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/*

 * Copyright © 2004 Carl Worth

 * Copyright © 2006 Red Hat, Inc.

 * Copyright © 2008 Chris Wilson

 *

 * 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 Carl Worth

 *

 * Contributor(s):

 *	Carl D. Worth 

 *	Chris Wilson 

 */

/* Provide definitions for standalone compilation */

#include "cairoint.h"

#include "cairo-error-private.h"

#include "cairo-freelist-private.h"

#include "cairo-combsort-inline.h"

typedef cairo_point_t cairo_bo_point32_t;

typedef struct _cairo_bo_intersect_ordinate {

    int32_t ordinate;

    enum { EXACT, INEXACT } exactness;

} cairo_bo_intersect_ordinate_t;

typedef struct _cairo_bo_intersect_point {

    cairo_bo_intersect_ordinate_t x;

    cairo_bo_intersect_ordinate_t y;

} cairo_bo_intersect_point_t;

typedef struct _cairo_bo_edge cairo_bo_edge_t;

typedef struct _cairo_bo_deferred {

    cairo_bo_edge_t *other;

    int32_t top;

} cairo_bo_deferred_t;

struct _cairo_bo_edge {

    int a_or_b;

    cairo_edge_t edge;

    cairo_bo_edge_t *prev;

    cairo_bo_edge_t *next;

    cairo_bo_deferred_t deferred;

};

/* the parent is always given by index/2 */

#define PQ_PARENT_INDEX(i) ((i) >> 1)

#define PQ_FIRST_ENTRY 1

/* left and right children are index * 2 and (index * 2) +1 respectively */

#define PQ_LEFT_CHILD_INDEX(i) ((i) << 1)

typedef enum {

    CAIRO_BO_EVENT_TYPE_STOP,

    CAIRO_BO_EVENT_TYPE_INTERSECTION,

    CAIRO_BO_EVENT_TYPE_START

} cairo_bo_event_type_t;

typedef struct _cairo_bo_event {

    cairo_bo_event_type_t type;

    cairo_point_t point;

} cairo_bo_event_t;

typedef struct _cairo_bo_start_event {

    cairo_bo_event_type_t type;

    cairo_point_t point;

    cairo_bo_edge_t edge;

} cairo_bo_start_event_t;

typedef struct _cairo_bo_queue_event {

    cairo_bo_event_type_t type;

    cairo_point_t point;

    cairo_bo_edge_t *e1;

    cairo_bo_edge_t *e2;

} cairo_bo_queue_event_t;

typedef struct _pqueue {

    int size, max_size;

    cairo_bo_event_t **elements;

    cairo_bo_event_t *elements_embedded[1024];

} pqueue_t;

typedef struct _cairo_bo_event_queue {

    cairo_freepool_t pool;

    pqueue_t pqueue;

    cairo_bo_event_t **start_events;

} cairo_bo_event_queue_t;

typedef struct _cairo_bo_sweep_line {

    cairo_bo_edge_t *head;

    int32_t current_y;

    cairo_bo_edge_t *current_edge;

} cairo_bo_sweep_line_t;

static cairo_fixed_t

_line_compute_intersection_x_for_y (const cairo_line_t *line,

				    cairo_fixed_t y)

{

    cairo_fixed_t x, dy;

    if (y == line->p1.y)

	return line->p1.x;

    if (y == line->p2.y)

	return line->p2.x;

    x = line->p1.x;

    dy = line->p2.y - line->p1.y;

    if (dy != 0) {

	x += _cairo_fixed_mul_div_floor (y - line->p1.y,

					 line->p2.x - line->p1.x,

					 dy);

    }

    return x;

}

static inline int

_cairo_bo_point32_compare (cairo_bo_point32_t const *a,

			   cairo_bo_point32_t const *b)

{

    int cmp;

    cmp = a->y - b->y;

    if (cmp)

	return cmp;

    return a->x - b->x;

}

/* Compare the slope of a to the slope of b, returning 1, 0, -1 if the

 * slope a is respectively greater than, equal to, or less than the

 * slope of b.

