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214f5a6f98
When we allocate a graphene_point_t on the stack, there's no guarantee that it will be aligned at an 8-byte boundary, which is an assumption made by gsk_pathop_encode() (which wants to use the lowest 3 bits to encode the operation). In the places where it matters, force the points on the stack and embedded in structs to be nicely aligned. By using a distinct type for this (a union with a suitable size and alignment), we ensure that the compiler will warn or error whenever we can't prove that a particular point is, in fact, suitably aligned. We can go from a `GskAlignedPoint *` to a `graphene_point_t *` (which is always valid, because the `GskAlignedPoint` is aligned) via &aligned_points[0].pt, but we cannot go back the other way (which is not always valid, because the `graphene_point_t` is not necessarily aligned nicely) without a cast. In practice, it seems that a graphene_point_t on x86_64 *is* usually placed at an 8-byte boundary, but this is not the case on 32-bit architectures or on s390x. In many cases we can avoid needing an explicit reference to the more complicated type by making use of a transparent union. There's already at least one transparent union in GSK's public API, so it's presumably portable enough to match GTK's requirements. Increasing the alignment of GskAlignedPoint also requires adjusting how a GskStandardContour is allocated and initialized. This data structure allocates extra memory to hold an array of GskAlignedPoint outside the bounds of the struct itself, and that array now needs to be aligned suitably. Previously the array started with at next byte after the flexible array of gskpathop, but the alignment of a gskpathop is only 4 bytes on 32-bit architectures, so depending on the number of gskpathop in the trailing flexible array, that pointer might be an unsuitable location to allocate a GskAlignedPoint. Resolves: https://gitlab.gnome.org/GNOME/gtk/-/issues/6395 Signed-off-by: Simon McVittie <smcv@debian.org>
2644 lines
80 KiB
C
2644 lines
80 KiB
C
/*
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* Copyright © 2020 Benjamin Otte
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*
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* This library is free software; you can redistribute it and/or
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* modify it under the terms of the GNU Lesser General Public
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* License as published by the Free Software Foundation; either
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* version 2.1 of the License, or (at your option) any later version.
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*
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* This library is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* Lesser General Public License for more details.
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*
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* You should have received a copy of the GNU Lesser General Public
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* License along with this library. If not, see <http://www.gnu.org/licenses/>.
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*
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* Authors: Benjamin Otte <otte@gnome.org>
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*/
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#include "config.h"
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#include "gskcurveprivate.h"
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#include "gskboundingboxprivate.h"
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/* GskCurve collects all the functionality we need for Bézier segments */
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#define MIN_PROGRESS (1/1024.f)
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typedef struct _GskCurveClass GskCurveClass;
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struct _GskCurveClass
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{
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void (* init) (GskCurve *curve,
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gskpathop op);
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void (* init_foreach) (GskCurve *curve,
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GskPathOperation op,
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const graphene_point_t *pts,
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gsize n_pts,
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float weight);
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void (* print) (const GskCurve *curve,
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GString *string);
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gskpathop (* pathop) (const GskCurve *curve);
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const graphene_point_t * (* get_start_point) (const GskCurve *curve);
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const graphene_point_t * (* get_end_point) (const GskCurve *curve);
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void (* get_start_tangent) (const GskCurve *curve,
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graphene_vec2_t *tangent);
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void (* get_end_tangent) (const GskCurve *curve,
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graphene_vec2_t *tangent);
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void (* get_point) (const GskCurve *curve,
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float t,
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graphene_point_t *pos);
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void (* get_tangent) (const GskCurve *curve,
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float t,
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graphene_vec2_t *tangent);
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void (* reverse) (const GskCurve *curve,
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GskCurve *reverse);
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float (* get_curvature) (const GskCurve *curve,
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float t);
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void (* split) (const GskCurve *curve,
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float progress,
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GskCurve *result1,
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GskCurve *result2);
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void (* segment) (const GskCurve *curve,
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float start,
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float end,
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GskCurve *segment);
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gboolean (* decompose) (const GskCurve *curve,
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float tolerance,
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GskCurveAddLineFunc add_line_func,
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gpointer user_data);
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gboolean (* decompose_curve) (const GskCurve *curve,
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GskPathForeachFlags flags,
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float tolerance,
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GskCurveAddCurveFunc add_curve_func,
