michael@0: /* -*- Mode: c; c-basic-offset: 4; tab-width: 8; indent-tabs-mode: t; -*- */
michael@0: /*
michael@0:  *
michael@0:  * Copyright © 2000 Keith Packard, member of The XFree86 Project, Inc.
michael@0:  * Copyright © 2000 SuSE, Inc.
michael@0:  *             2005 Lars Knoll & Zack Rusin, Trolltech
michael@0:  * Copyright © 2007 Red Hat, Inc.
michael@0:  *
michael@0:  *
michael@0:  * Permission to use, copy, modify, distribute, and sell this software and its
michael@0:  * documentation for any purpose is hereby granted without fee, provided that
michael@0:  * the above copyright notice appear in all copies and that both that
michael@0:  * copyright notice and this permission notice appear in supporting
michael@0:  * documentation, and that the name of Keith Packard not be used in
michael@0:  * advertising or publicity pertaining to distribution of the software without
michael@0:  * specific, written prior permission.  Keith Packard makes no
michael@0:  * representations about the suitability of this software for any purpose.  It
michael@0:  * is provided "as is" without express or implied warranty.
michael@0:  *
michael@0:  * THE COPYRIGHT HOLDERS DISCLAIM ALL WARRANTIES WITH REGARD TO THIS
michael@0:  * SOFTWARE, INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND
michael@0:  * FITNESS, IN NO EVENT SHALL THE COPYRIGHT HOLDERS BE LIABLE FOR ANY
michael@0:  * SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
michael@0:  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN
michael@0:  * AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING
michael@0:  * OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS
michael@0:  * SOFTWARE.
michael@0:  */
michael@0: 
michael@0: #ifdef HAVE_CONFIG_H
michael@0: #include <config.h>
michael@0: #endif
michael@0: #include <stdlib.h>
michael@0: #include <math.h>
michael@0: #include "pixman-private.h"
michael@0: 
michael@0: #include "pixman-dither.h"
michael@0: 
michael@0: static inline pixman_fixed_32_32_t
michael@0: dot (pixman_fixed_48_16_t x1,
michael@0:      pixman_fixed_48_16_t y1,
michael@0:      pixman_fixed_48_16_t z1,
michael@0:      pixman_fixed_48_16_t x2,
michael@0:      pixman_fixed_48_16_t y2,
michael@0:      pixman_fixed_48_16_t z2)
michael@0: {
michael@0:     /*
michael@0:      * Exact computation, assuming that the input values can
michael@0:      * be represented as pixman_fixed_16_16_t
michael@0:      */
michael@0:     return x1 * x2 + y1 * y2 + z1 * z2;
michael@0: }
michael@0: 
michael@0: static inline double
michael@0: fdot (double x1,
michael@0:       double y1,
michael@0:       double z1,
michael@0:       double x2,
michael@0:       double y2,
michael@0:       double z2)
michael@0: {
michael@0:     /*
michael@0:      * Error can be unbound in some special cases.
michael@0:      * Using clever dot product algorithms (for example compensated
michael@0:      * dot product) would improve this but make the code much less
michael@0:      * obvious
michael@0:      */
michael@0:     return x1 * x2 + y1 * y2 + z1 * z2;
michael@0: }
michael@0: 
michael@0: static uint32_t
michael@0: radial_compute_color (double                    a,
michael@0: 		      double                    b,
michael@0: 		      double                    c,
michael@0: 		      double                    inva,
michael@0: 		      double                    dr,
michael@0: 		      double                    mindr,
michael@0: 		      pixman_gradient_walker_t *walker,
michael@0: 		      pixman_repeat_t           repeat)
michael@0: {
michael@0:     /*
michael@0:      * In this function error propagation can lead to bad results:
michael@0:      *  - discr can have an unbound error (if b*b-a*c is very small),
michael@0:      *    potentially making it the opposite sign of what it should have been
michael@0:      *    (thus clearing a pixel that would have been colored or vice-versa)
michael@0:      *    or propagating the error to sqrtdiscr;
michael@0:      *    if discr has the wrong sign or b is very small, this can lead to bad
