Add mesh gradient rasterizer

Add an implementation of a fast and reasonably accurate
non-antialiased mesh gradient rasterizer.

1 files changed, 943 insertions, 0 deletions

diff --git a/src/cairo-mesh-pattern-rasterizer.c b/src/cairo-mesh-pattern-rasterizer.c
new file mode 100644
index 000000000..f95c484c2
--- /dev/null
+++ b/src/cairo-mesh-pattern-rasterizer.c
@@ -0,0 +1,943 @@
+/* -*- Mode: c; tab-width: 8; c-basic-offset: 4; indent-tabs-mode: t; -*- */
+/* cairo - a vector graphics library with display and print output
+ *
+ * Copyright 2009 Andrea Canciani
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it either under the terms of the GNU Lesser General Public
+ * License version 2.1 as published by the Free Software Foundation
+ * (the "LGPL") or, at your option, under the terms of the Mozilla
+ * Public License Version 1.1 (the "MPL"). If you do not alter this
+ * notice, a recipient may use your version of this file under either
+ * the MPL or the LGPL.
+ *
+ * You should have received a copy of the LGPL along with this library
+ * in the file COPYING-LGPL-2.1; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Suite 500, Boston, MA 02110-1335, USA
+ * You should have received a copy of the MPL along with this library
+ * in the file COPYING-MPL-1.1
+ *
+ * The contents of this file are subject to the Mozilla Public License
+ * Version 1.1 (the "License"); you may not use this file except in
+ * compliance with the License. You may obtain a copy of the License at
+ * http://www.mozilla.org/MPL/
+ *
+ * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
+ * OF ANY KIND, either express or implied. See the LGPL or the MPL for
+ * the specific language governing rights and limitations.
+ *
+ * The Original Code is the cairo graphics library.
+ *
+ * The Initial Developer of the Original Code is Andrea Canciani.
+ *
+ * Contributor(s):
+ *	Andrea Canciani <ranma42@gmail.com>
+ */
+
+#include "cairoint.h"
+
+/*
+ * Rasterizer for mesh patterns.
+ *
+ * This implementation is based on techniques derived from several
+ * papers (available from ACM):
+ *
+ * - Lien, Shantz and Pratt "Adaptive Forward Differencing for
+ *   Rendering Curves and Surfaces" (discussion of the AFD technique,
+ *   bound of 1/sqrt(2) on step length without proof)
+ *
+ * - Popescu and Rosen, "Forward rasterization" (description of
+ *   forward rasterization, proof of the previous bound)
+ *
+ * - Klassen, "Integer Forward Differencing of Cubic Polynomials:
+ *   Analysis and Algorithms"
+ *
+ * - Klassen, "Exact Integer Hybrid Subdivision and Forward
+ *   Differencing of Cubics" (improving the bound on the minimum
+ *   number of steps)
+ *
+ * - Chang, Shantz and Rocchetti, "Rendering Cubic Curves and Surfaces
+ *   with Integer Adaptive Forward Differencing" (analysis of forward
+ *   differencing applied to Bezier patches)
+ *
+ * Notes:
+ * - Poor performance expected in degenerate cases
+ *
+ * - Patches mostly outside the drawing area are drawn completely (and
+ *   clipped), wasting time
+ *
+ * - Both previous problems are greatly reduced by splitting until a
+ *   reasonably small size and clipping the new tiles: execution time
+ *   is quadratic in the convex-hull diameter instead than linear to
+ *   the painted area. Splitting the tiles doesn't change the painted
+ *   area but (usually) reduces the bounding box area (bbox area can
+ *   remain the same after splitting, but cannot grow)
+ *
+ * - The initial implementation used adaptive forward differencing,
+ *   but simple forward differencing scored better in benchmarks
+ *
+ * Idea:
+ *
+ * We do a sampling over the cubic patch with step du and dv (in the
+ * two parameters) that guarantees that any point of our sampling will
+ * be at most at 1/sqrt(2) from its adjacent points. In formulae
+ * (assuming B is the patch):
+ *
+ *   |B(u,v) - B(u+du,v)| < 1/sqrt(2)
+ *   |B(u,v) - B(u,v+dv)| < 1/sqrt(2)
+ *
+ * This means that every pixel covered by the patch will contain at
+ * least one of the samples, thus forward rasterization can be
+ * performed. Sketch of proof (from Popescu and Rosen):
+ *
+ * Let's take the P pixel we're interested into. If we assume it to be
+ * square, its boundaries define 9 regions on the plane:
+ *
+ * 1|2|3
+ * -+-+-
+ * 8|P|4
+ * -+-+-
+ * 7|6|5
+ *
+ * Let's check that the pixel P will contain at least one point
+ * assuming that it is covered by the patch.
