/* Copyright (C) 1998, 2000 Aladdin Enterprises.  All rights reserved.
  
  This file is part of GNU Ghostscript.
  
  GNU Ghostscript is distributed in the hope that it will be useful, but
  WITHOUT ANY WARRANTY.  No author or distributor accepts responsibility
  to anyone for the consequences of using it or for whether it serves any
  particular purpose or works at all, unless he says so in writing.  Refer
  to the GNU General Public License for full details.
  
  Everyone is granted permission to copy, modify and redistribute GNU
  Ghostscript, but only under the conditions described in the GNU General
  Public License.  A copy of this license is supposed to have been given
  to you along with GNU Ghostscript so you can know your rights and
  responsibilities.  It should be in a file named COPYING.  Among other
  things, the copyright notice and this notice must be preserved on all
  copies.
*/

/*$RCSfile: gxshade1.c,v $ $Revision: 1.5.2.5 $ */
/* Rendering for non-mesh shadings */
#include "math_.h"
#include "memory_.h"
#include "gx.h"
#include "gserrors.h"
#include "gsmatrix.h"		/* for gscoord.h */
#include "gscoord.h"
#include "gspath.h"
#include "gxcspace.h"
#include "gxdcolor.h"
#include "gxfarith.h"
#include "gxfixed.h"
#include "gxistate.h"
#include "gxpath.h"
#include "gxshade.h"
#include "gxdevcli.h"

/* ================ Utilities ================ */

/* Check whether 2 colors fall within the smoothness criterion. */
private bool
shade_colors2_converge(const gs_client_color cc[2],
		       const shading_fill_state_t * pfs)
{
    int ci;

    for (ci = pfs->num_components - 1; ci >= 0; --ci)
	if (fabs(cc[1].paint.values[ci] - cc[0].paint.values[ci]) >
	    pfs->cc_max_error[ci]
	    )
	    return false;
    return true;
}

/* Fill a user space rectangle that is also a device space rectangle. */
private int
shade_fill_device_rectangle(const shading_fill_state_t * pfs,
			    const gs_fixed_point * p0,
			    const gs_fixed_point * p1,
			    gx_device_color * pdevc)
{
    gs_imager_state *pis = pfs->pis;
    fixed xmin, ymin, xmax, ymax;
    int x, y;

    if (p0->x < p1->x)
	xmin = p0->x, xmax = p1->x;
    else
	xmin = p1->x, xmax = p0->x;
    if (p0->y < p1->y)
	ymin = p0->y, ymax = p1->y;
    else
	ymin = p1->y, ymax = p0->y;

    /* See gx_default_fill_path for an explanation of the tweak below. */
    xmin -= pis->fill_adjust.x;
    if (pis->fill_adjust.x == fixed_half)
	xmin += fixed_epsilon;
    xmax += pis->fill_adjust.x;
    ymin -= pis->fill_adjust.y;
    if (pis->fill_adjust.y == fixed_half)
	ymin += fixed_epsilon;
    ymax += pis->fill_adjust.y;
    x = fixed2int_var_pixround(xmin);
    y = fixed2int_var_pixround(ymin);
    return
	gx_fill_rectangle_device_rop(x, y,
				     fixed2int_var_pixround(xmax) - x,
				     fixed2int_var_pixround(ymax) - y,
				     pdevc, pfs->dev, pis->log_op);
}

/* ================ Specific shadings ================ */

/* ---------------- Function-based shading ---------------- */

#define Fb_max_depth 32		/* 16 bits each in X and Y */
typedef struct Fb_frame_s {	/* recursion frame */
    gs_rect region;
    gs_client_color cc[4];	/* colors at 4 corners */
    gs_paint_color c_min, c_max; /* estimated color range for the region */
    bool painted;
    bool devide_X;
    int state;
} Fb_frame_t;

typedef struct Fb_fill_state_s {
    shading_fill_state_common;
    const gs_shading_Fb_t *psh;
    gs_matrix_fixed ptm;	/* parameter space -> device space */
    bool orthogonal;		/* true iff ptm is xxyy or xyyx */
    int depth;			/* 1 <= depth < Fb_max_depth */
    bool painted;               /* the last region is painted */
    gs_paint_color c_min, c_max; /* estimated color range for the last region */
    Fb_frame_t frames[Fb_max_depth];
} Fb_fill_state_t;
/****** NEED GC DESCRIPTOR ******/