michael@0:      *    results
michael@0:      *
michael@0:      *  - the algorithm used to compute the solutions of the quadratic
michael@0:      *    equation is not numerically stable (but saves one division compared
michael@0:      *    to the numerically stable one);
michael@0:      *    this can be a problem if a*c is much smaller than b*b
michael@0:      *
michael@0:      *  - the above problems are worse if a is small (as inva becomes bigger)
michael@0:      */
michael@0:     double discr;
michael@0: 
michael@0:     if (a == 0)
michael@0:     {
michael@0: 	double t;
michael@0: 
michael@0: 	if (b == 0)
michael@0: 	    return 0;
michael@0: 
michael@0: 	t = pixman_fixed_1 / 2 * c / b;
michael@0: 	if (repeat == PIXMAN_REPEAT_NONE)
michael@0: 	{
michael@0: 	    if (0 <= t && t <= pixman_fixed_1)
michael@0: 		return _pixman_gradient_walker_pixel (walker, t);
michael@0: 	}
michael@0: 	else
michael@0: 	{
michael@0: 	    if (t * dr >= mindr)
michael@0: 		return _pixman_gradient_walker_pixel (walker, t);
michael@0: 	}
michael@0: 
michael@0: 	return 0;
michael@0:     }
michael@0: 
michael@0:     discr = fdot (b, a, 0, b, -c, 0);
michael@0:     if (discr >= 0)
michael@0:     {
michael@0: 	double sqrtdiscr, t0, t1;
michael@0: 
michael@0: 	sqrtdiscr = sqrt (discr);
michael@0: 	t0 = (b + sqrtdiscr) * inva;
michael@0: 	t1 = (b - sqrtdiscr) * inva;
michael@0: 
michael@0: 	/*
michael@0: 	 * The root that must be used is the biggest one that belongs
michael@0: 	 * to the valid range ([0,1] for PIXMAN_REPEAT_NONE, any
michael@0: 	 * solution that results in a positive radius otherwise).
michael@0: 	 *
michael@0: 	 * If a > 0, t0 is the biggest solution, so if it is valid, it
michael@0: 	 * is the correct result.
michael@0: 	 *
michael@0: 	 * If a < 0, only one of the solutions can be valid, so the
michael@0: 	 * order in which they are tested is not important.
michael@0: 	 */
michael@0: 	if (repeat == PIXMAN_REPEAT_NONE)
michael@0: 	{
michael@0: 	    if (0 <= t0 && t0 <= pixman_fixed_1)
michael@0: 		return _pixman_gradient_walker_pixel (walker, t0);
michael@0: 	    else if (0 <= t1 && t1 <= pixman_fixed_1)
michael@0: 		return _pixman_gradient_walker_pixel (walker, t1);
michael@0: 	}
michael@0: 	else
michael@0: 	{
michael@0: 	    if (t0 * dr >= mindr)
michael@0: 		return _pixman_gradient_walker_pixel (walker, t0);
michael@0: 	    else if (t1 * dr >= mindr)
michael@0: 		return _pixman_gradient_walker_pixel (walker, t1);
michael@0: 	}
michael@0:     }
michael@0: 
michael@0:     return 0;
michael@0: }
michael@0: 
michael@0: static uint32_t *
michael@0: radial_get_scanline_narrow (pixman_iter_t *iter, const uint32_t *mask)
michael@0: {
michael@0:     /*
michael@0:      * Implementation of radial gradients following the PDF specification.
michael@0:      * See section 8.7.4.5.4 Type 3 (Radial) Shadings of the PDF Reference
michael@0:      * Manual (PDF 32000-1:2008 at the time of this writing).
michael@0:      *
michael@0:      * In the radial gradient problem we are given two circles (c₁,r₁) and
michael@0:      * (c₂,r₂) that define the gradient itself.
michael@0:      *
michael@0:      * Mathematically the gradient can be defined as the family of circles
michael@0:      *
michael@0:      *     ((1-t)·c₁ + t·(c₂), (1-t)·r₁ + t·r₂)
michael@0:      *
michael@0:      * excluding those circles whose radius would be < 0. When a point
michael@0:      * belongs to more than one circle, the one with a bigger t is the only
michael@0:      * one that contributes to its color. When a point does not belong
michael@0:      * to any of the circles, it is transparent black, i.e. RGBA (0, 0, 0, 0).
michael@0:      * Further limitations on the range of values for t are imposed when
michael@0:      * the gradient is not repeated, namely t must belong to [0,1].
michael@0:      *
michael@0:      * The graphical result is the same as drawing the valid (radius > 0)
michael@0:      * circles with increasing t in [-inf, +inf] (or in [0,1] if the gradient
michael@0:      * is not repeated) using SOURCE operator composition.