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gpointer user_data);
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void (* get_bounds) (const GskCurve *curve,
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GskBoundingBox *bounds);
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void (* get_tight_bounds) (const GskCurve *curve,
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GskBoundingBox *bounds);
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void (* get_derivative_at) (const GskCurve *curve,
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float t,
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graphene_point_t *value);
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int (* get_crossing) (const GskCurve *curve,
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const graphene_point_t *point);
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float (* get_length_to) (const GskCurve *curve,
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float t);
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float (* get_at_length) (const GskCurve *curve,
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float distance,
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float epsilon);
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};
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/* {{{ Utilities */
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#define RAD_TO_DEG(r) ((r)*180.f/M_PI)
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#define DEG_TO_RAD(d) ((d)*M_PI/180.f)
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static void
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get_tangent (const graphene_point_t *p0,
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const graphene_point_t *p1,
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graphene_vec2_t *t)
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{
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graphene_vec2_init (t, p1->x - p0->x, p1->y - p0->y);
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graphene_vec2_normalize (t, t);
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}
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static int
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line_get_crossing (const graphene_point_t *p,
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const graphene_point_t *p1,
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const graphene_point_t *p2)
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{
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if (p1->y <= p->y)
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{
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if (p2->y > p->y)
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{
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if ((p2->x - p1->x) * (p->y - p1->y) - (p->x - p1->x) * (p2->y - p1->y) > 0)
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return 1;
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}
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}
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else if (p2->y <= p->y)
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{
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if ((p2->x - p1->x) * (p->y - p1->y) - (p->x - p1->x) * (p2->y - p1->y) < 0)
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return -1;
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}
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return 0;
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}
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static int
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get_crossing_by_bisection (const GskCurve *curve,
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const graphene_point_t *point)
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{
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GskBoundingBox bounds;
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GskCurve c1, c2;
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gsk_curve_get_bounds (curve, &bounds);
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if (bounds.max.y < point->y || bounds.min.y > point->y || bounds.max.x < point->x)
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return 0;
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if (bounds.min.x > point->x)
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return line_get_crossing (point, gsk_curve_get_start_point (curve), gsk_curve_get_end_point (curve));
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if (graphene_point_distance (&bounds.min, &bounds.max, NULL, NULL) < 0.001)
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return line_get_crossing (point, gsk_curve_get_start_point (curve), gsk_curve_get_end_point (curve));
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gsk_curve_split (curve, 0.5, &c1, &c2);
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return gsk_curve_get_crossing (&c1, point) + gsk_curve_get_crossing (&c2, point);
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}
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/* Replace a line by an equivalent quad,
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* and a quad by an equivalent cubic.
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*/
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static void
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gsk_curve_elevate (const GskCurve *curve,
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GskCurve *elevated)
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{
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if (curve->op == GSK_PATH_LINE)
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{
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GskAlignedPoint p[3];
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p[0].pt = curve->line.points[0];
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graphene_point_interpolate (&curve->line.points[0],
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&curve->line.points[1],
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0.5,
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&p[1].pt);
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p[2].pt = curve->line.points[1];
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gsk_curve_init (elevated, gsk_pathop_encode (GSK_PATH_QUAD, p));
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}
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else if (curve->op == GSK_PATH_QUAD)
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{
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GskAlignedPoint p[4];
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p[0].pt = curve->quad.points[0];
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graphene_point_interpolate (&curve->quad.points[0],
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&curve->quad.points[1],
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2/3.,
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&p[1].pt);
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graphene_point_interpolate (&curve->quad.points[2],
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&curve->quad.points[1],
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2/3.,
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&p[2].pt);
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p[3].pt = curve->quad.points[2];