 *

 * For each edge, consider the direction vector formed from:

 *

 *	top -> bottom

 *

 * which is:

 *

 *	(dx, dy) = (line.p2.x - line.p1.x, line.p2.y - line.p1.y)

 *

 * We then define the slope of each edge as dx/dy, (which is the

 * inverse of the slope typically used in math instruction). We never

 * compute a slope directly as the value approaches infinity, but we

 * can derive a slope comparison without division as follows, (where

 * the ? represents our compare operator).

 *

 * 1.	   slope(a) ? slope(b)

 * 2.	    adx/ady ? bdx/bdy

 * 3.	(adx * bdy) ? (bdx * ady)

 *

 * Note that from step 2 to step 3 there is no change needed in the

 * sign of the result since both ady and bdy are guaranteed to be

 * greater than or equal to 0.

 *

 * When using this slope comparison to sort edges, some care is needed

 * when interpreting the results. Since the slope compare operates on

 * distance vectors from top to bottom it gives a correct left to

 * right sort for edges that have a common top point, (such as two

 * edges with start events at the same location). On the other hand,

 * the sense of the result will be exactly reversed for two edges that

 * have a common stop point.

 */

static inline int

_slope_compare (const cairo_bo_edge_t *a,

		const cairo_bo_edge_t *b)

{

    /* XXX: We're assuming here that dx and dy will still fit in 32

     * bits. That's not true in general as there could be overflow. We

     * should prevent that before the tessellation algorithm

     * begins.

     */

    int32_t adx = a->edge.line.p2.x - a->edge.line.p1.x;

    int32_t bdx = b->edge.line.p2.x - b->edge.line.p1.x;

    /* Since the dy's are all positive by construction we can fast

     * path several common cases.

     */

    /* First check for vertical lines. */

    if (adx == 0)

	return -bdx;

    if (bdx == 0)

	return adx;

    /* Then where the two edges point in different directions wrt x. */

    if ((adx ^ bdx) < 0)

	return adx;

    /* Finally we actually need to do the general comparison. */

    {

	int32_t ady = a->edge.line.p2.y - a->edge.line.p1.y;

	int32_t bdy = b->edge.line.p2.y - b->edge.line.p1.y;

	cairo_int64_t adx_bdy = _cairo_int32x32_64_mul (adx, bdy);

	cairo_int64_t bdx_ady = _cairo_int32x32_64_mul (bdx, ady);

	return _cairo_int64_cmp (adx_bdy, bdx_ady);

    }

}

/*

 * We need to compare the x-coordinates of a pair of lines for a particular y,

 * without loss of precision.

 *

 * The x-coordinate along an edge for a given y is:

 *   X = A_x + (Y - A_y) * A_dx / A_dy

 *

 * So the inequality we wish to test is:

 *   A_x + (Y - A_y) * A_dx / A_dy ∘ B_x + (Y - B_y) * B_dx / B_dy,

 * where ∘ is our inequality operator.

 *

 * By construction, we know that A_dy and B_dy (and (Y - A_y), (Y - B_y)) are

 * all positive, so we can rearrange it thus without causing a sign change:

 *   A_dy * B_dy * (A_x - B_x) ∘ (Y - B_y) * B_dx * A_dy

 *                                 - (Y - A_y) * A_dx * B_dy

 *

 * Given the assumption that all the deltas fit within 32 bits, we can compute

 * this comparison directly using 128 bit arithmetic. For certain, but common,

 * input we can reduce this down to a single 32 bit compare by inspecting the

 * deltas.

 *

 * (And put the burden of the work on developing fast 128 bit ops, which are

 * required throughout the tessellator.)

 *

 * See the similar discussion for _slope_compare().

 */

static int

edges_compare_x_for_y_general (const cairo_bo_edge_t *a,

			       const cairo_bo_edge_t *b,

			       int32_t y)

{

    /* XXX: We're assuming here that dx and dy will still fit in 32

     * bits. That's not true in general as there could be overflow. We

     * should prevent that before the tessellation algorithm

     * begins.