michael@0:      *
michael@0:      * It looks like a cone pointing towards the viewer if the ending circle
michael@0:      * is smaller than the starting one, a cone pointing inside the page if
michael@0:      * the starting circle is the smaller one and like a cylinder if they
michael@0:      * have the same radius.
michael@0:      *
michael@0:      * What we actually do is, given the point whose color we are interested
michael@0:      * in, compute the t values for that point, solving for t in:
michael@0:      *
michael@0:      *     length((1-t)·c₁ + t·(c₂) - p) = (1-t)·r₁ + t·r₂
michael@0:      *
michael@0:      * Let's rewrite it in a simpler way, by defining some auxiliary
michael@0:      * variables:
michael@0:      *
michael@0:      *     cd = c₂ - c₁
michael@0:      *     pd = p - c₁
michael@0:      *     dr = r₂ - r₁
michael@0:      *     length(t·cd - pd) = r₁ + t·dr
michael@0:      *
michael@0:      * which actually means
michael@0:      *
michael@0:      *     hypot(t·cdx - pdx, t·cdy - pdy) = r₁ + t·dr
michael@0:      *
michael@0:      * or
michael@0:      *
michael@0:      *     ⎷((t·cdx - pdx)² + (t·cdy - pdy)²) = r₁ + t·dr.
michael@0:      *
michael@0:      * If we impose (as stated earlier) that r₁ + t·dr >= 0, it becomes:
michael@0:      *
michael@0:      *     (t·cdx - pdx)² + (t·cdy - pdy)² = (r₁ + t·dr)²
michael@0:      *
michael@0:      * where we can actually expand the squares and solve for t:
michael@0:      *
michael@0:      *     t²cdx² - 2t·cdx·pdx + pdx² + t²cdy² - 2t·cdy·pdy + pdy² =
michael@0:      *       = r₁² + 2·r₁·t·dr + t²·dr²
michael@0:      *
michael@0:      *     (cdx² + cdy² - dr²)t² - 2(cdx·pdx + cdy·pdy + r₁·dr)t +
michael@0:      *         (pdx² + pdy² - r₁²) = 0
michael@0:      *
michael@0:      *     A = cdx² + cdy² - dr²
michael@0:      *     B = pdx·cdx + pdy·cdy + r₁·dr
michael@0:      *     C = pdx² + pdy² - r₁²
michael@0:      *     At² - 2Bt + C = 0
michael@0:      *
michael@0:      * The solutions (unless the equation degenerates because of A = 0) are:
michael@0:      *
michael@0:      *     t = (B ± ⎷(B² - A·C)) / A
michael@0:      *
michael@0:      * The solution we are going to prefer is the bigger one, unless the
michael@0:      * radius associated to it is negative (or it falls outside the valid t
michael@0:      * range).
michael@0:      *
michael@0:      * Additional observations (useful for optimizations):
michael@0:      * A does not depend on p
michael@0:      *
michael@0:      * A < 0 <=> one of the two circles completely contains the other one
michael@0:      *   <=> for every p, the radiuses associated with the two t solutions
michael@0:      *       have opposite sign
michael@0:      */
michael@0:     pixman_image_t *image = iter->image;
michael@0:     int x = iter->x;
michael@0:     int y = iter->y;
michael@0:     int width = iter->width;
michael@0:     uint32_t *buffer = iter->buffer;
michael@0: 
michael@0:     gradient_t *gradient = (gradient_t *)image;
michael@0:     radial_gradient_t *radial = (radial_gradient_t *)image;
michael@0:     uint32_t *end = buffer + width;
michael@0:     pixman_gradient_walker_t walker;
michael@0:     pixman_vector_t v, unit;
michael@0: 
michael@0:     /* reference point is the center of the pixel */
michael@0:     v.vector[0] = pixman_int_to_fixed (x) + pixman_fixed_1 / 2;
michael@0:     v.vector[1] = pixman_int_to_fixed (y) + pixman_fixed_1 / 2;
michael@0:     v.vector[2] = pixman_fixed_1;
michael@0: 
michael@0:     _pixman_gradient_walker_init (&walker, gradient, image->common.repeat);
michael@0: 
michael@0:     if (image->common.transform)
michael@0:     {
michael@0: 	if (!pixman_transform_point_3d (image->common.transform, &v))
michael@0: 	    return iter->buffer;
michael@0: 
