve *self = &curve->quad;
const graphene_point_t *pts = self->points;
gsk_bounding_box_init (bounds, &pts[0], &pts[2]);
gsk_bounding_box_expand (bounds, &pts[1]);
}
/* Solve P' = 0 where P is
* P = (1-t)^2*pa + 2*t*(1-t)*pb + t^2*pc
*/
static int
get_quadratic_extrema (float pa, float pb, float pc, float t[1])
{
float d = pa - 2 * pb + pc;
if (fabs (d) > 0.0001)
{
t[0] = (pa - pb) / d;
return 1;
}
return 0;
}
static void
gsk_quad_curve_get_tight_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
const GskQuadCurve *self = &curve->quad;
const graphene_point_t *pts = self->points;
float t[4];
int n;
gsk_bounding_box_init (bounds, &pts[0], &pts[2]);
n = 0;
n += get_quadratic_extrema (pts[0].x, pts[1].x, pts[2].x, &t[n]);
n += get_quadratic_extrema (pts[0].y, pts[1].y, pts[2].y, &t[n]);
for (int i = 0; i < n; i++)
{
graphene_point_t p;
gsk_quad_curve_get_point (curve, t[i], &p);
gsk_bounding_box_expand (bounds, &p);
}
}
static void
gsk_quad_curve_get_derivative (const GskCurve *curve,
GskCurve *deriv)
{
const GskQuadCurve *self = &curve->quad;
graphene_point_t p[2];
p[0].x = 2.f * (self->points[1].x - self->points[0].x);
p[0].y = 2.f * (self->points[1].y - self->points[0].y);
p[1].x = 2.f * (self->points[2].x - self->points[1].x);
p[1].y = 2.f * (self->points[2].y - self->points[1].y);
gsk_line_curve_init_from_points (&deriv->line, GSK_PATH_LINE, &p[0], &p[1]);
}
static void
gsk_quad_curve_get_derivative_at (const GskCurve *curve,
float t,
graphene_point_t *value)
{
GskCurve d;
gsk_quad_curve_get_derivative (curve, &d);
gsk_curve_get_point (&d, t, value);
}
static int
gsk_quad_curve_get_crossing (const GskCurve *curve,
const graphene_point_t *point)
{
return get_crossing_by_bisection (curve, point);
}
static float
gsk_quad_curve_get_length_to (const GskCurve *curve,
float t)
{
return get_length_by_approximation (curve, t);
}
static const GskCurveClass GSK_QUAD_CURVE_CLASS = {
gsk_quad_curve_init,
gsk_quad_curve_init_foreach,
gsk_quad_curve_print,
gsk_quad_curve_pathop,
gsk_quad_curve_get_start_point,
gsk_quad_curve_get_end_point,
gsk_quad_curve_get_start_tangent,
gsk_quad_curve_get_end_tangent,
gsk_quad_curve_get_point,
gsk_quad_curve_get_tangent,
gsk_quad_curve_reverse,
gsk_quad_curve_get_curvature,
gsk_quad_curve_split,
gsk_quad_curve_segment,
gsk_quad_curve_decompose,
gsk_quad_curve_decompose_curve,
gsk_quad_curve_get_bounds,
gsk_quad_curve_get_tight_bounds,
gsk_quad_curve_get_derivative_at,
gsk_quad_curve_get_crossing,
gsk_quad_curve_get_length_to,
};
/* }}} */
/* {{{ Cubic */
static void
gsk_cubic_curve_ensure_coefficients (const GskCubicCurve *curve)
{
GskCubicCurve *self = (GskCubicCurve *) curve;
const graphene_point_t *pts = &self->points[0];
if (self->has_coefficients)
return;
self->coeffs[0] = GRAPHENE_POINT_INIT (pts[3].x - 3.0f * pts[2].x + 3.0f * pts[1].x - pts[0].x,
pts[3].y - 3.0f * pts[2].y + 3.0f * pts[1].y - pts[0].y);
self->coeffs[1] = GRAPHENE_POINT_INIT (3.0f * pts[2].x - 6.0f * pts[1].x + 3.0f * pts[0].x,
3.0f * pts[2].y - 6.0f * pts[1].y + 3.0f * pts[0].y);
self->coeffs[2] = GRAPHENE_POINT_INIT (3.0f * pts[1].x - 3.0f * pts[0].x,
3.0f * pts[1].y - 3.0f * pts[0].y);
self->coeffs[3] = pts[0];
self->has_coefficients = TRUE;
}
static void
gsk_cubic_curve_init_from_points (GskCubicCurve *self,
const graphene_point_t pts[4])
{
self->op = GSK_PATH_CUBIC;
self->has_coefficients = FALSE;
memcpy (self->points, pts, sizeof (graphene_point_t) * 4);
}
static void
gsk_cubic_curve_init (GskCurve *curve,
gskpathop op)
{
GskCubicCurve *self = &curve->cubic;
gsk_cubic_curve_init_from_points (self, gsk_pathop_points (op));
}
static void
gsk_cubic_curve_init_foreach (GskCurve *curve,
GskPathOperation op,
const graphene_point_t *pts,
gsize n_pts,
float weight)
{
GskCubicCurve *self = &curve->cubic;
g_assert (n_pts == 4);
gsk_cubic_curve_init_from_points (self, pts);
}
static void
gsk_cubic_curve_print (const GskCurve *curve,
GString *string)
{
g_string_append_printf (string,
"M %f %f C %f %f %f %f %f %f",
curve->cubic.points[0].x, curve->cubic.points[0].y,
curve->cubic.points[1].x, curve->cubic.points[1].y,
curve->cubic.points[2].x, curve->cubic.points[2].y,
curve->cubic.points[3].x, curve->cubic.points[3].y);
}
static gskpathop
gsk_cubic_curve_pathop (const GskCurve *curve)
{
const GskCubicCurve *self = &curve->cubic;
return gsk_pathop_encode (self->op, self->points);
}
static const graphene_point_t *
gsk_cubic_curve_get_start_point (const GskCurve *curve)
{
const GskCubicCurve *self = &curve->cubic;
return &self->points[0];
}
static const graphene_point_t *
gsk_cubic_curve_get_end_point (const GskCurve *curve)
{
const GskCubicCurve *self = &curve->cubic;
return &self->points[3];
}
static void
gsk_cubic_curve_get_start_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
const GskCubicCurve *self = &curve->cubic;
if (graphene_point_near (&self->points[0], &self->points[1], 0.0001))
{
if (graphene_point_near (&self->points[0], &self->points[2], 0.0001))
get_tangent (&self->points[0], &self->points[3], tangent);
else