     */

    int32_t dx;

    int32_t adx, ady;

    int32_t bdx, bdy;

    enum {

       HAVE_NONE    = 0x0,

       HAVE_DX      = 0x1,

       HAVE_ADX     = 0x2,

       HAVE_DX_ADX  = HAVE_DX | HAVE_ADX,

       HAVE_BDX     = 0x4,

       HAVE_DX_BDX  = HAVE_DX | HAVE_BDX,

       HAVE_ADX_BDX = HAVE_ADX | HAVE_BDX,

       HAVE_ALL     = HAVE_DX | HAVE_ADX | HAVE_BDX

    } have_dx_adx_bdx = HAVE_ALL;

    /* don't bother solving for abscissa if the edges' bounding boxes

     * can be used to order them. */

    {

           int32_t amin, amax;

           int32_t bmin, bmax;

           if (a->edge.line.p1.x < a->edge.line.p2.x) {

                   amin = a->edge.line.p1.x;

                   amax = a->edge.line.p2.x;

           } else {

                   amin = a->edge.line.p2.x;

                   amax = a->edge.line.p1.x;

           }

           if (b->edge.line.p1.x < b->edge.line.p2.x) {

                   bmin = b->edge.line.p1.x;

                   bmax = b->edge.line.p2.x;

           } else {

                   bmin = b->edge.line.p2.x;

                   bmax = b->edge.line.p1.x;

           }

           if (amax < bmin) return -1;

           if (amin > bmax) return +1;

    }

    ady = a->edge.line.p2.y - a->edge.line.p1.y;

    adx = a->edge.line.p2.x - a->edge.line.p1.x;

    if (adx == 0)

	have_dx_adx_bdx &= ~HAVE_ADX;

    bdy = b->edge.line.p2.y - b->edge.line.p1.y;

    bdx = b->edge.line.p2.x - b->edge.line.p1.x;

    if (bdx == 0)

	have_dx_adx_bdx &= ~HAVE_BDX;

    dx = a->edge.line.p1.x - b->edge.line.p1.x;

    if (dx == 0)

	have_dx_adx_bdx &= ~HAVE_DX;

#define L _cairo_int64x32_128_mul (_cairo_int32x32_64_mul (ady, bdy), dx)

#define A _cairo_int64x32_128_mul (_cairo_int32x32_64_mul (adx, bdy), y - a->edge.line.p1.y)

#define B _cairo_int64x32_128_mul (_cairo_int32x32_64_mul (bdx, ady), y - b->edge.line.p1.y)

    switch (have_dx_adx_bdx) {

    default:

    case HAVE_NONE:

	return 0;

    case HAVE_DX:

	/* A_dy * B_dy * (A_x - B_x) ∘ 0 */

	return dx; /* ady * bdy is positive definite */

    case HAVE_ADX:

	/* 0 ∘  - (Y - A_y) * A_dx * B_dy */

	return adx; /* bdy * (y - a->top.y) is positive definite */

    case HAVE_BDX:

	/* 0 ∘ (Y - B_y) * B_dx * A_dy */

	return -bdx; /* ady * (y - b->top.y) is positive definite */

    case HAVE_ADX_BDX:

	/*  0 ∘ (Y - B_y) * B_dx * A_dy - (Y - A_y) * A_dx * B_dy */

	if ((adx ^ bdx) < 0) {

	    return adx;

	} else if (a->edge.line.p1.y == b->edge.line.p1.y) { /* common origin */

	    cairo_int64_t adx_bdy, bdx_ady;

	    /* ∴ A_dx * B_dy ∘ B_dx * A_dy */

	    adx_bdy = _cairo_int32x32_64_mul (adx, bdy);

	    bdx_ady = _cairo_int32x32_64_mul (bdx, ady);

	    return _cairo_int64_cmp (adx_bdy, bdx_ady);

	} else

	    return _cairo_int128_cmp (A, B);

    case HAVE_DX_ADX:

	/* A_dy * (A_x - B_x) ∘ - (Y - A_y) * A_dx */

	if ((-adx ^ dx) < 0) {

	    return dx;

	} else {

	    cairo_int64_t ady_dx, dy_adx;

	    ady_dx = _cairo_int32x32_64_mul (ady, dx);

	    dy_adx = _cairo_int32x32_64_mul (a->edge.line.p1.y - y, adx);

	    return _cairo_int64_cmp (ady_dx, dy_adx);

	}

    case HAVE_DX_BDX:

	/* B_dy * (A_x - B_x) ∘ (Y - B_y) * B_dx */

	if ((bdx ^ dx) < 0) {

	    return dx;

	} else {

	    cairo_int64_t bdy_dx, dy_bdx;

	    bdy_dx = _cairo_int32x32_64_mul (bdy, dx);

	    dy_bdx = _cairo_int32x32_64_mul (y - b->edge.line.p1.y, bdx);

	    return _cairo_int64_cmp (bdy_dx, dy_bdx);

	}

    case HAVE_ALL:

	/* XXX try comparing (a->edge.line.p2.x - b->edge.line.p2.x) et al */

	return _cairo_int128_cmp (L, _cairo_int128_sub (B, A));

    }

#undef B

#undef A

#undef L

}

/*

 * We need to compare the x-coordinate of a line for a particular y wrt to a

 * given x, without loss of precision.