michael@0: 	unit.vector[0] = image->common.transform->matrix[0][0];
michael@0: 	unit.vector[1] = image->common.transform->matrix[1][0];
michael@0: 	unit.vector[2] = image->common.transform->matrix[2][0];
michael@0:     }
michael@0:     else
michael@0:     {
michael@0: 	unit.vector[0] = pixman_fixed_1;
michael@0: 	unit.vector[1] = 0;
michael@0: 	unit.vector[2] = 0;
michael@0:     }
michael@0: 
michael@0:     if (unit.vector[2] == 0 && v.vector[2] == pixman_fixed_1)
michael@0:     {
michael@0: 	/*
michael@0: 	 * Given:
michael@0: 	 *
michael@0: 	 * t = (B ± ⎷(B² - A·C)) / A
michael@0: 	 *
michael@0: 	 * where
michael@0: 	 *
michael@0: 	 * A = cdx² + cdy² - dr²
michael@0: 	 * B = pdx·cdx + pdy·cdy + r₁·dr
michael@0: 	 * C = pdx² + pdy² - r₁²
michael@0: 	 * det = B² - A·C
michael@0: 	 *
michael@0: 	 * Since we have an affine transformation, we know that (pdx, pdy)
michael@0: 	 * increase linearly with each pixel,
michael@0: 	 *
michael@0: 	 * pdx = pdx₀ + n·ux,
michael@0: 	 * pdy = pdy₀ + n·uy,
michael@0: 	 *
michael@0: 	 * we can then express B, C and det through multiple differentiation.
michael@0: 	 */
michael@0: 	pixman_fixed_32_32_t b, db, c, dc, ddc;
michael@0: 
michael@0: 	/* warning: this computation may overflow */
michael@0: 	v.vector[0] -= radial->c1.x;
michael@0: 	v.vector[1] -= radial->c1.y;
michael@0: 
michael@0: 	/*
michael@0: 	 * B and C are computed and updated exactly.
michael@0: 	 * If fdot was used instead of dot, in the worst case it would
michael@0: 	 * lose 11 bits of precision in each of the multiplication and
michael@0: 	 * summing up would zero out all the bit that were preserved,
michael@0: 	 * thus making the result 0 instead of the correct one.
michael@0: 	 * This would mean a worst case of unbound relative error or
michael@0: 	 * about 2^10 absolute error
michael@0: 	 */
michael@0: 	b = dot (v.vector[0], v.vector[1], radial->c1.radius,
michael@0: 		 radial->delta.x, radial->delta.y, radial->delta.radius);
michael@0: 	db = dot (unit.vector[0], unit.vector[1], 0,
michael@0: 		  radial->delta.x, radial->delta.y, 0);
michael@0: 
michael@0: 	c = dot (v.vector[0], v.vector[1],
michael@0: 		 -((pixman_fixed_48_16_t) radial->c1.radius),
michael@0: 		 v.vector[0], v.vector[1], radial->c1.radius);
michael@0: 	dc = dot (2 * (pixman_fixed_48_16_t) v.vector[0] + unit.vector[0],
michael@0: 		  2 * (pixman_fixed_48_16_t) v.vector[1] + unit.vector[1],
michael@0: 		  0,
michael@0: 		  unit.vector[0], unit.vector[1], 0);
michael@0: 	ddc = 2 * dot (unit.vector[0], unit.vector[1], 0,
michael@0: 		       unit.vector[0], unit.vector[1], 0);
michael@0: 
michael@0: 	while (buffer < end)
michael@0: 	{
michael@0: 	    if (!mask || *mask++)
michael@0: 	    {
michael@0: 		*buffer = radial_compute_color (radial->a, b, c,
michael@0: 						radial->inva,
michael@0: 						radial->delta.radius,
michael@0: 						radial->mindr,
michael@0: 						&walker,
michael@0: 						image->common.repeat);
michael@0: 	    }
michael@0: 
michael@0: 	    b += db;
michael@0: 	    c += dc;
michael@0: 	    dc += ddc;
michael@0: 	    ++buffer;
michael@0: 	}
michael@0:     }
michael@0:     else
michael@0:     {
michael@0: 	/* projective */
michael@0: 	/* Warning:
michael@0: 	 * error propagation guarantees are much looser than in the affine case
michael@0: 	 */
michael@0: 	while (buffer < end)
michael@0: 	{
michael@0: 	    if (!mask || *mask++)
michael@0: 	    {
michael@0: 		if (v.vector[2] != 0)
michael@0: 		{
michael@0: 		    double pdx, pdy, invv2, b, c;
michael@0: 
michael@0: 		    invv2 = 1. * pixman_fixed_1 / v.vector[2];
michael@0: 
michael@0: 		    pdx = v.vector[0] * invv2 - radial->c1.x;
michael@0: 		    /*    / pixman_fixed_1 */
michael@0: 
michael@0: 		    pdy = v.vector[1] * invv2 - radial->c1.y;
michael@0: 		    /*    / pixman_fixed_1 */