get_tangent (&self->points[0], &self->points[2], tangent);
}
else
get_tangent (&self->points[0], &self->points[1], tangent);
}
static void
gsk_cubic_curve_get_end_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
const GskCubicCurve *self = &curve->cubic;
if (graphene_point_near (&self->points[2], &self->points[3], 0.0001))
{
if (graphene_point_near (&self->points[1], &self->points[3], 0.0001))
get_tangent (&self->points[0], &self->points[3], tangent);
else
get_tangent (&self->points[1], &self->points[3], tangent);
}
else
get_tangent (&self->points[2], &self->points[3], tangent);
}
static void
gsk_cubic_curve_get_point (const GskCurve *curve,
float t,
graphene_point_t *pos)
{
const GskCubicCurve *self = &curve->cubic;
const graphene_point_t *c = self->coeffs;
gsk_cubic_curve_ensure_coefficients (self);
*pos = GRAPHENE_POINT_INIT (((c[0].x * t + c[1].x) * t +c[2].x) * t + c[3].x,
((c[0].y * t + c[1].y) * t +c[2].y) * t + c[3].y);
}
static void
gsk_cubic_curve_get_tangent (const GskCurve *curve,
float t,
graphene_vec2_t *tangent)
{
const GskCubicCurve *self = &curve->cubic;
const graphene_point_t *c = self->coeffs;
gsk_cubic_curve_ensure_coefficients (self);
graphene_vec2_init (tangent,
(3.0f * c[0].x * t + 2.0f * c[1].x) * t + c[2].x,
(3.0f * c[0].y * t + 2.0f * c[1].y) * t + c[2].y);
graphene_vec2_normalize (tangent, tangent);
}
static void
gsk_cubic_curve_reverse (const GskCurve *curve,
GskCurve *reverse)
{
const GskCubicCurve *self = &curve->cubic;
reverse->op = GSK_PATH_CUBIC;
reverse->cubic.points[0] = self->points[3];
reverse->cubic.points[1] = self->points[2];
reverse->cubic.points[2] = self->points[1];
reverse->cubic.points[3] = self->points[0];
reverse->cubic.has_coefficients = FALSE;
}
static inline float
cross (const graphene_vec2_t *v1,
const graphene_vec2_t *v2)
{
return graphene_vec2_get_x (v1) * graphene_vec2_get_y (v2)
- graphene_vec2_get_y (v1) * graphene_vec2_get_x (v2);
}
static inline float
pow3 (float w)
{
return w * w * w;
}
static void
gsk_cubic_curve_get_derivative (const GskCurve *curve,
GskCurve *deriv)
{
const GskCubicCurve *self = &curve->cubic;
graphene_point_t p[3];
p[0].x = 3.f * (self->points[1].x - self->points[0].x);
p[0].y = 3.f * (self->points[1].y - self->points[0].y);
p[1].x = 3.f * (self->points[2].x - self->points[1].x);
p[1].y = 3.f * (self->points[2].y - self->points[1].y);
p[2].x = 3.f * (self->points[3].x - self->points[2].x);
p[2].y = 3.f * (self->points[3].y - self->points[2].y);
gsk_quad_curve_init_from_points (&deriv->quad, p);
}
static float
gsk_cubic_curve_get_curvature (const GskCurve *curve,
float t)
{
GskCurve c1, c2;
graphene_point_t p, pp;
graphene_vec2_t d, dd;
float num, denom;
gsk_cubic_curve_get_derivative (curve, &c1);
gsk_quad_curve_get_derivative (&c1, &c2);
gsk_curve_get_point (&c1, t, &p);
gsk_curve_get_point (&c2, t, &pp);
graphene_vec2_init (&d, p.x, p.y);
graphene_vec2_init (&dd, pp.x, pp.y);
num = cross (&d, &dd);
if (num == 0)
return 0;
denom = pow3 (graphene_vec2_length (&d));
if (denom == 0)
return 0;
return num / denom;
}
static void
gsk_cubic_curve_split (const GskCurve *curve,
float progress,
GskCurve *start,
GskCurve *end)
{
const GskCubicCurve *self = &curve->cubic;
const graphene_point_t *pts = self->points;
graphene_point_t ab, bc, cd;
graphene_point_t abbc, bccd;
graphene_point_t final;
graphene_point_interpolate (&pts[0], &pts[1], progress, &ab);
graphene_point_interpolate (&pts[1], &pts[2], progress, &bc);
graphene_point_interpolate (&pts[2], &pts[3], progress, &cd);
graphene_point_interpolate (&ab, &bc, progress, &abbc);
graphene_point_interpolate (&bc, &cd, progress, &bccd);
graphene_point_interpolate (&abbc, &bccd, progress, &final);
if (start)
gsk_cubic_curve_init_from_points (&start->cubic, (graphene_point_t[4]) { pts[0], ab, abbc, final });
if (end)
gsk_cubic_curve_init_from_points (&end->cubic, (graphene_point_t[4]) { final, bccd, cd, pts[3] });
}
static void
gsk_cubic_curve_segment (const GskCurve *curve,
float start,
float end,
GskCurve *segment)
{
GskCurve tmp;
gsk_cubic_curve_split (curve, start, NULL, &tmp);
gsk_cubic_curve_split (&tmp, (end - start) / (1.0f - start), segment, NULL);
}
/* taken from Skia, including the very descriptive name */
static gboolean
gsk_cubic_curve_too_curvy (const GskCubicCurve *self,
float tolerance)
{
const graphene_point_t *pts = self->points;
graphene_point_t p;
graphene_point_interpolate (&pts[0], &pts[3], 1.0f / 3, &p);
if (MAX (ABS (p.x - pts[1].x), ABS (p.y - pts[1].y)) > tolerance)
return TRUE;
graphene_point_interpolate (&pts[0], &pts[3], 2.0f / 3, &p);
if (MAX (ABS (p.x - pts[2].x), ABS (p.y - pts[2].y)) > tolerance)
return TRUE;
return FALSE;
}
static gboolean
gsk_cubic_curve_decompose_step (const GskCurve *curve,
float start_progress,
float end_progress,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
const GskCubicCurve *self = &curve->cubic;
GskCurve left, right;
float mid_progress;
if (!gsk_cubic_curve_too_curvy (self, tolerance))
return add_line_func (&self->points[0], &self->points[3], start_progress, end_progress, GSK_CURVE_LINE_REASON_STRAIGHT, user_data);