+ *
+ * Since the pixel is covered by the patch, its center will belong to
+ * (at least) one of the quads:
+ *
+ *   {(B(u,v), B(u+du,v), B(u,v+dv), B(u+du,v+dv)) for u,v in [0,1]}
+ *
+ * If P doesn't contain any of the corners of the quad:
+ *
+ * - if one of the corners is in 1,3,5 or 7, other two of them have to
+ *   be in 2,4,6 or 8, thus if the last corner is not in P, the length
+ *   of one of the edges will be > 1/sqrt(2)
+ *
+ * - if none of the corners is in 1,3,5 or 7, all of them are in 2,4,6
+ *   and/or 8. If they are all in different regions, they can't
+ *   satisfy the distance constraint. If two of them are in the same
+ *   region (let's say 2), no point is in 6 and again it is impossible
+ *   to have the center of P in the quad respecting the distance
+ *   constraint (both these assertions can be checked by continuity
+ *   considering the length of the edges of a quad with the vertices
+ *   on the edges of P)
+ *
+ * Each of the cases led to a contradiction, so P contains at least
+ * one of the corners of the quad.
+ */
+
+/*
+ * Make sure that errors are less than 1 in fixed point math if you
+ * change these values.
+ *
+ * The error is amplified by about steps^3/4 times.
+ * The rasterizer always uses a number of steps that is a power of 2.
+ *
+ * 256 is the maximum allowed number of steps (to have error < 1)
+ * using 8.24 for the differences.
+ */
+#define STEPS_MAX_V 256.0
+#define STEPS_MAX_U 256.0
+
+/*
+ * If the patch/curve is only partially visible, split it to a finer
+ * resolution to get higher chances to clip (part of) it.
+ *
+ * These values have not been computed, but simply obtained
+ * empirically (by benchmarking some patches). They should never be
+ * greater than STEPS_MAX_V (or STEPS_MAX_U), but they can be as small
+ * as 1 (depending on how much you want to spend time in splitting the
+ * patch/curve when trying to save some rasterization time).
+ */
+#define STEPS_CLIP_V 64.0
+#define STEPS_CLIP_U 64.0
+
+
+/* Utils */
+static inline double
+sqlen (cairo_point_double_t p0, cairo_point_double_t p1)
+{
+    cairo_point_double_t delta;
+
+    delta.x = p0.x - p1.x;
+    delta.y = p0.y - p1.y;
+
+    return delta.x * delta.x + delta.y * delta.y;
+}
+
+static inline double
+max (double x, double y)
+{
+    return x > y ? x : y;
+}
+
+static inline double
+min (double x, double y)
+{
+    return x < y ? x : y;
+}
+
+static inline int16_t
+_color_delta_to_shifted_short (int32_t from, int32_t to, int shift)
+{
+    int32_t delta = to - from;
+
+    /* We need to round toward zero, because otherwise adding the
+     * delta 2^shift times can overflow */
+    if (delta >= 0)
+	return delta >> shift;
+    else
+	return -((-delta) >> shift);
+}
+
+/*
+ * Convert a number of steps to the equivalent shift.
+ *
+ * Input: the square of the minimum number of steps
+ *
+ * Output: the smallest integer x such that 2^x > steps
+ */
+static inline int
+sqsteps2shift (double steps_sq)
+{
+    int r;
+    frexp (max (1.0, steps_sq), &r);
+    return (r + 1) >> 1;
+}
+
+/*
+ * FD functions
+ *
+ * A Bezier curve is defined (with respect to a parameter t in
+ * [0,1]) from its nodes (x,y,z,w) like this:
+ *
+ *   B(t) = x(1-t)^3 + 3yt(1-t)^2 + 3zt^2(1-t) + wt^3
+ *
+ * To efficiently evaluate a Bezier curve, the rasterizer uses forward
+ * differences. Given x, y, z, w (the 4 nodes of the Bezier curve), it
+ * is possible to convert them to forward differences form and walk
+ * over the curve using fd_init (), fd_down () and fd_fwd ().
+ *
+ * f[0] is always the value of the Bezier curve for "current" t.
+ */
+
+/*
+ * Initialize the coefficient for forward differences.
+ *
+ * Input: x,y,z,w are the 4 nodes of the Bezier curve
+ *
+ * Output: f[i] is the i-th difference of the curve
+ *
+ * f[0] is the value of the curve for t==0, i.e. f[0]==x.
+ *
+ * The initial step is 1; this means that each step increases t by 1
+ * (so fd_init () immediately followed by fd_fwd (f) n times makes
+ * f[0] be the value of the curve for t==n).