#define CheckRET(a) { int code = a; if (code < 0) return code; }
#define Exch(t,a,b) { t x; x = a; a = b; b = x; }

private bool
Fb_build_color_range(const Fb_fill_state_t * pfs, const Fb_frame_t * fp, 
		     gs_paint_color * c_min, gs_paint_color * c_max)
{
    int ci;
    const gs_client_color *cc = fp->cc;
    bool big = false;
    for (ci = 0; ci < pfs->num_components; ++ci) {
	float c0 = cc[0].paint.values[ci], c1 = cc[1].paint.values[ci],
	      c2 = cc[2].paint.values[ci], c3 = cc[3].paint.values[ci];
	float min01, max01, min23, max23;
	if (c0 < c1)
	    min01 = c0, max01 = c1;
	else
	    min01 = c1, max01 = c0;
	if (c2 < c3)
	    min23 = c2, max23 = c3;
	else
	    min23 = c3, max23 = c2;
	c_max->values[ci] = max(max01, max23);
	c_min->values[ci] = min(min01, min23);
	big |= ((c_max->values[ci] - c_min->values[ci]) > pfs->cc_max_error[ci]);
    }
    return !big;
}

private bool 
Fb_unite_color_range(const Fb_fill_state_t * pfs, 
		     const gs_paint_color * c_min0, const gs_paint_color * c_max0, 
		     gs_paint_color * c_min1, gs_paint_color * c_max1)
{   int ci;
    bool big = false;
    for (ci = 0; ci < pfs->num_components; ++ci) {
	c_min1->values[ci] = min(c_min1->values[ci], c_min0->values[ci]);
	c_max1->values[ci] = max(c_max1->values[ci], c_max0->values[ci]);
	big |= ((c_max1->values[ci] - c_min1->values[ci]) > pfs->cc_max_error[ci]);
    }
    return !big;
}

private int
Fb_build_half_region(Fb_fill_state_t * pfs, int h, bool use_old)
{   Fb_frame_t * fp0 = &pfs->frames[pfs->depth];
    Fb_frame_t * fp1 = &pfs->frames[pfs->depth + 1];
    gs_function_t *pfn = pfs->psh->params.Function;
    const double x0 = fp0->region.p.x, y0 = fp0->region.p.y,
		 x1 = fp0->region.q.x, y1 = fp0->region.q.y;
    float v[2];
    int code;
    if (fp0->devide_X) {
	double xm = (x0 + x1) * 0.5;
	int h10 = (!h ? 1 : 0), h32 = (!h ? 3 : 2);
	int h01 = (!h ? 0 : 1), h23 = (!h ? 2 : 3);
	if_debug1('|', "[|]dividing at x=%g\n", xm);
	if (use_old) {
	    fp1->cc[h10].paint = fp1->cc[h01].paint;
	    fp1->cc[h32].paint = fp1->cc[h23].paint;
	} else {
	    v[0] = xm, v[1] = y0;
	    CheckRET(gs_function_evaluate(pfn, v, fp1->cc[h10].paint.values));
	    v[1] = y1;
	    CheckRET(gs_function_evaluate(pfn, v, fp1->cc[h32].paint.values));
	}
	fp1->cc[h01].paint = fp0->cc[h01].paint;
	fp1->cc[h23].paint = fp0->cc[h23].paint;
	if (!h) {
	    fp1->region.p.x = x0, fp1->region.p.y = y0;
	    fp1->region.q.x = xm, fp1->region.q.y = y1;
	} else {
	    fp1->region.p.x = xm, fp1->region.p.y = y0;
	    fp1->region.q.x = x1, fp1->region.q.y = y1;
	}
    } else {
	double ym = (y0 + y1) * 0.5;
	int h20 = (!h ? 2 : 0), h31 = (!h ? 3 : 1);
	int h02 = (!h ? 0 : 2), h13 = (!h ? 1 : 3);
	if_debug1('|', "[|]dividing at y=%g\n", ym);
	if (use_old) {
	    fp1->cc[h20].paint = fp1->cc[h02].paint;
	    fp1->cc[h31].paint = fp1->cc[h13].paint;
	} else {
	    v[0] = x0, v[1] = ym;
	    CheckRET(gs_function_evaluate(pfn, v, fp1->cc[h20].paint.values));
	    v[0] = x1;
	    CheckRET(gs_function_evaluate(pfn, v, fp1->cc[h31].paint.values));
	}
	fp1->cc[h02].paint = fp0->cc[h02].paint;
	fp1->cc[h13].paint = fp0->cc[h13].paint;
	if (!h) {
	    fp1->region.p.x = x0, fp1->region.p.y = y0;
	    fp1->region.q.x = x1, fp1->region.q.y = ym;
	} else {
	    fp1->region.p.x = x0, fp1->region.p.y = ym;
	    fp1->region.q.x = x1, fp1->region.q.y = y1;
	}
    }
    return 0;
}