if (end_progress - start_progress <= MIN_PROGRESS)
return add_line_func (&self->points[0], &self->points[3], start_progress, end_progress, GSK_CURVE_LINE_REASON_SHORT, user_data);
gsk_cubic_curve_split ((const GskCurve *) self, 0.5, &left, &right);
mid_progress = (start_progress + end_progress) / 2;
return gsk_cubic_curve_decompose_step (&left, start_progress, mid_progress, tolerance, add_line_func, user_data)
&& gsk_cubic_curve_decompose_step (&right, mid_progress, end_progress, tolerance, add_line_func, user_data);
}
static gboolean
gsk_cubic_curve_decompose (const GskCurve *curve,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
return gsk_cubic_curve_decompose_step (curve, 0.0, 1.0, tolerance, add_line_func, user_data);
}
static gboolean
gsk_cubic_curve_decompose_curve (const GskCurve *curve,
GskPathForeachFlags flags,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data)
{
const GskCubicCurve *self = &curve->cubic;
if (flags & GSK_PATH_FOREACH_ALLOW_CUBIC)
return add_curve_func (GSK_PATH_CUBIC, self->points, 4, 0.f, user_data);
/* FIXME: Quadratic or arc approximation */
return gsk_cubic_curve_decompose (curve,
tolerance,
gsk_curve_add_line_cb,
&(AddLineData) { add_curve_func, user_data });
}
static void
gsk_cubic_curve_get_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
const GskCubicCurve *self = &curve->cubic;
const graphene_point_t *pts = self->points;
gsk_bounding_box_init (bounds, &pts[0], &pts[3]);
gsk_bounding_box_expand (bounds, &pts[1]);
gsk_bounding_box_expand (bounds, &pts[2]);
}
static inline gboolean
acceptable (float t)
{
return 0 <= t && t <= 1;
}
/* Solve P' = 0 where P is
* P = (1-t)^3*pa + 3*t*(1-t)^2*pb + 3*t^2*(1-t)*pc + t^3*pd
*/
static int
get_cubic_extrema (float pa, float pb, float pc, float pd, float t[2])
{
float a, b, c;
float d, tt;
int n = 0;
a = 3 * (pd - 3*pc + 3*pb - pa);
b = 6 * (pc - 2*pb + pa);
c = 3 * (pb - pa);
if (fabs (a) > 0.0001)
{
if (b*b > 4*a*c)
{
d = sqrt (b*b - 4*a*c);
tt = (-b + d)/(2*a);
if (acceptable (tt))
t[n++] = tt;
tt = (-b - d)/(2*a);
if (acceptable (tt))
t[n++] = tt;
}
else
{
tt = -b / (2*a);
if (acceptable (tt))
t[n++] = tt;
}
}
else if (fabs (b) > 0.0001)
{
tt = -c / b;
if (acceptable (tt))
t[n++] = tt;
}
return n;
}
static void
gsk_cubic_curve_get_tight_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
const GskCubicCurve *self = &curve->cubic;
const graphene_point_t *pts = self->points;
float t[4];
int n;
gsk_bounding_box_init (bounds, &pts[0], &pts[3]);
n = 0;
n += get_cubic_extrema (pts[0].x, pts[1].x, pts[2].x, pts[3].x, &t[n]);
n += get_cubic_extrema (pts[0].y, pts[1].y, pts[2].y, pts[3].y, &t[n]);
for (int i = 0; i < n; i++)
{
graphene_point_t p;
gsk_cubic_curve_get_point (curve, t[i], &p);
gsk_bounding_box_expand (bounds, &p);
}
}
static void
gsk_cubic_curve_get_derivative_at (const GskCurve *curve,
float t,
graphene_point_t *value)
{
GskCurve d;
gsk_cubic_curve_get_derivative (curve, &d);
gsk_curve_get_point (&d, t, value);
}
static int
gsk_cubic_curve_get_crossing (const GskCurve *curve,
const graphene_point_t *point)
{
return get_crossing_by_bisection (curve, point);
}
static float
gsk_cubic_curve_get_length_to (const GskCurve *curve,
float t)
{
return get_length_by_approximation (curve, t);
}
static const GskCurveClass GSK_CUBIC_CURVE_CLASS = {
gsk_cubic_curve_init,
gsk_cubic_curve_init_foreach,
gsk_cubic_curve_print,
gsk_cubic_curve_pathop,
gsk_cubic_curve_get_start_point,
gsk_cubic_curve_get_end_point,
gsk_cubic_curve_get_start_tangent,
gsk_cubic_curve_get_end_tangent,
gsk_cubic_curve_get_point,
gsk_cubic_curve_get_tangent,
gsk_cubic_curve_reverse,
gsk_cubic_curve_get_curvature,
gsk_cubic_curve_split,
gsk_cubic_curve_segment,
gsk_cubic_curve_decompose,
gsk_cubic_curve_decompose_curve,
gsk_cubic_curve_get_bounds,
gsk_cubic_curve_get_tight_bounds,
gsk_cubic_curve_get_derivative_at,
gsk_cubic_curve_get_crossing,
gsk_cubic_curve_get_length_to,
};
/* }}} */
/* {{{ Conic */
static inline float
gsk_conic_curve_get_weight (const GskConicCurve *self)
{
return self->points[2].x;
}
static void
gsk_conic_curve_ensure_coefficents (const GskConicCurve *curve)
{
GskConicCurve *self = (GskConicCurve *) curve;
float w = gsk_conic_curve_get_weight (self);
const graphene_point_t *pts = self->points;
graphene_point_t pw = GRAPHENE_POINT_INIT (w * pts[1].x, w * pts[1].y);
if (self->has_coefficients)
return;
self->num[2] = pts[0];
self->num[1] = GRAPHENE_POINT_INIT (2 * (pw.x - pts[0].x),
2 * (pw.y - pts[0].y));
self->num[0] = GRAPHENE_POINT_INIT (pts[3].x - 2 * pw.x + pts[0].x,
pts[3].y - 2 * pw.y + pts[0].y);
self->denom[2] = GRAPHENE_POINT_INIT (1, 1);
self->denom[1] = GRAPHENE_POINT_INIT (2 * (w - 1), 2 * (w - 1));
self->denom[0] = GRAPHENE_POINT_INIT (-self->denom[1].x, -self->denom[1].y);
self->has_coefficients = TRUE;
}
static void
gsk_conic_curve_init_from_points (GskConicCurve *self,
const graphene_point_t pts[4])
{
self->op = GSK_PATH_CONIC;
self->has_coefficients = FALSE;
memcpy (self->points, pts, sizeof (graphene_point_t) * 4);