michael@0: 
michael@0: 		    b = fdot (pdx, pdy, radial->c1.radius,
michael@0: 			      radial->delta.x, radial->delta.y,
michael@0: 			      radial->delta.radius);
michael@0: 		    /*  / pixman_fixed_1 / pixman_fixed_1 */
michael@0: 
michael@0: 		    c = fdot (pdx, pdy, -radial->c1.radius,
michael@0: 			      pdx, pdy, radial->c1.radius);
michael@0: 		    /*  / pixman_fixed_1 / pixman_fixed_1 */
michael@0: 
michael@0: 		    *buffer = radial_compute_color (radial->a, b, c,
michael@0: 						    radial->inva,
michael@0: 						    radial->delta.radius,
michael@0: 						    radial->mindr,
michael@0: 						    &walker,
michael@0: 						    image->common.repeat);
michael@0: 		}
michael@0: 		else
michael@0: 		{
michael@0: 		    *buffer = 0;
michael@0: 		}
michael@0: 	    }
michael@0: 
michael@0: 	    ++buffer;
michael@0: 
michael@0: 	    v.vector[0] += unit.vector[0];
michael@0: 	    v.vector[1] += unit.vector[1];
michael@0: 	    v.vector[2] += unit.vector[2];
michael@0: 	}
michael@0:     }
michael@0: 
michael@0:     iter->y++;
michael@0:     return iter->buffer;
michael@0: }
michael@0: 
michael@0: static uint32_t *
michael@0: radial_get_scanline_16 (pixman_iter_t *iter, const uint32_t *mask)
michael@0: {
michael@0:     /*
michael@0:      * Implementation of radial gradients following the PDF specification.
michael@0:      * See section 8.7.4.5.4 Type 3 (Radial) Shadings of the PDF Reference
michael@0:      * Manual (PDF 32000-1:2008 at the time of this writing).
michael@0:      *
michael@0:      * In the radial gradient problem we are given two circles (c₁,r₁) and
michael@0:      * (c₂,r₂) that define the gradient itself.
michael@0:      *
michael@0:      * Mathematically the gradient can be defined as the family of circles
michael@0:      *
michael@0:      *     ((1-t)·c₁ + t·(c₂), (1-t)·r₁ + t·r₂)
michael@0:      *
michael@0:      * excluding those circles whose radius would be < 0. When a point
michael@0:      * belongs to more than one circle, the one with a bigger t is the only
michael@0:      * one that contributes to its color. When a point does not belong
michael@0:      * to any of the circles, it is transparent black, i.e. RGBA (0, 0, 0, 0).
michael@0:      * Further limitations on the range of values for t are imposed when
michael@0:      * the gradient is not repeated, namely t must belong to [0,1].
michael@0:      *
michael@0:      * The graphical result is the same as drawing the valid (radius > 0)
michael@0:      * circles with increasing t in [-inf, +inf] (or in [0,1] if the gradient
michael@0:      * is not repeated) using SOURCE operator composition.
michael@0:      *
michael@0:      * It looks like a cone pointing towards the viewer if the ending circle
michael@0:      * is smaller than the starting one, a cone pointing inside the page if
michael@0:      * the starting circle is the smaller one and like a cylinder if they
michael@0:      * have the same radius.
michael@0:      *
michael@0:      * What we actually do is, given the point whose color we are interested
michael@0:      * in, compute the t values for that point, solving for t in:
michael@0:      *
michael@0:      *     length((1-t)·c₁ + t·(c₂) - p) = (1-t)·r₁ + t·r₂
michael@0:      *
michael@0:      * Let's rewrite it in a simpler way, by defining some auxiliary
michael@0:      * variables:
michael@0:      *
michael@0:      *     cd = c₂ - c₁
michael@0:      *     pd = p - c₁
michael@0:      *     dr = r₂ - r₁
michael@0:      *     length(t·cd - pd) = r₁ + t·dr
michael@0:      *
michael@0:      * which actually means
michael@0:      *
michael@0:      *     hypot(t·cdx - pdx, t·cdy - pdy) = r₁ + t·dr
michael@0:      *
michael@0:      * or
michael@0:      *
michael@0:      *     ⎷((t·cdx - pdx)² + (t·cdy - pdy)²) = r₁ + t·dr.