private int
Fb_fill_region_with_constant_color(const Fb_fill_state_t * pfs, const Fb_frame_t * fp, gs_paint_color * c_min, gs_paint_color * c_max)
{
    const gs_shading_Fb_t * const psh = pfs->psh;
    gx_device_color dev_color;
    const gs_color_space *pcs = psh->params.ColorSpace;
    gs_client_color cc;
    gs_fixed_point pts[4];
    int code, ci;

    if_debug0('|', "[|]... filling region\n");
    cc = fp->cc[0];
    for (ci = 0; ci < pfs->num_components; ++ci)
	cc.paint.values[ci] = (c_min->values[ci] + c_max->values[ci]) / 2;
    (*pcs->type->restrict_color)(&cc, pcs);
    (*pcs->type->remap_color)(&cc, pcs, &dev_color, pfs->pis,
			      pfs->dev, gs_color_select_texture);
    gs_point_transform2fixed(&pfs->ptm, fp->region.p.x, fp->region.p.y, &pts[0]);
    gs_point_transform2fixed(&pfs->ptm, fp->region.q.x, fp->region.q.y, &pts[2]);
    if (pfs->orthogonal) {
	code = shade_fill_device_rectangle((const shading_fill_state_t *)pfs,
					   &pts[0], &pts[2], &dev_color);
    } else {
	gx_path *ppath = gx_path_alloc(pfs->pis->memory, "Fb_fill");

	gs_point_transform2fixed(&pfs->ptm, fp->region.q.x, fp->region.p.y, &pts[1]);
	gs_point_transform2fixed(&pfs->ptm, fp->region.p.x, fp->region.q.y, &pts[3]);
	gx_path_add_point(ppath, pts[0].x, pts[0].y);
	gx_path_add_lines(ppath, pts + 1, 3);
	code = shade_fill_path((const shading_fill_state_t *)pfs,
			       ppath, &dev_color);
	gx_path_free(ppath, "Fb_fill");
    }
    return code;
}