+ */
+static inline void
+fd_init (double x, double y, double z, double w, double f[4])
+{
+    f[0] = x;
+    f[1] = w - x;
+    f[2] = 6. * (w - 2. * z + y);
+    f[3] = 6. * (w - 3. * z + 3. * y - x);
+}
+
+/*
+ * Halve the step of the coefficients for forward differences.
+ *
+ * Input: f[i] is the i-th difference of the curve
+ *
+ * Output: f[i] is the i-th difference of the curve with half the
+ *         original step
+ *
+ * f[0] is not affected, so the current t is not changed.
+ *
+ * The other coefficients are changed so that the step is half the
+ * original step. This means that doing fd_fwd (f) n times with the
+ * input f results in the same f[0] as doing fd_fwd (f) 2n times with
+ * the output f.
+ */
+static inline void
+fd_down (double f[4])
+{
+    f[3] *= 0.125;
+    f[2] = f[2] * 0.25 - f[3];
+    f[1] = (f[1] - f[2]) * 0.5;
+}
+
+/*
+ * Perform one step of forward differences along the curve.
+ *
+ * Input: f[i] is the i-th difference of the curve
+ *
+ * Output: f[i] is the i-th difference of the curve after one step
+ */
+static inline void
+fd_fwd (double f[4])
+{
+    f[0] += f[1];
+    f[1] += f[2];
+    f[2] += f[3];
+}
+
+/*
+ * Transform to integer forward differences.
+ *
+ * Input: d[n] is the n-th difference (in double precision)
+ *
+ * Output: i[n] is the n-th difference (in fixed point precision)
+ *
+ * i[0] is 9.23 fixed point, other differences are 4.28 fixed point.
+ */
+static inline void
+fd_fixed (double d[4], int32_t i[4])
+{
+    i[0] = _cairo_fixed_16_16_from_double (256 *  2 * d[0]);
+    i[1] = _cairo_fixed_16_16_from_double (256 * 16 * d[1]);
+    i[2] = _cairo_fixed_16_16_from_double (256 * 16 * d[2]);
+    i[3] = _cairo_fixed_16_16_from_double (256 * 16 * d[3]);
+}
+
+/*
+ * Perform one step of integer forward differences along the curve.
+ *
+ * Input: f[n] is the n-th difference
+ *
+ * Output: f[n] is the n-th difference
+ *
+ * f[0] is 9.23 fixed point, other differences are 4.28 fixed point.
+ */
+static inline void
+fd_fixed_fwd (int32_t f[4])
+{
+    f[0] += (f[1] >> 5) + ((f[1] >> 4) & 1);
+    f[1] += f[2];
+    f[2] += f[3];
+}
+
+/*
+ * Compute the minimum number of steps that guarantee that walking
+ * over a curve will leave no holes.
+ *
+ * Input: p[0..3] the nodes of the Bezier curve
+ *
+ * Returns: the square of the number of steps
+ *
+ * Idea:
+ *
+ * We want to make sure that at every step we move by less than
+ * 1/sqrt(2).
+ *
+ * The derivative of the cubic Bezier with nodes (p0, p1, p2, p3) is
+ * the quadratic Bezier with nodes (p1-p0, p2-p1, p3-p2) scaled by 3,
+ * so (since a Bezier curve is always bounded by its convex hull), we
+ * can say that:
+ *
+ *  max(|B'(t)|) <= 3 max (|p1-p0|, |p2-p1|, |p3-p2|)
+ *
+ * We can improve this by noticing that a quadratic Bezier (a,b,c) is
+ * bounded by the quad (a,lerp(a,b,t),lerp(b,c,t),c) for any t, so
+ * (substituting the previous values, using t=0.5 and simplifying):
+ *
+ *  max(|B'(t)|) <= 3 max (|p1-p0|, |p2-p0|/2, |p3-p1|/2, |p3-p2|)
+ *
+ * So, to guarantee a maximum step lenght of 1/sqrt(2) we must do:
+ *
+ *   3 max (|p1-p0|, |p2-p0|/2, |p3-p1|/2, |p3-p2|) sqrt(2) steps
+ */
+static inline double
+bezier_steps_sq (cairo_point_double_t p[4])
+{
+    double tmp = sqlen (p[0], p[1]);
+    tmp = max (tmp, sqlen (p[2], p[3]));
+    tmp = max (tmp, sqlen (p[0], p[2]) * .25);
+    tmp = max (tmp, sqlen (p[1], p[3]) * .25);
+    return 18.0 * tmp;
+}
+
+/*
+ * Split a 1D Bezier cubic using de Casteljau's algorithm.
+ *
+ * Input: x,y,z,w the nodes of the Bezier curve
+ *
+ * Output: x0,y0,z0,w0 and x1,y1,z1,w1 are respectively the nodes of
+ *         the first half and of the second half of the curve
+ *
+ * The output control nodes have to be distinct.