michael@0:      *
michael@0:      * If we impose (as stated earlier) that r₁ + t·dr >= 0, it becomes:
michael@0:      *
michael@0:      *     (t·cdx - pdx)² + (t·cdy - pdy)² = (r₁ + t·dr)²
michael@0:      *
michael@0:      * where we can actually expand the squares and solve for t:
michael@0:      *
michael@0:      *     t²cdx² - 2t·cdx·pdx + pdx² + t²cdy² - 2t·cdy·pdy + pdy² =
michael@0:      *       = r₁² + 2·r₁·t·dr + t²·dr²
michael@0:      *
michael@0:      *     (cdx² + cdy² - dr²)t² - 2(cdx·pdx + cdy·pdy + r₁·dr)t +
michael@0:      *         (pdx² + pdy² - r₁²) = 0
michael@0:      *
michael@0:      *     A = cdx² + cdy² - dr²
michael@0:      *     B = pdx·cdx + pdy·cdy + r₁·dr
michael@0:      *     C = pdx² + pdy² - r₁²
michael@0:      *     At² - 2Bt + C = 0
michael@0:      *
michael@0:      * The solutions (unless the equation degenerates because of A = 0) are:
michael@0:      *
michael@0:      *     t = (B ± ⎷(B² - A·C)) / A
michael@0:      *
michael@0:      * The solution we are going to prefer is the bigger one, unless the
michael@0:      * radius associated to it is negative (or it falls outside the valid t
michael@0:      * range).
michael@0:      *
michael@0:      * Additional observations (useful for optimizations):
michael@0:      * A does not depend on p
michael@0:      *
michael@0:      * A < 0 <=> one of the two circles completely contains the other one
michael@0:      *   <=> for every p, the radiuses associated with the two t solutions
michael@0:      *       have opposite sign
michael@0:      */
michael@0:     pixman_image_t *image = iter->image;
michael@0:     int x = iter->x;
michael@0:     int y = iter->y;
michael@0:     int width = iter->width;
michael@0:     uint16_t *buffer = iter->buffer;
michael@0:     pixman_bool_t toggle = ((x ^ y) & 1);
michael@0: 
michael@0:     gradient_t *gradient = (gradient_t *)image;
michael@0:     radial_gradient_t *radial = (radial_gradient_t *)image;
michael@0:     uint16_t *end = buffer + width;
michael@0:     pixman_gradient_walker_t walker;
michael@0:     pixman_vector_t v, unit;
michael@0: 
michael@0:     /* reference point is the center of the pixel */
michael@0:     v.vector[0] = pixman_int_to_fixed (x) + pixman_fixed_1 / 2;
michael@0:     v.vector[1] = pixman_int_to_fixed (y) + pixman_fixed_1 / 2;
michael@0:     v.vector[2] = pixman_fixed_1;
michael@0: 
michael@0:     _pixman_gradient_walker_init (&walker, gradient, image->common.repeat);
michael@0: 
michael@0:     if (image->common.transform)
michael@0:     {
michael@0: 	if (!pixman_transform_point_3d (image->common.transform, &v))
michael@0: 	    return iter->buffer;
michael@0: 
michael@0: 	unit.vector[0] = image->common.transform->matrix[0][0];
michael@0: 	unit.vector[1] = image->common.transform->matrix[1][0];
michael@0: 	unit.vector[2] = image->common.transform->matrix[2][0];
michael@0:     }
michael@0:     else
michael@0:     {
michael@0: 	unit.vector[0] = pixman_fixed_1;
michael@0: 	unit.vector[1] = 0;
michael@0: 	unit.vector[2] = 0;
michael@0:     }
michael@0: 
michael@0:     if (unit.vector[2] == 0 && v.vector[2] == pixman_fixed_1)
michael@0:     {
michael@0: 	/*
michael@0: 	 * Given:
michael@0: 	 *
michael@0: 	 * t = (B ± ⎷(B² - A·C)) / A
michael@0: 	 *
michael@0: 	 * where
michael@0: 	 *
michael@0: 	 * A = cdx² + cdy² - dr²
michael@0: 	 * B = pdx·cdx + pdy·cdy + r₁·dr
michael@0: 	 * C = pdx² + pdy² - r₁²
michael@0: 	 * det = B² - A·C
michael@0: 	 *
michael@0: 	 * Since we have an affine transformation, we know that (pdx, pdy)
michael@0: 	 * increase linearly with each pixel,
michael@0: 	 *
michael@0: 	 * pdx = pdx₀ + n·ux,
michael@0: 	 * pdy = pdy₀ + n·uy,
michael@0: 	 *
michael@0: 	 * we can then express B, C and det through multiple differentiation.