private int
Fb_fill_region_lazy(Fb_fill_state_t * pfs)
{   fixed minsize = fixed_1 * 7 / 10; /* pixels */
    float min_extreme_dist = 4; /* points */
    fixed min_edist_x = float2fixed(min_extreme_dist * pfs->dev->HWResolution[0] / 72); 
    fixed min_edist_y = float2fixed(min_extreme_dist * pfs->dev->HWResolution[1] / 72); 
    min_edist_x = max(min_edist_x, minsize);
    min_edist_y = max(min_edist_y, minsize);
    while (pfs->depth >= 0) {
	Fb_frame_t * fp = &pfs->frames[pfs->depth];
	fixed size_x, size_y;
	bool single_extreme, single_pixel, small_color_diff = false;
	switch (fp->state) {
	    case 0:
		fp->painted = false;
		{   /* Region size in device space : */
		    gs_point p, q;
		    CheckRET(gs_distance_transform(fp->region.q.x - fp->region.p.x, 0, (gs_matrix *)&pfs->ptm, &p));
		    CheckRET(gs_distance_transform(0, fp->region.q.y - fp->region.p.y, (gs_matrix *)&pfs->ptm, &q));
		    size_x = float2fixed(hypot(p.x, p.y));
		    size_y = float2fixed(hypot(q.x, q.y));
		    single_extreme = (float2fixed(any_abs(p.x) + any_abs(q.x)) < min_edist_x && 
		                      float2fixed(any_abs(p.y) + any_abs(q.y)) < min_edist_y);
		    single_pixel = (size_x < minsize && size_y < minsize);
		    /* Note: single_pixel implies single_extreme. */
		}
		if (single_extreme || pfs->depth >= Fb_max_depth - 1)
		    small_color_diff = Fb_build_color_range(pfs, fp, &pfs->c_min, &pfs->c_max);
		if (single_extreme && small_color_diff || 
		    single_pixel ||
		    pfs->depth >= Fb_max_depth - 1) {
		    Fb_build_color_range(pfs, fp, &pfs->c_min, &pfs->c_max);
		    -- pfs->depth;
		    pfs->painted = false;
		    break;
		}
		fp->state = 1;
		fp->devide_X = (size_x > size_y);
		CheckRET(Fb_build_half_region(pfs, 0, false));
		++ pfs->depth; /* Do recur, left branch. */
		pfs->frames[pfs->depth].state = 0;
		break;
	    case 1:
		fp->painted = pfs->painted; /* Save return value. */
		fp->c_min = pfs->c_min;	    /* Save return value. */
		fp->c_max = pfs->c_max;	    /* Save return value. */
		CheckRET(Fb_build_half_region(pfs, 1, true));
		fp->state = 2;
		++ pfs->depth; /* Do recur, right branch. */
		pfs->frames[pfs->depth].state = 0;
		break;
	    case 2:
		if (fp->painted && pfs->painted) { 
		    /* Both branches are painted - do nothing */
		} else if (fp->painted && !pfs->painted) { 
		    CheckRET(Fb_fill_region_with_constant_color(pfs, fp + 1, &pfs->c_min, &pfs->c_max));
		    pfs->painted = true;
		} else if (!fp->painted && pfs->painted) { 
		    CheckRET(Fb_build_half_region(pfs, 0, true));
		    CheckRET(Fb_fill_region_with_constant_color(pfs, fp + 1, &fp->c_min, &fp->c_max));
		} else {
		    gs_paint_color c_min = pfs->c_min, c_max = pfs->c_max;
		    if (Fb_unite_color_range(pfs, &fp->c_min, &fp->c_max, &pfs->c_min, &pfs->c_max)) {
			/* Both halfs are not painted, and color range is small - do nothing */
			/* return: pfs->painted = false; pfs->c_min, pfs->c_max are united. */
		    } else {
			/* Color range too big, paint each part with its color : */
			CheckRET(Fb_fill_region_with_constant_color(pfs, fp + 1, &c_min, &c_max));
			CheckRET(Fb_build_half_region(pfs, 0, true));
			CheckRET(Fb_fill_region_with_constant_color(pfs, fp + 1, &fp->c_min, &fp->c_max));
			pfs->painted = true;
		    }
		}
		-- pfs->depth;
		break;
	}
    }
    return 0;
}

private int
Fb_fill_region(Fb_fill_state_t * pfs)
{   
    pfs->depth = 0;
    pfs->frames[0].state = 0;
    CheckRET(Fb_fill_region_lazy(pfs));
    if(!pfs->painted)
	CheckRET(Fb_fill_region_with_constant_color(pfs, &pfs->frames[0], &pfs->c_min, &pfs->c_max));
    return 0;
}

int
gs_shading_Fb_fill_rectangle(const gs_shading_t * psh0, const gs_rect * rect,
			     gx_device * dev, gs_imager_state * pis)
{
    const gs_shading_Fb_t * const psh = (const gs_shading_Fb_t *)psh0;
    gs_matrix save_ctm;
    int xi, yi;
    float x[2], y[2];
    Fb_fill_state_t state;

    shade_init_fill_state((shading_fill_state_t *) & state, psh0, dev, pis);
    state.psh = psh;
    /****** HACK FOR FIXED-POINT MATRIX MULTIPLY ******/
    gs_currentmatrix((gs_state *) pis, &save_ctm);
    gs_concat((gs_state *) pis, &psh->params.Matrix);
    state.ptm = pis->ctm;
    gs_setmatrix((gs_state *) pis, &save_ctm);
    state.orthogonal = is_xxyy(&state.ptm) || is_xyyx(&state.ptm);
    /* Compute the parameter X and Y ranges. */
    {
	gs_rect pbox;

	gs_bbox_transform_inverse(rect, &psh->params.Matrix, &pbox);
	x[0] = max(pbox.p.x, psh->params.Domain[0]);
	x[1] = min(pbox.q.x, psh->params.Domain[1]);
	y[0] = max(pbox.p.y, psh->params.Domain[2]);
	y[1] = min(pbox.q.y, psh->params.Domain[3]);
    }
    for (xi = 0; xi < 2; ++xi)
	for (yi = 0; yi < 2; ++yi) {
	    float v[2];