}
static void
gsk_conic_curve_init (GskCurve *curve,
gskpathop op)
{
GskConicCurve *self = &curve->conic;
gsk_conic_curve_init_from_points (self, gsk_pathop_points (op));
}
static void
gsk_conic_curve_init_foreach (GskCurve *curve,
GskPathOperation op,
const graphene_point_t *pts,
gsize n_pts,
float weight)
{
GskConicCurve *self = &curve->conic;
g_assert (n_pts == 3);
gsk_conic_curve_init_from_points (self,
(graphene_point_t[4]) {
pts[0],
pts[1],
GRAPHENE_POINT_INIT (weight, 0),
pts[2]
});
}
static void
gsk_conic_curve_print (const GskCurve *curve,
GString *string)
{
g_string_append_printf (string,
"M %g %g O %g %g %g %g %g",
curve->conic.points[0].x, curve->conic.points[0].y,
curve->conic.points[1].x, curve->conic.points[1].y,
curve->conic.points[3].x, curve->conic.points[3].y,
curve->conic.points[2].x);
}
static gskpathop
gsk_conic_curve_pathop (const GskCurve *curve)
{
const GskConicCurve *self = &curve->conic;
return gsk_pathop_encode (self->op, self->points);
}
static const graphene_point_t *
gsk_conic_curve_get_start_point (const GskCurve *curve)
{
const GskConicCurve *self = &curve->conic;
return &self->points[0];
}
static const graphene_point_t *
gsk_conic_curve_get_end_point (const GskCurve *curve)
{
const GskConicCurve *self = &curve->conic;
return &self->points[3];
}
static void
gsk_conic_curve_get_start_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
const GskConicCurve *self = &curve->conic;
get_tangent (&self->points[0], &self->points[1], tangent);
}
static void
gsk_conic_curve_get_end_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
const GskConicCurve *self = &curve->conic;
get_tangent (&self->points[1], &self->points[3], tangent);
}
static inline void
gsk_curve_eval_quad (const graphene_point_t quad[3],
float progress,
graphene_point_t *result)
{
*result = GRAPHENE_POINT_INIT ((quad[0].x * progress + quad[1].x) * progress + quad[2].x,
(quad[0].y * progress + quad[1].y) * progress + quad[2].y);
}
static inline void
gsk_conic_curve_eval_point (const GskConicCurve *self,
float progress,
graphene_point_t *point)
{
graphene_point_t num, denom;
gsk_curve_eval_quad (self->num, progress, &num);
gsk_curve_eval_quad (self->denom, progress, &denom);
*point = GRAPHENE_POINT_INIT (num.x / denom.x, num.y / denom.y);
}
static void
gsk_conic_curve_get_point (const GskCurve *curve,
float t,
graphene_point_t *pos)
{
const GskConicCurve *self = &curve->conic;
gsk_conic_curve_ensure_coefficents (self);
gsk_conic_curve_eval_point (self, t, pos);
}
/* See M. Floater, Derivatives of rational Bezier curves */
static void
gsk_conic_curve_get_derivative_at (const GskCurve *curve,
float t,
graphene_point_t *value)
{
const GskConicCurve *self = &curve->conic;
float w = gsk_conic_curve_get_weight (self);
const graphene_point_t *pts = self->points;
graphene_point_t p[3], p1[2];
float w1[2], w2;
/* The tangent will be 0 in these corner cases, just
* treat it like a line here.
*/
if ((t <= 0.f && graphene_point_equal (&pts[0], &pts[1])) ||
(t >= 1.f && graphene_point_equal (&pts[1], &pts[3])))
{
graphene_point_init (value, pts[3].x - pts[0].x, pts[3].y - pts[0].y);
return;
}
p[0] = pts[0];
p[1] = pts[1];
p[2] = pts[3];
w1[0] = (1 - t) + t*w;
w1[1] = (1 - t)*w + t;
w2 = (1 - t) * w1[0] + t * w1[1];
p1[0].x = ((1 - t)*p[0].x + t*w*p[1].x)/w1[0];
p1[0].y = ((1 - t)*p[0].y + t*w*p[1].y)/w1[0];
p1[1].x = ((1 - t)*w*p[1].x + t*p[2].x)/w1[1];
p1[1].y = ((1 - t)*w*p[1].y + t*p[2].y)/w1[1];
value->x = 2 * (w1[0] * w1[1])/(w2*w2) * (p1[1].x - p1[0].x);
value->y = 2 * (w1[0] * w1[1])/(w2*w2) * (p1[1].y - p1[0].y);
}
static void
gsk_conic_curve_get_tangent (const GskCurve *curve,
float t,
graphene_vec2_t *tangent)
{
graphene_point_t tmp;
gsk_conic_curve_get_derivative_at (curve, t, &tmp);
graphene_vec2_init (tangent, tmp.x, tmp.y);
graphene_vec2_normalize (tangent, tangent);
}
/* See M. Floater, Derivatives of rational Bezier curves */
static float
gsk_conic_curve_get_curvature (const GskCurve *curve,
float t)
{
graphene_point_t p[3], p1[2];
float w, w1[2], w2;
graphene_vec2_t t1, t2, t3;
w = curve->conic.points[2].x;
p[0] = curve->conic.points[0];
p[1] = curve->conic.points[1];
p[2] = curve->conic.points[3];
w1[0] = (1 - t) + t*w;
w1[1] = (1 - t)*w + t;
w2 = (1 - t)*w1[0] + t*w1[1];
p1[0].x = ((1 - t)*p[0].x + t*w*p[1].x)/w1[0];
p1[0].y = ((1 - t)*p[0].y + t*w*p[1].y)/w1[0];
p1[1].x = ((1 - t)*w*p[1].x + t*p[2].x)/w1[1];
p1[1].y = ((1 - t)*w*p[1].y + t*p[2].y)/w1[1];
graphene_vec2_init (&t1, p[1].x - p[0].x, p[1].y - p[0].y);
graphene_vec2_init (&t2, p[2].x - p[1].x, p[2].y - p[1].y);
graphene_vec2_init (&t3, p1[1].x - p1[0].x, p1[1].y - p1[0].y);
return 0.5 * ((w*pow3 (w2))/(pow3 (w1[0])*pow3 (w1[1]))) * (cross (&t1, &t2) / pow3 (graphene_vec2_length (&t3)));
}
static void
gsk_conic_curve_reverse (const GskCurve *curve,
GskCurve *reverse)
{
const GskConicCurve *self = &curve->conic;
reverse->op = GSK_PATH_CONIC;
reverse->conic.points[0] = self->points[3];
reverse->conic.points[1] = self->points[1];
reverse->conic.points[2] = self->points[2];