michael@0: 	 */
michael@0: 	pixman_fixed_32_32_t b, db, c, dc, ddc;
michael@0: 
michael@0: 	/* warning: this computation may overflow */
michael@0: 	v.vector[0] -= radial->c1.x;
michael@0: 	v.vector[1] -= radial->c1.y;
michael@0: 
michael@0: 	/*
michael@0: 	 * B and C are computed and updated exactly.
michael@0: 	 * If fdot was used instead of dot, in the worst case it would
michael@0: 	 * lose 11 bits of precision in each of the multiplication and
michael@0: 	 * summing up would zero out all the bit that were preserved,
michael@0: 	 * thus making the result 0 instead of the correct one.
michael@0: 	 * This would mean a worst case of unbound relative error or
michael@0: 	 * about 2^10 absolute error
michael@0: 	 */
michael@0: 	b = dot (v.vector[0], v.vector[1], radial->c1.radius,
michael@0: 		 radial->delta.x, radial->delta.y, radial->delta.radius);
michael@0: 	db = dot (unit.vector[0], unit.vector[1], 0,
michael@0: 		  radial->delta.x, radial->delta.y, 0);
michael@0: 
michael@0: 	c = dot (v.vector[0], v.vector[1],
michael@0: 		 -((pixman_fixed_48_16_t) radial->c1.radius),
michael@0: 		 v.vector[0], v.vector[1], radial->c1.radius);
michael@0: 	dc = dot (2 * (pixman_fixed_48_16_t) v.vector[0] + unit.vector[0],
michael@0: 		  2 * (pixman_fixed_48_16_t) v.vector[1] + unit.vector[1],
michael@0: 		  0,
michael@0: 		  unit.vector[0], unit.vector[1], 0);
michael@0: 	ddc = 2 * dot (unit.vector[0], unit.vector[1], 0,
michael@0: 		       unit.vector[0], unit.vector[1], 0);
michael@0: 
michael@0: 	while (buffer < end)
michael@0: 	{
michael@0: 	    if (!mask || *mask++)
michael@0: 	    {
michael@0: 		*buffer = dither_8888_to_0565(
michael@0: 			  radial_compute_color (radial->a, b, c,
michael@0: 						radial->inva,
michael@0: 						radial->delta.radius,
michael@0: 						radial->mindr,
michael@0: 						&walker,
michael@0: 						image->common.repeat),
michael@0: 			  toggle);
michael@0: 	    }
michael@0: 
michael@0: 	    toggle ^= 1;
michael@0: 	    b += db;
michael@0: 	    c += dc;
michael@0: 	    dc += ddc;
michael@0: 	    ++buffer;
michael@0: 	}
michael@0:     }
michael@0:     else
michael@0:     {
michael@0: 	/* projective */
michael@0: 	/* Warning:
michael@0: 	 * error propagation guarantees are much looser than in the affine case
michael@0: 	 */
michael@0: 	while (buffer < end)
michael@0: 	{
michael@0: 	    if (!mask || *mask++)
michael@0: 	    {
michael@0: 		if (v.vector[2] != 0)
michael@0: 		{
michael@0: 		    double pdx, pdy, invv2, b, c;
michael@0: 
michael@0: 		    invv2 = 1. * pixman_fixed_1 / v.vector[2];
michael@0: 
michael@0: 		    pdx = v.vector[0] * invv2 - radial->c1.x;
michael@0: 		    /*    / pixman_fixed_1 */
michael@0: 
michael@0: 		    pdy = v.vector[1] * invv2 - radial->c1.y;
michael@0: 		    /*    / pixman_fixed_1 */
michael@0: 
michael@0: 		    b = fdot (pdx, pdy, radial->c1.radius,
michael@0: 			      radial->delta.x, radial->delta.y,
michael@0: 			      radial->delta.radius);
michael@0: 		    /*  / pixman_fixed_1 / pixman_fixed_1 */
michael@0: 
michael@0: 		    c = fdot (pdx, pdy, -radial->c1.radius,
michael@0: 			      pdx, pdy, radial->c1.radius);
michael@0: 		    /*  / pixman_fixed_1 / pixman_fixed_1 */
michael@0: 
michael@0: 		    *buffer = dither_8888_to_0565 (
michael@0: 			      radial_compute_color (radial->a, b, c,
michael@0: 						    radial->inva,
michael@0: 						    radial->delta.radius,
michael@0: 						    radial->mindr,
michael@0: 						    &walker,
michael@0: 						    image->common.repeat),