+ */
+static inline void
+split_bezier_1D (double  x,  double  y,  double  z,  double  w,
+		 double *x0, double *y0, double *z0, double *w0,
+		 double *x1, double *y1, double *z1, double *w1)
+{
+    double tmp;
+
+    *x0 = x;
+    *w1 = w;
+
+    tmp = 0.5 * (y + z);
+    *y0 = 0.5 * (x + y);
+    *z1 = 0.5 * (z + w);
+
+    *z0 = 0.5 * (*y0 + tmp);
+    *y1 = 0.5 * (tmp + *z1);
+
+    *w0 = *x1 = 0.5 * (*z0 + *y1);
+}
+
+/*
+ * Split a Bezier curve using de Casteljau's algorithm.
+ *
+ * Input: p[0..3] the nodes of the Bezier curve
+ *
+ * Output: fst_half[0..3] and snd_half[0..3] are respectively the
+ *         nodes of the first and of the second half of the curve
+ *
+ * fst_half and snd_half must be different, but they can be the same as
+ * nodes.
+ */
+static void
+split_bezier (cairo_point_double_t p[4],
+	      cairo_point_double_t fst_half[4],
+	      cairo_point_double_t snd_half[4])
+{
+    split_bezier_1D (p[0].x, p[1].x, p[2].x, p[3].x,
+		     &fst_half[0].x, &fst_half[1].x, &fst_half[2].x, &fst_half[3].x,
+		     &snd_half[0].x, &snd_half[1].x, &snd_half[2].x, &snd_half[3].x);
+
+    split_bezier_1D (p[0].y, p[1].y, p[2].y, p[3].y,
+		     &fst_half[0].y, &fst_half[1].y, &fst_half[2].y, &fst_half[3].y,
+		     &snd_half[0].y, &snd_half[1].y, &snd_half[2].y, &snd_half[3].y);
+}
+
+
+typedef enum _intersection {
+    INSIDE = -1, /* the interval is entirely contained in the reference interval */
+    OUTSIDE = 0, /* the interval has no intersection with the reference interval */
+    PARTIAL = 1  /* the interval intersects the reference interval (but is not fully inside it) */
+} intersection_t;
+
+/*
+ * Check if an interval if inside another.
+ *
+ * Input: a,b are the extrema of the first interval
+ *        c,d are the extrema of the second interval
+ *
+ * Returns: INSIDE  iff [a,b) intersection [c,d) = [a,b)
+ *          OUTSIDE iff [a,b) intersection [c,d) = {}
+ *          PARTIAL otherwise
+ *
+ * The function assumes a < b and c < d
+ *
+ * Note: Bitwise-anding the results along each component gives the
+ *       expected result for [a,b) x [A,B) intersection [c,d) x [C,D).
+ */
+static inline int
+intersect_interval (double a, double b, double c, double d)
+{
+    if (c <= a && b <= d)
+	return INSIDE;
+    else if (a >= d || b <= c)
+	return OUTSIDE;
+    else
+	return PARTIAL;
+}
+
+/*
+ * Set the color of a pixel.
+ *
+ * Input: data is the base pointer of the image
+ *        width, height are the dimensions of the image
+ *        stride is the stride in bytes between adjacent rows
+ *        x, y are the coordinates of the pixel to be colored
+ *        r,g,b,a are the color components of the color to be set
+ *
+ * Output: the (x,y) pixel in data has the (r,g,b,a) color
+ *
+ * The input color components are not premultiplied, but the data
+ * stored in the image is assumed to be in CAIRO_FORMAT_ARGB32 (8 bpc,
+ * premultiplied).
+ *
+ * If the pixel to be set is outside the image, this function does
+ * nothing.
+ */
+static inline void
+draw_pixel (unsigned char *data, int width, int height, int stride,
+	    int x, int y, uint16_t r, uint16_t g, uint16_t b, uint16_t a)
+{
+    if (likely (0 <= x && 0 <= y && x < width && y < height)) {
+	uint32_t tr, tg, tb, ta;
+
+	/* Premultiply and round */
+	ta = a;
+	tr = r * ta + 0x8000;
+	tg = g * ta + 0x8000;
+	tb = b * ta + 0x8000;
+
+	tr += tr >> 16;
+	tg += tg >> 16;
+	tb += tb >> 16;
+
+	*((uint32_t*) (data + y*stride + 4*x)) = ((ta << 16) & 0xff000000) |
+	    ((tr >> 8) & 0xff0000) | ((tg >> 16) & 0xff00) | (tb >> 24);
+    }
+}
+
+/*
+ * Forward-rasterize a cubic curve using forward differences.