	    v[0] = x[xi], v[1] = y[yi];
	    gs_function_evaluate(psh->params.Function, v,
				 state.frames[0].cc[yi * 2 + xi].paint.values);
	}
    state.frames[0].region.p.x = x[0];
    state.frames[0].region.p.y = y[0];
    state.frames[0].region.q.x = x[1];
    state.frames[0].region.q.y = y[1];
    return Fb_fill_region(&state);
}

/* ---------------- Axial shading ---------------- */

/*
 * Note that the max recursion depth and the frame structure are shared
 * with radial shading.
 */
#define AR_max_depth 16
typedef struct AR_frame_s {	/* recursion frame */
    double t0, t1;
    gs_client_color cc[2];	/* color at t0, t1 */
} AR_frame_t;

#define A_max_depth AR_max_depth
typedef AR_frame_t A_frame_t;

typedef struct A_fill_state_s {
    shading_fill_state_common;
    const gs_shading_A_t *psh;
    bool orthogonal;		/* true iff ctm is xxyy or xyyx */
    gs_rect rect;		/* bounding rectangle in user space */
    gs_point delta;
    double length, dd;
    int depth;
    A_frame_t frames[A_max_depth];
} A_fill_state_t;
/****** NEED GC DESCRIPTOR ******/

/* Note t0 and t1 vary over [0..1], not the Domain. */

private int
A_fill_stripe(const A_fill_state_t * pfs, gs_client_color *pcc,
	      floatp t0, floatp t1)
{
    const gs_shading_A_t * const psh = pfs->psh;
    gx_device_color dev_color;
    const gs_color_space *pcs = psh->params.ColorSpace;
    gs_imager_state *pis = pfs->pis;
    double
	x0 = psh->params.Coords[0] + pfs->delta.x * t0,
	y0 = psh->params.Coords[1] + pfs->delta.y * t0;
    double
	x1 = psh->params.Coords[0] + pfs->delta.x * t1,
	y1 = psh->params.Coords[1] + pfs->delta.y * t1;
    gs_fixed_point pts[4];
    int code;

    (*pcs->type->restrict_color)(pcc, pcs);
    (*pcs->type->remap_color)(pcc, pcs, &dev_color, pis,
			      pfs->dev, gs_color_select_texture);
    if (x0 == x1 && pfs->orthogonal) {
	/*
	 * Stripe is horizontal in user space and horizontal or vertical
	 * in device space.
	 */
	x0 = pfs->rect.p.x;
	x1 = pfs->rect.q.x;
    } else if (y0 == y1 && pfs->orthogonal) {
	/*
	 * Stripe is vertical in user space and horizontal or vertical
	 * in device space.
	 */
	y0 = pfs->rect.p.y;
	y1 = pfs->rect.q.y;
    } else {
	/*
	 * Stripe is neither horizontal nor vertical in user space.
	 * Extend it to the edges of the (user-space) rectangle.
	 */
	gx_path *ppath = gx_path_alloc(pis->memory, "A_fill");
	if (fabs(pfs->delta.x) < fabs(pfs->delta.y)) {
	    /*
	     * Calculate intersections with vertical sides of rect.
	     */
	    double slope = pfs->delta.x / pfs->delta.y;
	    double yi = y0 - slope * (pfs->rect.p.x - x0);

	    gs_point_transform2fixed(&pis->ctm, pfs->rect.p.x, yi, &pts[0]);
	    yi = y1 - slope * (pfs->rect.p.x - x1);
	    gs_point_transform2fixed(&pis->ctm, pfs->rect.p.x, yi, &pts[1]);
	    yi = y1 - slope * (pfs->rect.q.x - x1);
	    gs_point_transform2fixed(&pis->ctm, pfs->rect.q.x, yi, &pts[2]);
	    yi = y0 - slope * (pfs->rect.q.x - x0);
	    gs_point_transform2fixed(&pis->ctm, pfs->rect.q.x, yi, &pts[3]);
	}
	else {
	    /*
	     * Calculate intersections with horizontal sides of rect.
	     */
	    double slope = pfs->delta.y / pfs->delta.x;
	    double xi = x0 - slope * (pfs->rect.p.y - y0);