michael@0: 			      toggle);
michael@0: 		}
michael@0: 		else
michael@0: 		{
michael@0: 		    *buffer = 0;
michael@0: 		}
michael@0: 	    }
michael@0: 
michael@0: 	    ++buffer;
michael@0: 	    toggle ^= 1;
michael@0: 
michael@0: 	    v.vector[0] += unit.vector[0];
michael@0: 	    v.vector[1] += unit.vector[1];
michael@0: 	    v.vector[2] += unit.vector[2];
michael@0: 	}
michael@0:     }
michael@0: 
michael@0:     iter->y++;
michael@0:     return iter->buffer;
michael@0: }
michael@0: static uint32_t *
michael@0: radial_get_scanline_wide (pixman_iter_t *iter, const uint32_t *mask)
michael@0: {
michael@0:     uint32_t *buffer = radial_get_scanline_narrow (iter, NULL);
michael@0: 
michael@0:     pixman_expand_to_float (
michael@0: 	(argb_t *)buffer, buffer, PIXMAN_a8r8g8b8, iter->width);
michael@0: 
michael@0:     return buffer;
michael@0: }
michael@0: 
michael@0: void
michael@0: _pixman_radial_gradient_iter_init (pixman_image_t *image, pixman_iter_t *iter)
michael@0: {
michael@0:     if (iter->iter_flags & ITER_16)
michael@0: 	iter->get_scanline = radial_get_scanline_16;
michael@0:     else if (iter->iter_flags & ITER_NARROW)
michael@0: 	iter->get_scanline = radial_get_scanline_narrow;
michael@0:     else
michael@0: 	iter->get_scanline = radial_get_scanline_wide;
michael@0: }
michael@0: 
michael@0: 
michael@0: PIXMAN_EXPORT pixman_image_t *
michael@0: pixman_image_create_radial_gradient (const pixman_point_fixed_t *  inner,
michael@0:                                      const pixman_point_fixed_t *  outer,
michael@0:                                      pixman_fixed_t                inner_radius,
michael@0:                                      pixman_fixed_t                outer_radius,
michael@0:                                      const pixman_gradient_stop_t *stops,
michael@0:                                      int                           n_stops)
michael@0: {
michael@0:     pixman_image_t *image;
michael@0:     radial_gradient_t *radial;
michael@0: 
michael@0:     image = _pixman_image_allocate ();
michael@0: 
michael@0:     if (!image)
michael@0: 	return NULL;
michael@0: 
michael@0:     radial = &image->radial;
michael@0: 
michael@0:     if (!_pixman_init_gradient (&radial->common, stops, n_stops))
michael@0:     {
michael@0: 	free (image);
michael@0: 	return NULL;
michael@0:     }
michael@0: 
michael@0:     image->type = RADIAL;
michael@0: 
michael@0:     radial->c1.x = inner->x;
michael@0:     radial->c1.y = inner->y;
michael@0:     radial->c1.radius = inner_radius;
michael@0:     radial->c2.x = outer->x;
michael@0:     radial->c2.y = outer->y;
michael@0:     radial->c2.radius = outer_radius;
michael@0: 
michael@0:     /* warning: this computations may overflow */
michael@0:     radial->delta.x = radial->c2.x - radial->c1.x;
michael@0:     radial->delta.y = radial->c2.y - radial->c1.y;
michael@0:     radial->delta.radius = radial->c2.radius - radial->c1.radius;
michael@0: 
michael@0:     /* computed exactly, then cast to double -> every bit of the double
michael@0:        representation is correct (53 bits) */
michael@0:     radial->a = dot (radial->delta.x, radial->delta.y, -radial->delta.radius,
michael@0: 		     radial->delta.x, radial->delta.y, radial->delta.radius);
michael@0:     if (radial->a != 0)
michael@0: 	radial->inva = 1. * pixman_fixed_1 / radial->a;
michael@0: 
michael@0:     radial->mindr = -1. * pixman_fixed_1 * radial->c1.radius;
michael@0: 
michael@0:     return image;
michael@0: }