+ *
+ * Input: data is the base pointer of the image
+ *        width, height are the dimensions of the image
+ *        stride is the stride in bytes between adjacent rows
+ *        ushift is log2(n) if n is the number of desired steps
+ *        dxu[i], dyu[i] are the x,y forward differences of the curve
+ *        r0,g0,b0,a0 are the color components of the start point
+ *        r3,g3,b3,a3 are the color components of the end point
+ *
+ * Output: data will be changed to have the requested curve drawn in
+ *         the specified colors
+ *
+ * The input color components are not premultiplied, but the data
+ * stored in the image is assumed to be in CAIRO_FORMAT_ARGB32 (8 bpc,
+ * premultiplied).
+ *
+ * The function draws n+1 pixels, that is from the point at step 0 to
+ * the point at step n, both included. This is the discrete equivalent
+ * to drawing the curve for values of the interpolation parameter in
+ * [0,1] (including both extremes).
+ */
+static inline void
+rasterize_bezier_curve (unsigned char *data, int width, int height, int stride,
+			int ushift, double dxu[4], double dyu[4],
+			uint16_t r0, uint16_t g0, uint16_t b0, uint16_t a0,
+			uint16_t r3, uint16_t g3, uint16_t b3, uint16_t a3)
+{
+    int32_t xu[4], yu[4];
+    int x0, y0, u, usteps = 1 << ushift;
+
+    uint16_t r = r0, g = g0, b = b0, a = a0;
+    int16_t dr = _color_delta_to_shifted_short (r0, r3, ushift);
+    int16_t dg = _color_delta_to_shifted_short (g0, g3, ushift);
+    int16_t db = _color_delta_to_shifted_short (b0, b3, ushift);
+    int16_t da = _color_delta_to_shifted_short (a0, a3, ushift);
+
+    fd_fixed (dxu, xu);
+    fd_fixed (dyu, yu);
+
+    /*
+     * Use (dxu[0],dyu[0]) as origin for the forward differences.
+     *
+     * This makes it possible to handle much larger coordinates (the
+     * ones that can be represented as cairo_fixed_t)
+     */
+    x0 = _cairo_fixed_from_double (dxu[0]);
+    y0 = _cairo_fixed_from_double (dyu[0]);
+    xu[0] = 0;
+    yu[0] = 0;
+
+    for (u = 0; u <= usteps; ++u) {
+	/*
+	 * This rasterizer assumes that pixels are integer aligned
+	 * squares, so a generic (x,y) point belongs to the pixel with
+	 * top-left coordinates (floor(x), floor(y))
+	 */
+
+	int x = _cairo_fixed_integer_floor (x0 + (xu[0] >> 15) + ((xu[0] >> 14) & 1));
+	int y = _cairo_fixed_integer_floor (y0 + (yu[0] >> 15) + ((yu[0] >> 14) & 1));
+
+	draw_pixel (data, width, height, stride, x, y, r, g, b, a);
+
+	fd_fixed_fwd (xu);
+	fd_fixed_fwd (yu);
+	r += dr;
+	g += dg;
+	b += db;
+	a += da;
+    }
+}
+
+/*
+ * Clip, split and rasterize a Bezier curve.
+ *
+ * Input: data is the base pointer of the image
+ *        width, height are the dimensions of the image
+ *        stride is the stride in bytes between adjacent rows
+ *        p[i] is the i-th node of the Bezier curve
+ *        c0[i] is the i-th color component at the start point
+ *        c3[i] is the i-th color component at the end point
+ *
+ * Output: data will be changed to have the requested curve drawn in
+ *         the specified colors
+ *
+ * The input color components are not premultiplied, but the data
+ * stored in the image is assumed to be in CAIRO_FORMAT_ARGB32 (8 bpc,
+ * premultiplied).
+ *
+ * The color components are red, green, blue and alpha, in this order.
+ *
+ * The function guarantees that it will draw the curve with a step
+ * small enough to never have a distance above 1/sqrt(2) between two
+ * consecutive points (which is needed to ensure that no hole can
+ * appear when using this function to rasterize a patch).