 *

 * The x-coordinate along an edge for a given y is:

 *   X = A_x + (Y - A_y) * A_dx / A_dy

 *

 * So the inequality we wish to test is:

 *   A_x + (Y - A_y) * A_dx / A_dy ∘ X

 * where ∘ is our inequality operator.

 *

 * By construction, we know that A_dy (and (Y - A_y)) are

 * all positive, so we can rearrange it thus without causing a sign change:

 *   (Y - A_y) * A_dx ∘ (X - A_x) * A_dy

 *

 * Given the assumption that all the deltas fit within 32 bits, we can compute

 * this comparison directly using 64 bit arithmetic.

 *

 * See the similar discussion for _slope_compare() and

 * edges_compare_x_for_y_general().

 */

static int

edge_compare_for_y_against_x (const cairo_bo_edge_t *a,

			      int32_t y,

			      int32_t x)

{

    int32_t adx, ady;

    int32_t dx, dy;

    cairo_int64_t L, R;

    if (x < a->edge.line.p1.x && x < a->edge.line.p2.x)

	return 1;

    if (x > a->edge.line.p1.x && x > a->edge.line.p2.x)

	return -1;

    adx = a->edge.line.p2.x - a->edge.line.p1.x;

    dx = x - a->edge.line.p1.x;

    if (adx == 0)

	return -dx;

    if (dx == 0 || (adx ^ dx) < 0)

	return adx;

    dy = y - a->edge.line.p1.y;

    ady = a->edge.line.p2.y - a->edge.line.p1.y;

    L = _cairo_int32x32_64_mul (dy, adx);

    R = _cairo_int32x32_64_mul (dx, ady);

    return _cairo_int64_cmp (L, R);

}

static int

edges_compare_x_for_y (const cairo_bo_edge_t *a,

		       const cairo_bo_edge_t *b,

		       int32_t y)

{

    /* If the sweep-line is currently on an end-point of a line,

     * then we know its precise x value (and considering that we often need to

     * compare events at end-points, this happens frequently enough to warrant

     * special casing).

     */

    enum {

       HAVE_NEITHER = 0x0,

       HAVE_AX      = 0x1,

       HAVE_BX      = 0x2,

       HAVE_BOTH    = HAVE_AX | HAVE_BX

    } have_ax_bx = HAVE_BOTH;

    int32_t ax, bx;

    if (y == a->edge.line.p1.y)

	ax = a->edge.line.p1.x;

    else if (y == a->edge.line.p2.y)

	ax = a->edge.line.p2.x;

    else

	have_ax_bx &= ~HAVE_AX;

    if (y == b->edge.line.p1.y)

	bx = b->edge.line.p1.x;

    else if (y == b->edge.line.p2.y)

	bx = b->edge.line.p2.x;

    else

	have_ax_bx &= ~HAVE_BX;

    switch (have_ax_bx) {

    default:

    case HAVE_NEITHER:

	return edges_compare_x_for_y_general (a, b, y);

    case HAVE_AX:

	return -edge_compare_for_y_against_x (b, y, ax);

    case HAVE_BX:

	return edge_compare_for_y_against_x (a, y, bx);

    case HAVE_BOTH:

	return ax - bx;

    }

}

static inline int

_line_equal (const cairo_line_t *a, const cairo_line_t *b)

{

    return a->p1.x == b->p1.x && a->p1.y == b->p1.y &&

           a->p2.x == b->p2.x && a->p2.y == b->p2.y;

}

static int

_cairo_bo_sweep_line_compare_edges (cairo_bo_sweep_line_t	*sweep_line,

				    const cairo_bo_edge_t	*a,

				    const cairo_bo_edge_t	*b)

{

    int cmp;