reverse->conic.points[3] = self->points[0];
reverse->conic.has_coefficients = FALSE;
}
static void
split_bezier3d_recurse (const graphene_point3d_t *p,
int l,
float t,
graphene_point3d_t *left,
graphene_point3d_t *right,
int *lpos,
int *rpos)
{
if (l == 1)
{
left[*lpos] = p[0];
right[*rpos] = p[0];
}
else
{
graphene_point3d_t *np;
int i;
np = g_alloca (sizeof (graphene_point3d_t) * (l - 1));
for (i = 0; i < l - 1; i++)
{
if (i == 0)
{
left[*lpos] = p[i];
(*lpos)++;
}
if (i + 1 == l - 1)
{
right[*rpos] = p[i + 1];
(*rpos)--;
}
graphene_point3d_interpolate (&p[i], &p[i + 1], t, &np[i]);
}
split_bezier3d_recurse (np, l - 1, t, left, right, lpos, rpos);
}
}
static void
split_bezier3d (const graphene_point3d_t *p,
int l,
float t,
graphene_point3d_t *left,
graphene_point3d_t *right)
{
int lpos = 0;
int rpos = l - 1;
split_bezier3d_recurse (p, l, t, left, right, &lpos, &rpos);
}
static void
gsk_conic_curve_split (const GskCurve *curve,
float progress,
GskCurve *start,
GskCurve *end)
{
const GskConicCurve *self = &curve->conic;
graphene_point3d_t p[3];
graphene_point3d_t l[3], r[3];
graphene_point_t left[4], right[4];
float w;
/* do de Casteljau in homogeneous coordinates... */
w = self->points[2].x;
p[0] = GRAPHENE_POINT3D_INIT (self->points[0].x, self->points[0].y, 1);
p[1] = GRAPHENE_POINT3D_INIT (self->points[1].x * w, self->points[1].y * w, w);
p[2] = GRAPHENE_POINT3D_INIT (self->points[3].x, self->points[3].y, 1);
split_bezier3d (p, 3, progress, l, r);
/* then project the control points down */
left[0] = GRAPHENE_POINT_INIT (l[0].x / l[0].z, l[0].y / l[0].z);
left[1] = GRAPHENE_POINT_INIT (l[1].x / l[1].z, l[1].y / l[1].z);
left[3] = GRAPHENE_POINT_INIT (l[2].x / l[2].z, l[2].y / l[2].z);
right[0] = GRAPHENE_POINT_INIT (r[0].x / r[0].z, r[0].y / r[0].z);
right[1] = GRAPHENE_POINT_INIT (r[1].x / r[1].z, r[1].y / r[1].z);
right[3] = GRAPHENE_POINT_INIT (r[2].x / r[2].z, r[2].y / r[2].z);
/* normalize the outer weights to be 1 by using
* the fact that weights w_i and c*w_i are equivalent
* for any nonzero constant c
*/
for (int i = 0; i < 3; i++)
{
l[i].z /= l[0].z;
r[i].z /= r[2].z;
}
/* normalize the inner weight to be 1 by using
* the fact that w_0*w_2/w_1^2 is a constant for
* all equivalent weights.
*/
left[2] = GRAPHENE_POINT_INIT (l[1].z / sqrt (l[2].z), 0);
right[2] = GRAPHENE_POINT_INIT (r[1].z / sqrt (r[0].z), 0);
if (start)
gsk_curve_init (start, gsk_pathop_encode (GSK_PATH_CONIC, left));
if (end)
gsk_curve_init (end, gsk_pathop_encode (GSK_PATH_CONIC, right));
}
static void
gsk_conic_curve_segment (const GskCurve *curve,
float start,
float end,
GskCurve *segment)
{
const GskConicCurve *self = &curve->conic;
graphene_point_t start_num, start_denom;
graphene_point_t mid_num, mid_denom;
graphene_point_t end_num, end_denom;
graphene_point_t ctrl_num, ctrl_denom;
float mid;
if (start <= 0.0f)
return gsk_conic_curve_split (curve, end, segment, NULL);
else if (end >= 1.0f)
return gsk_conic_curve_split (curve, start, NULL, segment);
gsk_conic_curve_ensure_coefficents (self);
gsk_curve_eval_quad (self->num, start, &start_num);
gsk_curve_eval_quad (self->denom, start, &start_denom);
mid = (start + end) / 2;
gsk_curve_eval_quad (self->num, mid, &mid_num);
gsk_curve_eval_quad (self->denom, mid, &mid_denom);
gsk_curve_eval_quad (self->num, end, &end_num);
gsk_curve_eval_quad (self->denom, end, &end_denom);
ctrl_num = GRAPHENE_POINT_INIT (2 * mid_num.x - (start_num.x + end_num.x) / 2,
2 * mid_num.y - (start_num.y + end_num.y) / 2);
ctrl_denom = GRAPHENE_POINT_INIT (2 * mid_denom.x - (start_denom.x + end_denom.x) / 2,
2 * mid_denom.y - (start_denom.y + end_denom.y) / 2);
gsk_conic_curve_init_from_points (&segment->conic,
(graphene_point_t[4]) {
GRAPHENE_POINT_INIT (start_num.x / start_denom.x,
start_num.y / start_denom.y),
GRAPHENE_POINT_INIT (ctrl_num.x / ctrl_denom.x,
ctrl_num.y / ctrl_denom.y),
GRAPHENE_POINT_INIT (ctrl_denom.x / sqrtf (start_denom.x * end_denom.x),
0),
GRAPHENE_POINT_INIT (end_num.x / end_denom.x,
end_num.y / end_denom.y)
});
}
/* taken from Skia, including the very descriptive name */
static gboolean
gsk_conic_curve_too_curvy (const graphene_point_t *start,
const graphene_point_t *mid,
const graphene_point_t *end,
float tolerance)
{
return fabs ((start->x + end->x) * 0.5 - mid->x) > tolerance
|| fabs ((start->y + end->y) * 0.5 - mid->y) > tolerance;
}
static gboolean
gsk_conic_curve_decompose_subdivide (const GskConicCurve *self,
float tolerance,
const graphene_point_t *start,
float start_progress,
const graphene_point_t *end,
float end_progress,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
graphene_point_t mid;
float mid_progress;
mid_progress = (start_progress + end_progress) / 2;
gsk_conic_curve_eval_point (self, mid_progress, &mid);
if (!gsk_conic_curve_too_curvy (start, &mid, end, tolerance))
return add_line_func (start, end, start_progress, end_progress, GSK_CURVE_LINE_REASON_STRAIGHT, user_data);
if (end_progress - start_progress <= MIN_PROGRESS)
return add_line_func (start, end, start_progress, end_progress, GSK_CURVE_LINE_REASON_SHORT, user_data);