+ */
+static void
+draw_bezier_curve (unsigned char *data, int width, int height, int stride,
+		   cairo_point_double_t p[4], double c0[4], double c3[4])
+{
+    double steps_sq;
+    int v;
+
+    /* Check visibility */
+    v = intersect_interval (min (min (p[0].y, p[1].y), min (p[2].y, p[3].y)),
+			    max (max (p[0].y, p[1].y), max (p[2].y, p[3].y)),
+			    0,
+			    height);
+    if (v == OUTSIDE)
+	return;
+
+    v &= intersect_interval (min (min (p[0].x, p[1].x), min (p[2].x, p[3].x)),
+			     max (max (p[0].x, p[1].x), max (p[2].x, p[3].x)),
+			     0,
+			     width);
+    if (v == OUTSIDE)
+	return;
+
+    steps_sq = bezier_steps_sq (p);
+    if (steps_sq >= (v == INSIDE ? STEPS_MAX_U * STEPS_MAX_U : STEPS_CLIP_U * STEPS_CLIP_U)) {
+	/*
+	 * The number of steps is greater than the threshold. This
+	 * means that either the error would become too big if we
+	 * directly rasterized it or that we can probably save some
+	 * time by splitting the curve and clipping part of it
+	 */
+	cairo_point_double_t first[4], second[4];
+	double midc[4];
+	split_bezier (p, first, second);
+	midc[0] = (c0[0] + c3[0]) * 0.5;
+	midc[1] = (c0[1] + c3[1]) * 0.5;
+	midc[2] = (c0[2] + c3[2]) * 0.5;
+	midc[3] = (c0[3] + c3[3]) * 0.5;
+	draw_bezier_curve (data, width, height, stride, first, c0, midc);
+	draw_bezier_curve (data, width, height, stride, second, midc, c3);
+    } else {
+	double xu[4], yu[4];
+	int ushift = sqsteps2shift (steps_sq), k;
+
+	fd_init (p[0].x, p[1].x, p[2].x, p[3].x, xu);
+	fd_init (p[0].y, p[1].y, p[2].y, p[3].y, yu);
+
+	for (k = 0; k < ushift; ++k) {
+	    fd_down (xu);
+	    fd_down (yu);
+	}
+
+	rasterize_bezier_curve (data, width, height, stride, ushift,
+				xu, yu,
+				_cairo_color_double_to_short (c0[0]),
+				_cairo_color_double_to_short (c0[1]),
+				_cairo_color_double_to_short (c0[2]),
+				_cairo_color_double_to_short (c0[3]),
+				_cairo_color_double_to_short (c3[0]),
+				_cairo_color_double_to_short (c3[1]),
+				_cairo_color_double_to_short (c3[2]),
+				_cairo_color_double_to_short (c3[3]));
+
+	/* Draw the end point, to make sure that we didn't leave it
+	 * out because of rounding */
+	draw_pixel (data, width, height, stride,
+		    _cairo_fixed_integer_floor (_cairo_fixed_from_double (p[3].x)),
+		    _cairo_fixed_integer_floor (_cairo_fixed_from_double (p[3].y)),
+		    _cairo_color_double_to_short (c3[0]),
+		    _cairo_color_double_to_short (c3[1]),
+		    _cairo_color_double_to_short (c3[2]),
+		    _cairo_color_double_to_short (c3[3]));
+    }
+}
+
+/*
+ * Forward-rasterize a cubic Bezier patch using forward differences.
+ *
+ * Input: data is the base pointer of the image
+ *        width, height are the dimensions of the image
+ *        stride is the stride in bytes between adjacent rows
+ *        vshift is log2(n) if n is the number of desired steps
+ *        p[i][j], p[i][j] are the the nodes of the Bezier patch
+ *        col[i][j] is the j-th color component of the i-th corner
+ *
+ * Output: data will be changed to have the requested patch drawn in
+ *         the specified colors
+ *
+ * The nodes of the patch are as follows:
+ *
+ * u\v 0    - >    1
+ * 0  p00 p01 p02 p03
+ * |  p10 p11 p12 p13
+ * v  p20 p21 p22 p23
+ * 1  p30 p31 p32 p33
+ *
+ * i.e. u varies along the first component (rows), v varies along the
+ * second one (columns).
+ *
+ * The color components are red, green, blue and alpha, in this order.
+ * c[0..3] are the colors in p00, p30, p03, p33 respectively
+ *
+ * The input color components are not premultiplied, but the data
+ * stored in the image is assumed to be in CAIRO_FORMAT_ARGB32 (8 bpc,
+ * premultiplied).
+ *
+ * If the patch folds over itself, the part with the highest v
+ * parameter is considered above. If both have the same v, the one
+ * with the highest u parameter is above.
+ *
+ * The function draws n+1 curves, that is from the curve at step 0 to
+ * the curve at step n, both included. This is the discrete equivalent
+ * to drawing the patch for values of the interpolation parameter in
+ * [0,1] (including both extremes).
+ */
+static inline void
+rasterize_bezier_patch (unsigned char *data, int width, int height, int stride, int vshift,
+			cairo_point_double_t p[4][4], double col[4][4])
+{
+    double pv[4][2][4], cstart[4], cend[4], dcstart[4], dcend[4];
+    int vsteps, v, i, k;
+
+    vsteps = 1 << vshift;
+
+    /*
+     * pv[i][0] is the function (represented using forward
+     * differences) mapping v to the x coordinate of the i-th node of
+     * the Bezier curve with parameter u.