	    gs_point_transform2fixed(&pis->ctm, xi, pfs->rect.p.y, &pts[0]);
	    xi = x1 - slope * (pfs->rect.p.y - y1);
	    gs_point_transform2fixed(&pis->ctm, xi, pfs->rect.p.y, &pts[1]);
	    xi = x1 - slope * (pfs->rect.q.y - y1);
	    gs_point_transform2fixed(&pis->ctm, xi, pfs->rect.q.y, &pts[2]);
	    xi = x0 - slope * (pfs->rect.q.y - y0);
	    gs_point_transform2fixed(&pis->ctm, xi, pfs->rect.q.y, &pts[3]);
	}
	gx_path_add_point(ppath, pts[0].x, pts[0].y);
	gx_path_add_lines(ppath, pts + 1, 3);
	code = shade_fill_path((const shading_fill_state_t *)pfs,
			       ppath, &dev_color);
	gx_path_free(ppath, "A_fill");
	return code;
    }
    /* Stripe is horizontal or vertical in both user and device space. */
    gs_point_transform2fixed(&pis->ctm, x0, y0, &pts[0]);
    gs_point_transform2fixed(&pis->ctm, x1, y1, &pts[1]);
    return
	shade_fill_device_rectangle((const shading_fill_state_t *)pfs,
				    &pts[0], &pts[1], &dev_color);
}

private int
A_fill_region(A_fill_state_t * pfs)
{
    const gs_shading_A_t * const psh = pfs->psh;
    gs_function_t * const pfn = psh->params.Function;
    A_frame_t *fp = &pfs->frames[pfs->depth - 1];

    for (;;) {
	double t0 = fp->t0, t1 = fp->t1;
	float ft0, ft1;

	if ((!(pfn->head.is_monotonic > 0 ||
	       (ft0 = (float)t0, ft1 = (float)t1,
		gs_function_is_monotonic(pfn, &ft0, &ft1, EFFORT_MODERATE) > 0)) ||
	     !shade_colors2_converge(fp->cc,
				     (const shading_fill_state_t *)pfs)) &&
	     /*
	      * The function isn't monotonic, or the colors don't converge.
	      * Is the stripe less than 1 pixel wide?
	      */
	    pfs->length * (t1 - t0) > 1 &&
	    fp < &pfs->frames[countof(pfs->frames) - 1]
	    ) {
	    /* Subdivide the interval and recur.  */
	    double tm = (t0 + t1) * 0.5;
	    float dm = tm * pfs->dd + psh->params.Domain[0];

	    gs_function_evaluate(pfn, &dm, fp[1].cc[1].paint.values);
	    fp[1].cc[0].paint = fp->cc[0].paint;
	    fp[1].t0 = t0;
	    fp[1].t1 = fp->t0 = tm;
	    fp->cc[0].paint = fp[1].cc[1].paint;
	    ++fp;
	} else {
	    /* Fill the region with the color. */
	    int code = A_fill_stripe(pfs, &fp->cc[0], t0, t1);

	    if (code < 0 || fp == &pfs->frames[0])
		return code;
	    --fp;
	}
    }
}

int
gs_shading_A_fill_rectangle(const gs_shading_t * psh0, const gs_rect * rect,
			    gx_device * dev, gs_imager_state * pis)
{
    const gs_shading_A_t *const psh = (const gs_shading_A_t *)psh0;
    gs_matrix cmat;
    gs_rect t_rect;
    A_fill_state_t state;
    gs_client_color rcc[2];
    float d0 = psh->params.Domain[0], d1 = psh->params.Domain[1];
    float dd = d1 - d0;
    float t0, t1;
    float t[2];
    gs_point dist;
    int i;
    int code;