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gsk_curve_init (elevated, gsk_pathop_encode (GSK_PATH_CUBIC, p));
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}
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else
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g_assert_not_reached ();
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}
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/* Compute arclength by using Gauss quadrature on
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*
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* \int_0^z \sqrt{ (dx/dt)^2 + (dy/dt)^2 } dt
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*/
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#include "gskcurve-ct-values.c"
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static float
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get_length_by_approximation (const GskCurve *curve,
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float t)
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{
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double z = t / 2;
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double sum = 0;
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graphene_point_t d;
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for (unsigned int i = 0; i < G_N_ELEMENTS (T); i++)
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{
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gsk_curve_get_derivative_at (curve, z * T[i] + z, &d);
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sum += C[i] * sqrt (d.x * d.x + d.y * d.y);
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}
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return z * sum;
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}
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/* Compute the inverse of the arclength using bisection,
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* to a given precision
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*/
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static float
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get_t_by_bisection (const GskCurve *curve,
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float length,
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float epsilon)
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{
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float t1, t2, t, l;
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GskCurve c1;
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g_assert (epsilon >= FLT_EPSILON);
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t1 = 0;
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t2 = 1;
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do
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{
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t = (t1 + t2) / 2;
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if (t == t1 || t == t2)
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break;
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gsk_curve_split (curve, t, &c1, NULL);
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l = gsk_curve_get_length (&c1);
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if (fabsf (length - l) < epsilon)
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break;
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else if (l < length)
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t1 = t;
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else
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t2 = t;
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}
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while (t1 < t2);
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return t;
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}
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/* }}} */
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/* {{{ Line */
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static void
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gsk_line_curve_init_from_points (GskLineCurve *self,
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GskPathOperation op,
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const graphene_point_t *start,
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const graphene_point_t *end)
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{
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self->op = op;
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self->points[0] = *start;
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self->points[1] = *end;
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}
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static void
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gsk_line_curve_init (GskCurve *curve,
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gskpathop op)
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{
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GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = gsk_pathop_points (op);
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gsk_line_curve_init_from_points (self, gsk_pathop_op (op), &pts[0], &pts[1]);
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}
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static void
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gsk_line_curve_init_foreach (GskCurve *curve,
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GskPathOperation op,
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const graphene_point_t *pts,
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gsize n_pts,
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float weight)
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{
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GskLineCurve *self = &curve->line;
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g_assert (n_pts == 2);
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gsk_line_curve_init_from_points (self, op, &pts[0], &pts[1]);
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}
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static void
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gsk_line_curve_print (const GskCurve *curve,
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GString *string)
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{
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g_string_append_printf (string,
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"M %g %g L %g %g",
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curve->line.points[0].x, curve->line.points[0].y,
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curve->line.points[1].x, curve->line.points[1].y);
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}
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static gskpathop
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gsk_line_curve_pathop (const GskCurve *curve)
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{
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const GskLineCurve *self = &curve->line;
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return gsk_pathop_encode (self->op, self->aligned_points);
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}
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static const graphene_point_t *
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gsk_line_curve_get_start_point (const GskCurve *curve)