+     * (Likewise p[i][0] gives the y coordinate).
+     *
+     * This means that (pv[0][0][0],pv[0][1][0]),
+     * (pv[1][0][0],pv[1][1][0]), (pv[2][0][0],pv[2][1][0]) and
+     * (pv[3][0][0],pv[3][1][0]) are the nodes of the Bezier curve for
+     * the "current" v value (see the FD comments for more details).
+     */
+    for (i = 0; i < 4; ++i) {
+	fd_init (p[i][0].x, p[i][1].x, p[i][2].x, p[i][3].x, pv[i][0]);
+	fd_init (p[i][0].y, p[i][1].y, p[i][2].y, p[i][3].y, pv[i][1]);
+	for (k = 0; k < vshift; ++k) {
+	    fd_down (pv[i][0]);
+	    fd_down (pv[i][1]);
+	}
+    }
+
+    for (i = 0; i < 4; ++i) {
+	cstart[i]  = col[0][i];
+	cend[i]    = col[1][i];
+	dcstart[i] = (col[2][i] - col[0][i]) / vsteps;
+	dcend[i]   = (col[3][i] - col[1][i]) / vsteps;
+    }
+
+    for (v = 0; v <= vsteps; ++v) {
+	cairo_point_double_t nodes[4];
+	for (i = 0; i < 4; ++i) {
+	    nodes[i].x = pv[i][0][0];
+	    nodes[i].y = pv[i][1][0];
+	}
+
+	draw_bezier_curve (data, width, height, stride, nodes, cstart, cend);
+
+	for (i = 0; i < 4; ++i) {
+	    fd_fwd (pv[i][0]);
+	    fd_fwd (pv[i][1]);
+	    cstart[i] += dcstart[i];
+	    cend[i] += dcend[i];
+	}
+    }
+}
+
+/*
+ * Clip, split and rasterize a Bezier cubic patch.
+ *
+ * Input: data is the base pointer of the image
+ *        width, height are the dimensions of the image
+ *        stride is the stride in bytes between adjacent rows
+ *        p[i][j], p[i][j] are the nodes of the patch
+ *        col[i][j] is the j-th color component of the i-th corner
+ *
+ * Output: data will be changed to have the requested patch drawn in
+ *         the specified colors
+ *
+ * The nodes of the patch are as follows:
+ *
+ * u\v 0    - >    1
+ * 0  p00 p01 p02 p03
+ * |  p10 p11 p12 p13
+ * v  p20 p21 p22 p23
+ * 1  p30 p31 p32 p33
+ *
+ * i.e. u varies along the first component (rows), v varies along the
+ * second one (columns).
+ *
+ * The color components are red, green, blue and alpha, in this order.
+ * c[0..3] are the colors in p00, p30, p03, p33 respectively
+ *
+ * The input color components are not premultiplied, but the data
+ * stored in the image is assumed to be in CAIRO_FORMAT_ARGB32 (8 bpc,
+ * premultiplied).
+ *
+ * If the patch folds over itself, the part with the highest v
+ * parameter is considered above. If both have the same v, the one
+ * with the highest u parameter is above.
+ *
+ * The function guarantees that it will draw the patch with a step
+ * small enough to never have a distance above 1/sqrt(2) between two
+ * adjacent points (which guarantees that no hole can appear).
+ *
+ * This function can be used to rasterize a tile of PDF type 7
+ * shadings (see http://www.adobe.com/devnet/pdf/pdf_reference.html).