    /* compare the edges if not identical */

    if (! _line_equal (&a->edge.line, &b->edge.line)) {

	cmp = edges_compare_x_for_y (a, b, sweep_line->current_y);

	if (cmp)

	    return cmp;

	/* The two edges intersect exactly at y, so fall back on slope

	 * comparison. We know that this compare_edges function will be

	 * called only when starting a new edge, (not when stopping an

	 * edge), so we don't have to worry about conditionally inverting

	 * the sense of _slope_compare. */

	cmp = _slope_compare (a, b);

	if (cmp)

	    return cmp;

    }

    /* We've got two collinear edges now. */

    return b->edge.bottom - a->edge.bottom;

}

static inline cairo_int64_t

det32_64 (int32_t a, int32_t b,

	  int32_t c, int32_t d)

{

    /* det = a * d - b * c */

    return _cairo_int64_sub (_cairo_int32x32_64_mul (a, d),

			     _cairo_int32x32_64_mul (b, c));

}

static inline cairo_int128_t

det64x32_128 (cairo_int64_t a, int32_t       b,

	      cairo_int64_t c, int32_t       d)

{

    /* det = a * d - b * c */

    return _cairo_int128_sub (_cairo_int64x32_128_mul (a, d),

			      _cairo_int64x32_128_mul (c, b));

}

/* Compute the intersection of two lines as defined by two edges. The

 * result is provided as a coordinate pair of 128-bit integers.

 *

 * Returns %CAIRO_BO_STATUS_INTERSECTION if there is an intersection or

 * %CAIRO_BO_STATUS_PARALLEL if the two lines are exactly parallel.

 */

static cairo_bool_t

intersect_lines (cairo_bo_edge_t		*a,

		 cairo_bo_edge_t		*b,

		 cairo_bo_intersect_point_t	*intersection)

{

    cairo_int64_t a_det, b_det;

    /* XXX: We're assuming here that dx and dy will still fit in 32

     * bits. That's not true in general as there could be overflow. We

     * should prevent that before the tessellation algorithm begins.

     * What we're doing to mitigate this is to perform clamping in

     * cairo_bo_tessellate_polygon().

     */

    int32_t dx1 = a->edge.line.p1.x - a->edge.line.p2.x;

    int32_t dy1 = a->edge.line.p1.y - a->edge.line.p2.y;

    int32_t dx2 = b->edge.line.p1.x - b->edge.line.p2.x;

    int32_t dy2 = b->edge.line.p1.y - b->edge.line.p2.y;

    cairo_int64_t den_det;

    cairo_int64_t R;

    cairo_quorem64_t qr;

    den_det = det32_64 (dx1, dy1, dx2, dy2);

     /* Q: Can we determine that the lines do not intersect (within range)

      * much more cheaply than computing the intersection point i.e. by

      * avoiding the division?

      *

      *   X = ax + t * adx = bx + s * bdx;

      *   Y = ay + t * ady = by + s * bdy;

      *   ∴ t * (ady*bdx - bdy*adx) = bdx * (by - ay) + bdy * (ax - bx)

      *   => t * L = R

      *

      * Therefore we can reject any intersection (under the criteria for

      * valid intersection events) if:

      *   L^R < 0 => t < 0, or

      *   L t > 1

      *

      * (where top/bottom must at least extend to the line endpoints).

      *

      * A similar substitution can be performed for s, yielding:

      *   s * (ady*bdx - bdy*adx) = ady * (ax - bx) - adx * (ay - by)

      */

    R = det32_64 (dx2, dy2,

		  b->edge.line.p1.x - a->edge.line.p1.x,

		  b->edge.line.p1.y - a->edge.line.p1.y);

    if (_cairo_int64_negative (den_det)) {

	if (_cairo_int64_ge (den_det, R))

	    return FALSE;

    } else {

	if (_cairo_int64_le (den_det, R))

	    return FALSE;

    }

    R = det32_64 (dy1, dx1,

		  a->edge.line.p1.y - b->edge.line.p1.y,

		  a->edge.line.p1.x - b->edge.line.p1.x);

    if (_cairo_int64_negative (den_det)) {

	if (_cairo_int64_ge (den_det, R))

	    return FALSE;

    } else {

	if (_cairo_int64_le (den_det, R))

	    return FALSE;