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{
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const GskLineCurve *self = &curve->line;
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return &self->points[0];
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}
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static const graphene_point_t *
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gsk_line_curve_get_end_point (const GskCurve *curve)
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{
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const GskLineCurve *self = &curve->line;
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return &self->points[1];
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}
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static void
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gsk_line_curve_get_start_end_tangent (const GskCurve *curve,
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graphene_vec2_t *tangent)
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{
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const GskLineCurve *self = &curve->line;
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get_tangent (&self->points[0], &self->points[1], tangent);
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}
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static void
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gsk_line_curve_get_point (const GskCurve *curve,
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float t,
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graphene_point_t *pos)
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{
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const GskLineCurve *self = &curve->line;
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graphene_point_interpolate (&self->points[0], &self->points[1], t, pos);
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}
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static void
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gsk_line_curve_get_tangent (const GskCurve *curve,
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float t,
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graphene_vec2_t *tangent)
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{
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const GskLineCurve *self = &curve->line;
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get_tangent (&self->points[0], &self->points[1], tangent);
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}
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static float
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gsk_line_curve_get_curvature (const GskCurve *curve,
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float t)
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{
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return 0;
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}
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static void
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gsk_line_curve_reverse (const GskCurve *curve,
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GskCurve *reverse)
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{
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const GskLineCurve *self = &curve->line;
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reverse->op = GSK_PATH_LINE;
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reverse->line.points[0] = self->points[1];
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reverse->line.points[1] = self->points[0];
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}
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static void
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gsk_line_curve_split (const GskCurve *curve,
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float progress,
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GskCurve *start,
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GskCurve *end)
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{
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const GskLineCurve *self = &curve->line;
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graphene_point_t point;
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graphene_point_interpolate (&self->points[0], &self->points[1], progress, &point);
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if (start)
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gsk_line_curve_init_from_points (&start->line, GSK_PATH_LINE, &self->points[0], &point);
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if (end)
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gsk_line_curve_init_from_points (&end->line, GSK_PATH_LINE, &point, &self->points[1]);
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}
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static void
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gsk_line_curve_segment (const GskCurve *curve,
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float start,
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float end,
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GskCurve *segment)
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{
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const GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = self->points;
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graphene_point_t p0, p1;
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graphene_point_interpolate (&pts[0], &pts[1], start, &p0);
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graphene_point_interpolate (&pts[0], &pts[1], end, &p1);
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gsk_line_curve_init_from_points (&segment->line, GSK_PATH_LINE, &p0, &p1);
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}
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static gboolean
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gsk_line_curve_decompose (const GskCurve *curve,
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float tolerance,
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GskCurveAddLineFunc add_line_func,
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gpointer user_data)
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{
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const GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = self->points;
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return add_line_func (&pts[0], &pts[1], 0.f, 1.f, GSK_CURVE_LINE_REASON_STRAIGHT, user_data);
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}
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static gboolean
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gsk_line_curve_decompose_curve (const GskCurve *curve,
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GskPathForeachFlags flags,
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float tolerance,
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GskCurveAddCurveFunc add_curve_func,
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gpointer user_data)
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{
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const GskLineCurve *self = &curve->line;
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return add_curve_func (GSK_PATH_LINE, self->points, 2, 0.f, user_data);
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}
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static void