+ */
+static void
+draw_bezier_patch (unsigned char *data, int width, int height, int stride,
+		     cairo_point_double_t p[4][4], double c[4][4])
+{
+    double top, bottom, left, right, steps_sq;
+    int i, j, v;
+
+    top = bottom = p[0][0].y;
+    for (i = 0; i < 4; ++i) {
+	for (j= 0; j < 4; ++j) {
+	    top    = min (top,    p[i][j].y);
+	    bottom = max (bottom, p[i][j].y);
+	}
+    }
+
+    v = intersect_interval (top, bottom, 0, height);
+    if (v == OUTSIDE)
+	return;
+
+    left = right = p[0][0].x;
+    for (i = 0; i < 4; ++i) {
+	for (j= 0; j < 4; ++j) {
+	    left  = min (left,  p[i][j].x);
+	    right = max (right, p[i][j].x);
+	}
+    }
+
+    v &= intersect_interval (left, right, 0, width);
+    if (v == OUTSIDE)
+	return;
+
+    steps_sq = 0;
+    for (i = 0; i < 4; ++i)
+	steps_sq = max (steps_sq, bezier_steps_sq (p[i]));
+
+    if (steps_sq >= (v == INSIDE ? STEPS_MAX_V * STEPS_MAX_V : STEPS_CLIP_V * STEPS_CLIP_V)) {
+	/* The number of steps is greater than the threshold. This
+	 * means that either the error would become too big if we
+	 * directly rasterized it or that we can probably save some
+	 * time by splitting the curve and clipping part of it. The
+	 * patch is only split in the v direction to guarantee that
+	 * rasterizing each part will overwrite parts with low v with
+	 * overlapping parts with higher v. */
+
+	cairo_point_double_t first[4][4], second[4][4];
+	double subc[4][4];
+
+	for (i = 0; i < 4; ++i)
+	    split_bezier (p[i], first[i], second[i]);
+
+	for (i = 0; i < 4; ++i) {
+	    subc[0][i] = c[0][i];
+	    subc[1][i] = c[1][i];
+	    subc[2][i] = 0.5 * (c[0][i] + c[2][i]);
+	    subc[3][i] = 0.5 * (c[1][i] + c[3][i]);
+	}
+
+	draw_bezier_patch (data, width, height, stride, first, subc);
+
+	for (i = 0; i < 4; ++i) {
+	    subc[0][i] = subc[2][i];
+	    subc[1][i] = subc[3][i];
+	    subc[2][i] = c[2][i];
+	    subc[3][i] = c[3][i];
+	}
+	draw_bezier_patch (data, width, height, stride, second, subc);
+    } else {
+	rasterize_bezier_patch (data, width, height, stride, sqsteps2shift (steps_sq), p, c);
+    }
+}
+
+/*
+ * Draw a tensor product shading pattern.
+ *
+ * Input: mesh is the mesh pattern
+ *        data is the base pointer of the image
+ *        width, height are the dimensions of the image
+ *        stride is the stride in bytes between adjacent rows
+ *
+ * Output: data will be changed to have the pattern drawn on it
+ *
+ * data is assumed to be clear and its content is assumed to be in
+ * CAIRO_FORMAT_ARGB32 (8 bpc, premultiplied).
+ *
+ * This function can be used to rasterize a PDF type 7 shading (see
+ * http://www.adobe.com/devnet/pdf/pdf_reference.html).
+ */
+void
+_cairo_mesh_pattern_rasterize (const cairo_mesh_pattern_t *mesh,
+			       void                       *data,
+			       int                         width,
+			       int                         height,
+			       int                         stride,
+			       double                      x_offset,
+			       double                      y_offset)
+{
+    cairo_point_double_t nodes[4][4];
+    double colors[4][4];
+    cairo_matrix_t p2u;
+    unsigned int i, j, k, n;
+    cairo_status_t status;
+    const cairo_mesh_patch_t *patch;
+    const cairo_color_t *c;
+
+    assert (mesh->base.status == CAIRO_STATUS_SUCCESS);
+    assert (mesh->current_patch == NULL);
+
+    p2u = mesh->base.matrix;
+    status = cairo_matrix_invert (&p2u);
+    assert (status == CAIRO_STATUS_SUCCESS);
+
+    n = _cairo_array_num_elements (&mesh->patches);
+    patch = _cairo_array_index_const (&mesh->patches, 0);
+    for (i = 0; i < n; i++) {
+	for (j = 0; j < 4; j++) {
+	    for (k = 0; k < 4; k++) {
+		nodes[j][k] = patch->points[j][k];
+		cairo_matrix_transform_point (&p2u, &nodes[j][k].x, &nodes[j][k].y);
+		nodes[j][k].x += x_offset;
+		nodes[j][k].y += y_offset;
+	    }
+	}
+
+	c = &patch->colors[0];
+	colors[0][0] = c->red;
+	colors[0][1] = c->green;
+	colors[0][2] = c->blue;
+	colors[0][3] = c->alpha;
+
+	c = &patch->colors[3];
+	colors[1][0] = c->red;
+	colors[1][1] = c->green;
+	colors[1][2] = c->blue;
+	colors[1][3] = c->alpha;
+
+	c = &patch->colors[1];
+	colors[2][0] = c->red;
+	colors[2][1] = c->green;
+	colors[2][2] = c->blue;
+	colors[2][3] = c->alpha;
+
+	c = &patch->colors[2];
+	colors[3][0] = c->red;
+	colors[3][1] = c->green;
+	colors[3][2] = c->blue;
+	colors[3][3] = c->alpha;
+
+	draw_bezier_patch (data, width, height, stride, nodes, colors);
+	patch++;
+    }
+}