return gsk_conic_curve_decompose_subdivide (self, tolerance,
start, start_progress, &mid, mid_progress,
add_line_func, user_data)
&& gsk_conic_curve_decompose_subdivide (self, tolerance,
&mid, mid_progress, end, end_progress,
add_line_func, user_data);
}
static gboolean
gsk_conic_curve_decompose (const GskCurve *curve,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
const GskConicCurve *self = &curve->conic;
graphene_point_t mid;
gsk_conic_curve_ensure_coefficents (self);
gsk_conic_curve_eval_point (self, 0.5, &mid);
return gsk_conic_curve_decompose_subdivide (self,
tolerance,
&self->points[0],
0.0f,
&mid,
0.5f,
add_line_func,
user_data)
&& gsk_conic_curve_decompose_subdivide (self,
tolerance,
&mid,
0.5f,
&self->points[3],
1.0f,
add_line_func,
user_data);
}
/* See Floater, M: An analysis of cubic approximation schemes
* for conic sections
*/
static void
cubic_approximation (const GskCurve *curve,
GskCurve *cubic)
{
const GskConicCurve *self = &curve->conic;
graphene_point_t p[4];
float w = self->points[2].x;
float w2 = w*w;
float lambda;
lambda = 2 * (6*w2 + 1 - sqrt (3*w2 + 1)) / (12*w2 + 3);
p[0] = self->points[0];
p[3] = self->points[3];
graphene_point_interpolate (&self->points[0], &self->points[1], lambda, &p[1]);
graphene_point_interpolate (&self->points[3], &self->points[1], lambda, &p[2]);
gsk_curve_init (cubic, gsk_pathop_encode (GSK_PATH_CUBIC, p));
}
static gboolean
gsk_conic_is_close_to_cubic (const GskCurve *conic,
const GskCurve *cubic,
float tolerance)
{
float t[] = { 0.1, 0.5, 0.9 };
graphene_point_t p0, p1;
for (int i = 0; i < G_N_ELEMENTS (t); i++)
{
gsk_curve_get_point (conic, t[i], &p0);
gsk_curve_get_point (cubic, t[i], &p1);
if (graphene_point_distance (&p0, &p1, NULL, NULL) > tolerance)
return FALSE;
}
return TRUE;
}
static gboolean gsk_conic_curve_decompose_curve (const GskCurve *curve,
GskPathForeachFlags flags,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data);
static gboolean
gsk_conic_curve_decompose_or_add (const GskCurve *curve,
const GskCurve *cubic,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data)
{
if (graphene_point_equal (&curve->conic.points[0], &curve->conic.points[1]) ||
graphene_point_equal (&curve->conic.points[1], &curve->conic.points[3]))
{
if (!graphene_point_equal (&curve->conic.points[0], &curve->conic.points[3]))
return add_curve_func (GSK_PATH_LINE,
(graphene_point_t[2]) {
curve->conic.points[0],
curve->conic.points[3],
},
2, 0.f, user_data);
else
return TRUE;
}
else if (gsk_conic_is_close_to_cubic (curve, cubic, tolerance))
return add_curve_func (GSK_PATH_CUBIC, cubic->cubic.points, 4, 0.f, user_data);
else
{
GskCurve c1, c2;
GskCurve cc1, cc2;
gsk_conic_curve_split (curve, 0.5, &c1, &c2);
cubic_approximation (&c1, &cc1);
cubic_approximation (&c2, &cc2);
return gsk_conic_curve_decompose_or_add (&c1, &cc1, tolerance, add_curve_func, user_data) &&
gsk_conic_curve_decompose_or_add (&c2, &cc2, tolerance, add_curve_func, user_data);
}
}
static gboolean
gsk_conic_curve_decompose_curve (const GskCurve *curve,
GskPathForeachFlags flags,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data)
{
const GskConicCurve *self = &curve->conic;
GskCurve c;
if (flags & GSK_PATH_FOREACH_ALLOW_CONIC)
return add_curve_func (GSK_PATH_CONIC,
(const graphene_point_t[3]) { self->points[0],
self->points[1],
self->points[3] },
3,
self->points[2].x,
user_data);
if (flags & GSK_PATH_FOREACH_ALLOW_CUBIC)
{
cubic_approximation (curve, &c);
return gsk_conic_curve_decompose_or_add (curve, &c, tolerance, add_curve_func, user_data);
}
/* FIXME: Quadratic approximation */
return gsk_conic_curve_decompose (curve,
tolerance,
gsk_curve_add_line_cb,
&(AddLineData) { add_curve_func, user_data });
}
static void
gsk_conic_curve_get_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
const GskConicCurve *self = &curve->conic;
const graphene_point_t *pts = self->points;
gsk_bounding_box_init (bounds, &pts[0], &pts[3]);
gsk_bounding_box_expand (bounds, &pts[1]);
}
/* Solve N = 0 where N is the numerator of (P/Q)', with
* P = (1-t)^2*a + 2*t*(1-t)*w*b + t^2*c
* Q = (1-t)^2 + 2*t*(1-t)*w + t^2
*/
static int
get_conic_extrema (float a, float b, float c, float w, float t[4])
{
float q, tt;
int n = 0;
float w2 = w*w;
float wac = (w - 1)*(a - c);
if (wac != 0)
{
q = - sqrt (a*a - 4*a*b*w2 + 4*a*c*w2 - 2*a*c + 4*b*b*w2 - 4*b*c*w2 + c*c);
tt = (- q + 2*a*w - a - 2*b*w + c)/(2*wac);
if (acceptable (tt))
t[n++] = tt;
tt = (q + 2*a*w - a - 2*b*w + c)/(2*wac);
if (acceptable (tt))
t[n++] = tt;
}
if (w * (b - c) != 0 && a == c)
t[n++] = 0.5;
if (w == 1 && a - 2*b + c != 0)
{
tt = (a - b) / (a - 2*b + c);
if (acceptable (tt))
t[n++] = tt;
}
return n;
}
static void
gsk_conic_curve_get_tight_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
const GskConicCurve *self = &curve->conic;
float w = gsk_conic_curve_get_weight (self);
const graphene_point_t *pts = self->points;
float t[8];
int n;
gsk_bounding_box_init (bounds, &pts[0], &pts[3]);
n = 0;
n += get_conic_extrema (pts[0].x, pts[1].x, pts[3].x, w, &t[n]);