    shade_init_fill_state((shading_fill_state_t *)&state, psh0, dev, pis);
    state.psh = psh;
    state.orthogonal = is_xxyy(&pis->ctm) || is_xyyx(&pis->ctm);
    state.rect = *rect;
    /*
     * Compute the parameter range.  We construct a matrix in which
     * (0,0) corresponds to t = 0 and (0,1) corresponds to t = 1,
     * and use it to inverse-map the rectangle to be filled.
     */
    cmat.tx = psh->params.Coords[0];
    cmat.ty = psh->params.Coords[1];
    state.delta.x = psh->params.Coords[2] - psh->params.Coords[0];
    state.delta.y = psh->params.Coords[3] - psh->params.Coords[1];
    cmat.yx = state.delta.x;
    cmat.yy = state.delta.y;
    cmat.xx = cmat.yy;
    cmat.xy = -cmat.yx;
    gs_bbox_transform_inverse(rect, &cmat, &t_rect);
    state.frames[0].t0 = t0 = max(t_rect.p.y, 0);
    t[0] = t0 * dd + d0;
    state.frames[0].t1 = t1 = min(t_rect.q.y, 1);
    t[1] = t1 * dd + d0;
    for (i = 0; i < 2; ++i) {
	gs_function_evaluate(psh->params.Function, &t[i],
			     rcc[i].paint.values);
    }
    memcpy(state.frames[0].cc, rcc, sizeof(rcc[0]) * 2);
    gs_distance_transform(state.delta.x, state.delta.y, &ctm_only(pis),
			  &dist);
    state.length = hypot(dist.x, dist.y);	/* device space line length */
    state.dd = dd;
    state.depth = 1;
    code = A_fill_region(&state);
    if (psh->params.Extend[0] && t0 > t_rect.p.y) {
	if (code < 0)
	    return code;
	code = A_fill_stripe(&state, &rcc[0], t_rect.p.y, t0);
    }
    if (psh->params.Extend[1] && t1 < t_rect.q.y) {
	if (code < 0)
	    return code;
	code = A_fill_stripe(&state, &rcc[1], t1, t_rect.q.y);
    }
    return code;
}

/* ---------------- Radial shading ---------------- */

#define R_max_depth AR_max_depth
typedef AR_frame_t R_frame_t;

typedef struct R_fill_state_s {
    shading_fill_state_common;
    const gs_shading_R_t *psh;
    gs_rect rect;
    gs_point delta;
    double dr, width, dd;
    int depth;
    R_frame_t frames[R_max_depth];
} R_fill_state_t;
/****** NEED GC DESCRIPTOR ******/

/* Note t0 and t1 vary over [0..1], not the Domain. */

private int
R_fill_annulus(const R_fill_state_t * pfs, gs_client_color *pcc,
	       floatp t0, floatp t1, floatp r0, floatp r1)
{
    const gs_shading_R_t * const psh = pfs->psh;
    gx_device_color dev_color;
    const gs_color_space *pcs = psh->params.ColorSpace;
    gs_imager_state *pis = pfs->pis;
    double
	x0 = psh->params.Coords[0] + pfs->delta.x * t0,
	y0 = psh->params.Coords[1] + pfs->delta.y * t0;
    double
	x1 = psh->params.Coords[0] + pfs->delta.x * t1,
	y1 = psh->params.Coords[1] + pfs->delta.y * t1;
    gx_path *ppath = gx_path_alloc(pis->memory, "R_fill");
    int code;

    (*pcs->type->restrict_color)(pcc, pcs);
    (*pcs->type->remap_color)(pcc, pcs, &dev_color, pis,
			      pfs->dev, gs_color_select_texture);
    if ((code = gs_imager_arc_add(ppath, pis, false, x0, y0, r0,
				  0.0, 360.0, false)) >= 0 &&
	(code = gs_imager_arc_add(ppath, pis, true, x1, y1, r1,
				  360.0, 0.0, false)) >= 0
	) {
	code = shade_fill_path((const shading_fill_state_t *)pfs,
			       ppath, &dev_color);
    }
    gx_path_free(ppath, "R_fill");
    return code;
}

private int
R_fill_region(R_fill_state_t * pfs)
{
    const gs_shading_R_t * const psh = pfs->psh;
    gs_function_t *pfn = psh->params.Function;
    R_frame_t *fp = &pfs->frames[pfs->depth - 1];

    for (;;) {
	double t0 = fp->t0, t1 = fp->t1;
	float ft0, ft1;