    }

    /* We now know that the two lines should intersect within range. */

    a_det = det32_64 (a->edge.line.p1.x, a->edge.line.p1.y,

		      a->edge.line.p2.x, a->edge.line.p2.y);

    b_det = det32_64 (b->edge.line.p1.x, b->edge.line.p1.y,

		      b->edge.line.p2.x, b->edge.line.p2.y);

    /* x = det (a_det, dx1, b_det, dx2) / den_det */

    qr = _cairo_int_96by64_32x64_divrem (det64x32_128 (a_det, dx1,

						       b_det, dx2),

					 den_det);

    if (_cairo_int64_eq (qr.rem, den_det))

	return FALSE;

#if 0

    intersection->x.exactness = _cairo_int64_is_zero (qr.rem) ? EXACT : INEXACT;

#else

    intersection->x.exactness = EXACT;

    if (! _cairo_int64_is_zero (qr.rem)) {

	if (_cairo_int64_negative (den_det) ^ _cairo_int64_negative (qr.rem))

	    qr.rem = _cairo_int64_negate (qr.rem);

	qr.rem = _cairo_int64_mul (qr.rem, _cairo_int32_to_int64 (2));

	if (_cairo_int64_ge (qr.rem, den_det)) {

	    qr.quo = _cairo_int64_add (qr.quo,

				       _cairo_int32_to_int64 (_cairo_int64_negative (qr.quo) ? -1 : 1));

	} else

	    intersection->x.exactness = INEXACT;

    }

#endif

    intersection->x.ordinate = _cairo_int64_to_int32 (qr.quo);

    /* y = det (a_det, dy1, b_det, dy2) / den_det */

    qr = _cairo_int_96by64_32x64_divrem (det64x32_128 (a_det, dy1,

						       b_det, dy2),

					 den_det);

    if (_cairo_int64_eq (qr.rem, den_det))

	return FALSE;

#if 0

    intersection->y.exactness = _cairo_int64_is_zero (qr.rem) ? EXACT : INEXACT;

#else

    intersection->y.exactness = EXACT;

    if (! _cairo_int64_is_zero (qr.rem)) {

	if (_cairo_int64_negative (den_det) ^ _cairo_int64_negative (qr.rem))

	    qr.rem = _cairo_int64_negate (qr.rem);

	qr.rem = _cairo_int64_mul (qr.rem, _cairo_int32_to_int64 (2));

	if (_cairo_int64_ge (qr.rem, den_det)) {

	    qr.quo = _cairo_int64_add (qr.quo,

				       _cairo_int32_to_int64 (_cairo_int64_negative (qr.quo) ? -1 : 1));

	} else

	    intersection->y.exactness = INEXACT;

    }

#endif

    intersection->y.ordinate = _cairo_int64_to_int32 (qr.quo);

    return TRUE;

}

static int

_cairo_bo_intersect_ordinate_32_compare (cairo_bo_intersect_ordinate_t	a,

					 int32_t			b)

{

    /* First compare the quotient */

    if (a.ordinate > b)

	return +1;

    if (a.ordinate < b)

	return -1;

    /* With quotient identical, if remainder is 0 then compare equal */

    /* Otherwise, the non-zero remainder makes a > b */

    return INEXACT == a.exactness;

}

/* Does the given edge contain the given point. The point must already

 * be known to be contained within the line determined by the edge,

 * (most likely the point results from an intersection of this edge

 * with another).

 *

 * If we had exact arithmetic, then this function would simply be a

 * matter of examining whether the y value of the point lies within

 * the range of y values of the edge. But since intersection points

 * are not exact due to being rounded to the nearest integer within

 * the available precision, we must also examine the x value of the

 * point.

 *

 * The definition of "contains" here is that the given intersection

 * point will be seen by the sweep line after the start event for the

 * given edge and before the stop event for the edge. See the comments

 * in the implementation for more details.

 */

static cairo_bool_t

_cairo_bo_edge_contains_intersect_point (cairo_bo_edge_t		*edge,

					 cairo_bo_intersect_point_t	*point)

{

    int cmp_top, cmp_bottom;

    /* XXX: When running the actual algorithm, we don't actually need to

     * compare against edge->top at all here, since any intersection above

     * top is eliminated early via a slope comparison. We're leaving these

     * here for now only for the sake of the quadratic-time intersection

     * finder which needs them.