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gsk_line_curve_get_bounds (const GskCurve *curve,
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GskBoundingBox *bounds)
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{
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const GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = self->points;
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gsk_bounding_box_init (bounds, &pts[0], &pts[1]);
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}
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static void
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gsk_line_curve_get_derivative_at (const GskCurve *curve,
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float t,
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graphene_point_t *value)
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{
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const GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = self->points;
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value->x = pts[1].x - pts[0].x;
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value->y = pts[1].y - pts[0].y;
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}
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static int
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gsk_line_curve_get_crossing (const GskCurve *curve,
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const graphene_point_t *point)
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{
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const GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = self->points;
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return line_get_crossing (point, &pts[0], &pts[1]);
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}
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static float
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gsk_line_curve_get_length_to (const GskCurve *curve,
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float t)
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{
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const GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = self->points;
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return t * graphene_point_distance (&pts[0], &pts[1], NULL, NULL);
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}
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static float
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gsk_line_curve_get_at_length (const GskCurve *curve,
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float distance,
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float epsilon)
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{
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const GskLineCurve *self = &curve->line;
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const graphene_point_t *pts = self->points;
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float length;
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length = graphene_point_distance (&pts[0], &pts[1], NULL, NULL);
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if (length == 0)
|
|
return 0;
|
|
|
|
return CLAMP (distance / length, 0, 1);
|
|
}
|
|
|
|
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,
|
|
gsk_line_curve_get_at_length,
|
|
};
|
|
|
|
/* }}} */
|
|
/* {{{ 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->aligned_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 float
|
|
gsk_quad_curve_get_at_length (const GskCurve *curve,
|
|
float t,
|
|
float epsilon)
|
|
{
|
|
return get_t_by_bisection (curve, t, epsilon);
|
|
}
|
|
|
|
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,
|
|
gsk_quad_curve_get_at_length,
|
|
};
|
|
|
|
/* }}} */
|
|
/* {{{ 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->aligned_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 float
|
|
gsk_cubic_curve_get_at_length (const GskCurve *curve,
|
|
float t,
|
|
float epsilon)
|
|
{
|
|
return get_t_by_bisection (curve, t, epsilon);
|
|
}
|
|
|
|
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,
|
|
gsk_cubic_curve_get_at_length,
|
|
};
|
|
|
|
/* }}} */
|
|
/* {{{ 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->aligned_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];
|
|
GskAlignedPoint 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].pt = GRAPHENE_POINT_INIT (l[0].x / l[0].z, l[0].y / l[0].z);
|
|
left[1].pt = GRAPHENE_POINT_INIT (l[1].x / l[1].z, l[1].y / l[1].z);
|
|
left[3].pt = GRAPHENE_POINT_INIT (l[2].x / l[2].z, l[2].y / l[2].z);
|
|
|
|
right[0].pt = GRAPHENE_POINT_INIT (r[0].x / r[0].z, r[0].y / r[0].z);
|
|
right[1].pt = GRAPHENE_POINT_INIT (r[1].x / r[1].z, r[1].y / r[1].z);
|
|
right[3].pt = 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].pt = GRAPHENE_POINT_INIT (l[1].z / sqrt (l[2].z), 0);
|
|
right[2].pt = 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 || end >= 1.0f)
|
|
{
|
|
if (start <= 0.0f)
|
|
gsk_conic_curve_split (curve, end, segment, NULL);
|
|
else if (end >= 1.0f)
|
|
gsk_conic_curve_split (curve, start, NULL, segment);
|
|
|
|
return;
|
|
}
|
|
|
|
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;
|
|
GskAlignedPoint 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].pt = self->points[0];
|
|
p[3].pt = self->points[3];
|
|
graphene_point_interpolate (&self->points[0], &self->points[1], lambda, &p[1].pt);
|
|
graphene_point_interpolate (&self->points[3], &self->points[1], lambda, &p[2].pt);
|
|
|
|
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 float
|
|
gsk_conic_curve_get_at_length (const GskCurve *curve,
|
|
float t,
|
|
float epsilon)
|
|
{
|
|
return get_t_by_bisection (curve, t, epsilon);
|
|
}
|
|
|
|
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,
|
|
gsk_conic_curve_get_at_length,
|
|
};
|
|
|
|
/* }}} */
|
|
/* {{{ 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, ¢er);
|
|
if (graphene_point_distance (¢er, 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)
|
|
{
|
|
return get_class (curve->op)->get_at_length (curve, length, epsilon);
|
|
}
|
|
|
|
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);
|
|
}
|
|
|
|
/* Find cusps inside the open interval from 0 to 1.
|
|
*
|
|
* According to Stone & deRose, A Geometric Characterization
|
|
* of Parametric Cubic curves, a necessary and sufficient
|
|
* condition is that the first derivative vanishes.
|
|
*/
|
|
int
|
|
gsk_curve_get_cusps (const GskCurve *curve,
|
|
float t[2])
|
|
{
|
|
const graphene_point_t *pts = curve->cubic.points;
|
|
graphene_point_t p[3];
|
|
float ax, bx, cx;
|
|
float ay, by, cy;
|
|
float tx[3];
|
|
int nx;
|
|
int n = 0;
|
|
|
|
if (curve->op != GSK_PATH_CUBIC)
|
|
return 0;
|
|
|
|
p[0].x = 3 * (pts[1].x - pts[0].x);
|
|
p[0].y = 3 * (pts[1].y - pts[0].y);
|
|
p[1].x = 3 * (pts[2].x - pts[1].x);
|
|
p[1].y = 3 * (pts[2].y - pts[1].y);
|
|
p[2].x = 3 * (pts[3].x - pts[2].x);
|
|
p[2].y = 3 * (pts[3].y - pts[2].y);
|
|
|
|
ax = p[0].x - 2 * p[1].x + p[2].x;
|
|
bx = - 2 * p[0].x + 2 * p[1].x;
|
|
cx = p[0].x;
|
|
|
|
nx = solve_quadratic (ax, bx, cx, tx);
|
|
nx = filter_allowable (tx, nx);
|
|
|
|
ay = p[0].y - 2 * p[1].y + p[2].y;
|
|
by = - 2 * p[0].y + 2 * p[1].y;
|
|
cy = p[0].y;
|
|
|
|
for (int i = 0; i < nx; i++)
|
|
{
|
|
float ti = tx[i];
|
|
|
|
if (0 < ti && ti < 1 &&
|
|
fabsf (ay * ti * ti + by * ti + cy) < 0.001)
|
|
t[n++] = ti;
|
|
}
|
|
|
|
return n;
|
|
}
|
|
|
|
/* }}} */
|
|
|
|
/* vim:set foldmethod=marker expandtab: */
|