n += get_conic_extrema (pts[0].y, pts[1].y, pts[3].y, w, &t[n]);
for (int i = 0; i < n; i++)
{
graphene_point_t p;
gsk_conic_curve_get_point (curve, t[i], &p);
gsk_bounding_box_expand (bounds, &p);
}
}
static int
gsk_conic_curve_get_crossing (const GskCurve *curve,
const graphene_point_t *point)
{
return get_crossing_by_bisection (curve, point);
}
static float
gsk_conic_curve_get_length_to (const GskCurve *curve,
float t)
{
return get_length_by_approximation (curve, t);
}
static const GskCurveClass GSK_CONIC_CURVE_CLASS = {
gsk_conic_curve_init,
gsk_conic_curve_init_foreach,
gsk_conic_curve_print,
gsk_conic_curve_pathop,
gsk_conic_curve_get_start_point,
gsk_conic_curve_get_end_point,
gsk_conic_curve_get_start_tangent,
gsk_conic_curve_get_end_tangent,
gsk_conic_curve_get_point,
gsk_conic_curve_get_tangent,
gsk_conic_curve_reverse,
gsk_conic_curve_get_curvature,
gsk_conic_curve_split,
gsk_conic_curve_segment,
gsk_conic_curve_decompose,
gsk_conic_curve_decompose_curve,
gsk_conic_curve_get_bounds,
gsk_conic_curve_get_tight_bounds,
gsk_conic_curve_get_derivative_at,
gsk_conic_curve_get_crossing,
gsk_conic_curve_get_length_to,
};
/* }}} */
/* {{{ API */
static const GskCurveClass *
get_class (GskPathOperation op)
{
const GskCurveClass *klasses[] = {
[GSK_PATH_CLOSE] = &GSK_LINE_CURVE_CLASS,
[GSK_PATH_LINE] = &GSK_LINE_CURVE_CLASS,
[GSK_PATH_QUAD] = &GSK_QUAD_CURVE_CLASS,
[GSK_PATH_CUBIC] = &GSK_CUBIC_CURVE_CLASS,
[GSK_PATH_CONIC] = &GSK_CONIC_CURVE_CLASS,
};
g_assert (op < G_N_ELEMENTS (klasses) && klasses[op] != NULL);
return klasses[op];
}
void
gsk_curve_init (GskCurve *curve,
gskpathop op)
{
memset (curve, 0, sizeof (GskCurve));
get_class (gsk_pathop_op (op))->init (curve, op);
}
void
gsk_curve_init_foreach (GskCurve *curve,
GskPathOperation op,
const graphene_point_t *pts,
gsize n_pts,
float weight)
{
memset (curve, 0, sizeof (GskCurve));
get_class (op)->init_foreach (curve, op, pts, n_pts, weight);
}
void
gsk_curve_print (const GskCurve *curve,
GString *string)
{
get_class (curve->op)->print (curve, string);
}
char *
gsk_curve_to_string (const GskCurve *curve)
{
GString *s = g_string_new ("");
gsk_curve_print (curve, s);
return g_string_free (s, FALSE);
}
gskpathop
gsk_curve_pathop (const GskCurve *curve)
{
return get_class (curve->op)->pathop (curve);
}
const graphene_point_t *
gsk_curve_get_start_point (const GskCurve *curve)
{
return get_class (curve->op)->get_start_point (curve);
}
const graphene_point_t *
gsk_curve_get_end_point (const GskCurve *curve)
{
return get_class (curve->op)->get_end_point (curve);
}
void
gsk_curve_get_start_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
get_class (curve->op)->get_start_tangent (curve, tangent);
}
void
gsk_curve_get_end_tangent (const GskCurve *curve,
graphene_vec2_t *tangent)
{
get_class (curve->op)->get_end_tangent (curve, tangent);
}
void
gsk_curve_get_point (const GskCurve *curve,
float progress,
graphene_point_t *pos)
{
get_class (curve->op)->get_point (curve, progress, pos);
}
void
gsk_curve_get_tangent (const GskCurve *curve,
float progress,
graphene_vec2_t *tangent)
{
get_class (curve->op)->get_tangent (curve, progress, tangent);
}
float
gsk_curve_get_curvature (const GskCurve *curve,
float t,
graphene_point_t *center)
{
float k;
k = get_class (curve->op)->get_curvature (curve, t);
if (center != NULL && k != 0)
{
graphene_point_t p;
graphene_vec2_t tangent;
float r;
r = 1/k;
gsk_curve_get_point (curve, t, &p);
gsk_curve_get_tangent (curve, t, &tangent);
center->x = p.x - r * graphene_vec2_get_y (&tangent);
center->y = p.y + r * graphene_vec2_get_x (&tangent);
}
return k;
}
void
gsk_curve_reverse (const GskCurve *curve,
GskCurve *reverse)
{
get_class (curve->op)->reverse (curve, reverse);
}
void
gsk_curve_split (const GskCurve *curve,
float progress,
GskCurve *start,
GskCurve *end)
{
get_class (curve->op)->split (curve, progress, start, end);
}
void
gsk_curve_segment (const GskCurve *curve,
float start,
float end,
GskCurve *segment)
{
if (start <= 0 && end >= 1)
{
*segment = *curve;
return;
}
get_class (curve->op)->segment (curve, start, end, segment);
}
gboolean
gsk_curve_decompose (const GskCurve *curve,
float tolerance,
GskCurveAddLineFunc add_line_func,
gpointer user_data)
{
return get_class (curve->op)->decompose (curve, tolerance, add_line_func, user_data);
}
gboolean
gsk_curve_decompose_curve (const GskCurve *curve,
GskPathForeachFlags flags,
float tolerance,
GskCurveAddCurveFunc add_curve_func,
gpointer user_data)
{
return get_class (curve->op)->decompose_curve (curve, flags, tolerance, add_curve_func, user_data);
}
void
gsk_curve_get_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
get_class (curve->op)->get_bounds (curve, bounds);
}
void
gsk_curve_get_tight_bounds (const GskCurve *curve,
GskBoundingBox *bounds)
{
get_class (curve->op)->get_tight_bounds (curve, bounds);
}
void
gsk_curve_get_derivative_at (const GskCurve *curve,
float t,
graphene_point_t *value)
{
get_class (curve->op)->get_derivative_at (curve, t, value);
}
int
gsk_curve_get_crossing (const GskCurve *curve,