     */

    cmp_top = _cairo_bo_intersect_ordinate_32_compare (point->y,

						       edge->edge.top);

    cmp_bottom = _cairo_bo_intersect_ordinate_32_compare (point->y,

							  edge->edge.bottom);

    if (cmp_top < 0 || cmp_bottom > 0)

    {

	return FALSE;

    }

    if (cmp_top > 0 && cmp_bottom < 0)

    {

	return TRUE;

    }

    /* At this stage, the point lies on the same y value as either

     * edge->top or edge->bottom, so we have to examine the x value in

     * order to properly determine containment. */

    /* If the y value of the point is the same as the y value of the

     * top of the edge, then the x value of the point must be greater

     * to be considered as inside the edge. Similarly, if the y value

     * of the point is the same as the y value of the bottom of the

     * edge, then the x value of the point must be less to be

     * considered as inside. */

    if (cmp_top == 0) {

	cairo_fixed_t top_x;

	top_x = _line_compute_intersection_x_for_y (&edge->edge.line,

						    edge->edge.top);

	return _cairo_bo_intersect_ordinate_32_compare (point->x, top_x) > 0;

    } else { /* cmp_bottom == 0 */

	cairo_fixed_t bot_x;

	bot_x = _line_compute_intersection_x_for_y (&edge->edge.line,

						    edge->edge.bottom);

	return _cairo_bo_intersect_ordinate_32_compare (point->x, bot_x) < 0;

    }

}

/* Compute the intersection of two edges. The result is provided as a

 * coordinate pair of 128-bit integers.

 *

 * Returns %CAIRO_BO_STATUS_INTERSECTION if there is an intersection

 * that is within both edges, %CAIRO_BO_STATUS_NO_INTERSECTION if the

 * intersection of the lines defined by the edges occurs outside of

 * one or both edges, and %CAIRO_BO_STATUS_PARALLEL if the two edges

 * are exactly parallel.

 *

 * Note that when determining if a candidate intersection is "inside"

 * an edge, we consider both the infinitesimal shortening and the

 * infinitesimal tilt rules described by John Hobby. Specifically, if

 * the intersection is exactly the same as an edge point, it is

 * effectively outside (no intersection is returned). Also, if the

 * intersection point has the same

 */

static cairo_bool_t

_cairo_bo_edge_intersect (cairo_bo_edge_t	*a,

			  cairo_bo_edge_t	*b,

			  cairo_bo_point32_t	*intersection)

{

    cairo_bo_intersect_point_t quorem;

    if (! intersect_lines (a, b, &quorem))

	return FALSE;

    if (! _cairo_bo_edge_contains_intersect_point (a, &quorem))

	return FALSE;

    if (! _cairo_bo_edge_contains_intersect_point (b, &quorem))

	return FALSE;

    /* Now that we've correctly compared the intersection point and

     * determined that it lies within the edge, then we know that we

     * no longer need any more bits of storage for the intersection

     * than we do for our edge coordinates. We also no longer need the

     * remainder from the division. */

    intersection->x = quorem.x.ordinate;

    intersection->y = quorem.y.ordinate;

    return TRUE;

}

static inline int

cairo_bo_event_compare (const cairo_bo_event_t *a,

			const cairo_bo_event_t *b)

{

    int cmp;

    cmp = _cairo_bo_point32_compare (&a->point, &b->point);

    if (cmp)

	return cmp;

    cmp = a->type - b->type;

    if (cmp)

	return cmp;

    return a - b;

}

static inline void

_pqueue_init (pqueue_t *pq)

{

    pq->max_size = ARRAY_LENGTH (pq->elements_embedded);

    pq->size = 0;

    pq->elements = pq->elements_embedded;

}

static inline void

_pqueue_fini (pqueue_t *pq)

{

    if (pq->elements != pq->elements_embedded)

	free (pq->elements);

}

static cairo_status_t

_pqueue_grow (pqueue_t *pq)

{

    cairo_bo_event_t **new_elements;

    pq->max_size *= 2;

    if (pq->elements == pq->elements_embedded) {

	new_elements = _cairo_malloc_ab (pq->max_size,

					 sizeof (cairo_bo_event_t *));

	if (unlikely (new_elements == NULL))

	    return _cairo_error (CAIRO_STATUS_NO_MEMORY);

	memcpy (new_elements, pq->elements_embedded,

		sizeof (pq->elements_embedded));