	if ((!(pfn->head.is_monotonic > 0 ||
	       (ft0 = (float)t0, ft1 = (float)t1,
		gs_function_is_monotonic(pfn, &ft0, &ft1, EFFORT_MODERATE) > 0)) ||
	     !shade_colors2_converge(fp->cc,
				     (const shading_fill_state_t *)pfs)) &&
	    /*
	     * The function isn't monotonic, or the colors don't converge.
	     * Is the annulus less than 1 pixel wide?
	     */
	    pfs->width * (t1 - t0) > 1 &&
	    fp < &pfs->frames[countof(pfs->frames) - 1]
	   ) {
	    /* Subdivide the interval and recur.  */
	    float tm = (t0 + t1) * 0.5;
	    float dm = tm * pfs->dd + psh->params.Domain[0];

	    gs_function_evaluate(pfn, &dm, fp[1].cc[1].paint.values);
	    fp[1].cc[0].paint = fp->cc[0].paint;
	    fp[1].t0 = t0;
	    fp[1].t1 = fp->t0 = tm;
	    fp->cc[0].paint = fp[1].cc[1].paint;
	    ++fp;
	} else {
	    /* Fill the region with the color. */
	    int code = R_fill_annulus(pfs, &fp->cc[0], t0, t1,
				      psh->params.Coords[2] + pfs->dr * t0,
				      psh->params.Coords[2] + pfs->dr * t1);

	    if (code < 0 || fp == &pfs->frames[0])
		return code;
	    --fp;
	}
    }
}

private double
R_compute_radius(floatp x, floatp y, const gs_rect *rect)
{
    double x0 = rect->p.x - x, y0 = rect->p.y - y,
	x1 = rect->q.x - x, y1 = rect->q.y - y;
    double r00 = hypot(x0, y0), r01 = hypot(x0, y1),
	r10 = hypot(x1, y0), r11 = hypot(x1, y1);
    double rm0 = max(r00, r01), rm1 = max(r10, r11);

    return max(rm0, rm1);
}

int
gs_shading_R_fill_rectangle(const gs_shading_t * psh0, const gs_rect * rect,
			    gx_device * dev, gs_imager_state * pis)
{
    const gs_shading_R_t *const psh = (const gs_shading_R_t *)psh0;
    R_fill_state_t state;
    gs_client_color rcc[2];
    float d0 = psh->params.Domain[0], d1 = psh->params.Domain[1];
    float dd = d1 - d0;
    float x0 = psh->params.Coords[0], y0 = psh->params.Coords[1];
    floatp r0 = psh->params.Coords[2];
    float x1 = psh->params.Coords[3], y1 = psh->params.Coords[4];
    floatp r1 = psh->params.Coords[5];
    float t[2];
    int i;
    int code;

    shade_init_fill_state((shading_fill_state_t *)&state, psh0, dev, pis);
    state.psh = psh;
    state.rect = *rect;
    /* Compute the parameter range. */
    t[0] = d0;
    t[1] = d1;
    for (i = 0; i < 2; ++i)
	gs_function_evaluate(psh->params.Function, &t[i],
			     rcc[i].paint.values);
    memcpy(state.frames[0].cc, rcc, sizeof(rcc[0]) * 2);
    state.delta.x = x1 - x0;
    state.delta.y = y1 - y0;
    state.dr = r1 - r0;
    /*
     * Compute the annulus width in its thickest direction.  This is
     * only used for a conservative check, so it can be pretty crude
     * (and it is!).
     */
    state.width =
	(fabs(pis->ctm.xx) + fabs(pis->ctm.xy) + fabs(pis->ctm.yx) +
	 fabs(pis->ctm.yy)) * fabs(state.dr);
    state.dd = dd;
    if (psh->params.Extend[0]) {
	if (r0 < r1)
	    code = R_fill_annulus(&state, &rcc[0], 0.0, 0.0, 0.0, r0);
	else
	    code = R_fill_annulus(&state, &rcc[0], 0.0, 0.0, r0,
				  R_compute_radius(x0, y0, rect));
	if (code < 0)
	    return code;
    }
    state.depth = 1;
    state.frames[0].t0 = (t[0] - d0) / dd;
    state.frames[0].t1 = (t[1] - d0) / dd;
    code = R_fill_region(&state);
    if (psh->params.Extend[1]) {
	if (code < 0)
	    return code;
	if (r0 < r1)
	    code = R_fill_annulus(&state, &rcc[1], 1.0, 1.0, r1,
				  R_compute_radius(x1, y1, rect));
	else
	    code = R_fill_annulus(&state, &rcc[1], 1.0, 1.0, 0.0, r1);
    }
    return code;
}