const graphene_point_t *point)
{
return get_class (curve->op)->get_crossing (curve, point);
}
static gboolean
project_point_onto_line (const GskCurve *curve,
const graphene_point_t *point,
float threshold,
float *out_distance,
float *out_t)
{
const graphene_point_t *a = gsk_curve_get_start_point (curve);
const graphene_point_t *b = gsk_curve_get_end_point (curve);
graphene_vec2_t n, ap;
graphene_point_t p;
if (graphene_point_equal (a, b))
{
*out_t = 0;
*out_distance = graphene_point_distance (point, a, NULL, NULL);
}
else
{
graphene_vec2_init (&n, b->x - a->x, b->y - a->y);
graphene_vec2_init (&ap, point->x - a->x, point->y - a->y);
*out_t = graphene_vec2_dot (&n, &ap) / graphene_vec2_dot (&n, &n);
*out_t = CLAMP (*out_t, 0, 1);
graphene_point_interpolate (a, b, *out_t, &p);
*out_distance = graphene_point_distance (point, &p, NULL, NULL);
}
return *out_distance <= threshold;
}
static float
get_segment_bounding_sphere (const GskCurve *curve,
float t1,
float t2,
graphene_point_t *center)
{
GskCurve c;
GskBoundingBox bounds;
gsk_curve_segment (curve, t1, t2, &c);
gsk_curve_get_tight_bounds (&c, &bounds);
graphene_point_interpolate (&bounds.min, &bounds.max, 0.5, center);
return graphene_point_distance (center, &bounds.min, NULL, NULL);
}
static gboolean
find_closest_point (const GskCurve *curve,
const graphene_point_t *point,
float threshold,
float t1,
float t2,
float *out_distance,
float *out_t)
{
graphene_point_t center;
float radius;
float t, d, nt;
radius = get_segment_bounding_sphere (curve, t1, t2, &center);
if (graphene_point_distance (&center, point, NULL, NULL) > threshold + radius)
return FALSE;
d = INFINITY;
t = (t1 + t2) / 2;
if (fabs (t1 - t2) < 0.001)
{
graphene_point_t p;
gsk_curve_get_point (curve, t, &p);
d = graphene_point_distance (point, &p, NULL, NULL);
nt = t;
}
else
{
float dd, tt;
dd = INFINITY;
nt = 0;
if (find_closest_point (curve, point, threshold, t1, t, &dd, &tt))
{
d = dd;
nt = tt;
}
if (find_closest_point (curve, point, MIN (dd, threshold), t, t2, &dd, &tt))
{
d = dd;
nt = tt;
}
}
if (d < threshold)
{
*out_distance = d;
*out_t = nt;
return TRUE;
}
else
{
*out_distance = INFINITY;
*out_t = 0;
return FALSE;
}
}
gboolean
gsk_curve_get_closest_point (const GskCurve *curve,
const graphene_point_t *point,
float threshold,
float *out_dist,
float *out_t)
{
if (curve->op == GSK_PATH_LINE || curve->op == GSK_PATH_CLOSE)
return project_point_onto_line (curve, point, threshold, out_dist, out_t);
else
return find_closest_point (curve, point, threshold, 0, 1, out_dist, out_t);
}
float
gsk_curve_get_length_to (const GskCurve *curve,
float t)
{
return get_class (curve->op)->get_length_to (curve, t);
}
float
gsk_curve_get_length (const GskCurve *curve)
{
return gsk_curve_get_length_to (curve, 1);
}
/* Compute the inverse of the arclength using bisection,
* to a given precision
*/
float
gsk_curve_at_length (const GskCurve *curve,
float length,
float epsilon)
{
float t1, t2, t, l;
GskCurve c1;
g_assert (epsilon >= FLT_EPSILON);
t1 = 0;
t2 = 1;
while (t1 < t2)
{
t = (t1 + t2) / 2;
if (t == t1 || t == t2)
break;
gsk_curve_split (curve, t, &c1, NULL);
l = gsk_curve_get_length (&c1);
if (fabsf (length - l) < epsilon)
break;
else if (l < length)
t1 = t;
else
t2 = t;
}
return t;
}
static inline void
_sincosf (float angle,
float *out_s,
float *out_c)
{
#ifdef HAVE_SINCOSF
sincosf (angle, out_s, out_c);
#else
*out_s = sinf (angle);
*out_c = cosf (angle);
#endif
}
static void
align_points (const graphene_point_t *p,
const graphene_point_t *a,
const graphene_point_t *b,
graphene_point_t *q,
int n)
{
graphene_vec2_t n1;
float angle;
float s, c;
get_tangent (a, b, &n1);
angle = - atan2f (graphene_vec2_get_y (&n1), graphene_vec2_get_x (&n1));
_sincosf (angle, &s, &c);
for (int i = 0; i < n; i++)
{
q[i].x = (p[i].x - a->x) * c - (p[i].y - a->y) * s;
q[i].y = (p[i].x - a->x) * s + (p[i].y - a->y) * c;
}
}
static int
filter_allowable (float t[3],
int n)
{
float g[3];
int j = 0;
for (int i = 0; i < n; i++)
if (0 < t[i] && t[i] < 1)
g[j++] = t[i];
for (int i = 0; i < j; i++)
t[i] = g[i];
return j;
}
/* find solutions for at^2 + bt + c = 0 */
static int
solve_quadratic (float a, float b, float c, float t[2])
{
float d;
int n = 0;
if (fabsf (a) > 0.0001)
{
if (b*b > 4*a*c)
{
d = sqrtf (b*b - 4*a*c);
t[n++] = (-b + d)/(2*a);
t[n++] = (-b - d)/(2*a);
}
else
{
t[n++] = -b / (2*a);
}
}
else if (fabsf (b) > 0.0001)
{
t[n++] = -c / b;
}
return n;
}
int
gsk_curve_get_curvature_points (const GskCurve *curve,
float t[3])
{
const graphene_point_t *pts = curve->cubic.points;
graphene_point_t p[4];
float a, b, c, d;
float x, y, z;
int n;
if (curve->op != GSK_PATH_CUBIC)
return 0; /* FIXME */
align_points (pts, &pts[0], &pts[3], p, 4);
a = p[2].x * p[1].y;
b = p[3].x * p[1].y;
c = p[1].x * p[2].y;
d = p[3].x * p[2].y;
x = - 3*a + 2*b + 3*c - d;
y = 3*a - b - 3*c;
z = c - a;
n = solve_quadratic (x, y, z, t);
return filter_allowable (t, n);
}
/* }}} */
/* vim:set foldmethod=marker expandtab: */
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