    } else {

	new_elements = _cairo_realloc_ab (pq->elements,

					  pq->max_size,

					  sizeof (cairo_bo_event_t *));

	if (unlikely (new_elements == NULL))

	    return _cairo_error (CAIRO_STATUS_NO_MEMORY);

    }

    pq->elements = new_elements;

    return CAIRO_STATUS_SUCCESS;

}

static inline cairo_status_t

_pqueue_push (pqueue_t *pq, cairo_bo_event_t *event)

{

    cairo_bo_event_t **elements;

    int i, parent;

    if (unlikely (pq->size + 1 == pq->max_size)) {

	cairo_status_t status;

	status = _pqueue_grow (pq);

	if (unlikely (status))

	    return status;

    }

    elements = pq->elements;

    for (i = ++pq->size;

	 i != PQ_FIRST_ENTRY &&

	 cairo_bo_event_compare (event,

				 elements[parent = PQ_PARENT_INDEX (i)]) < 0;

	 i = parent)

    {

	elements[i] = elements[parent];

    }

    elements[i] = event;

    return CAIRO_STATUS_SUCCESS;

}

static inline void

_pqueue_pop (pqueue_t *pq)

{

    cairo_bo_event_t **elements = pq->elements;

    cairo_bo_event_t *tail;

    int child, i;

    tail = elements[pq->size--];

    if (pq->size == 0) {

	elements[PQ_FIRST_ENTRY] = NULL;

	return;

    }

    for (i = PQ_FIRST_ENTRY;

	 (child = PQ_LEFT_CHILD_INDEX (i)) <= pq->size;

	 i = child)

    {

	if (child != pq->size &&

	    cairo_bo_event_compare (elements[child+1],

				    elements[child]) < 0)

	{

	    child++;

	}

	if (cairo_bo_event_compare (elements[child], tail) >= 0)

	    break;

	elements[i] = elements[child];

    }

    elements[i] = tail;

}

static inline cairo_status_t

_cairo_bo_event_queue_insert (cairo_bo_event_queue_t	*queue,

			      cairo_bo_event_type_t	 type,

			      cairo_bo_edge_t		*e1,

			      cairo_bo_edge_t		*e2,

			      const cairo_point_t	 *point)

{

    cairo_bo_queue_event_t *event;

    event = _cairo_freepool_alloc (&queue->pool);

    if (unlikely (event == NULL))

	return _cairo_error (CAIRO_STATUS_NO_MEMORY);

    event->type = type;

    event->e1 = e1;

    event->e2 = e2;

    event->point = *point;

    return _pqueue_push (&queue->pqueue, (cairo_bo_event_t *) event);

}

static void

_cairo_bo_event_queue_delete (cairo_bo_event_queue_t *queue,

			      cairo_bo_event_t	     *event)

{

    _cairo_freepool_free (&queue->pool, event);

}

static cairo_bo_event_t *

_cairo_bo_event_dequeue (cairo_bo_event_queue_t *event_queue)

{

    cairo_bo_event_t *event, *cmp;

    event = event_queue->pqueue.elements[PQ_FIRST_ENTRY];

    cmp = *event_queue->start_events;

    if (event == NULL ||

	(cmp != NULL && cairo_bo_event_compare (cmp, event) < 0))

    {

	event = cmp;

	event_queue->start_events++;

    }

    else

    {

	_pqueue_pop (&event_queue->pqueue);

    }

    return event;

}

CAIRO_COMBSORT_DECLARE (_cairo_bo_event_queue_sort,

			cairo_bo_event_t *,

			cairo_bo_event_compare)

static void

_cairo_bo_event_queue_init (cairo_bo_event_queue_t	 *event_queue,

			    cairo_bo_event_t		**start_events,

			    int				  num_events)

{

    _cairo_bo_event_queue_sort (start_events, num_events);

    start_events[num_events] = NULL;

    event_queue->start_events = start_events;

    _cairo_freepool_init (&event_queue->pool,

			  sizeof (cairo_bo_queue_event_t));

    _pqueue_init (&event_queue->pqueue);

    event_queue->pqueue.elements[PQ_FIRST_ENTRY] = NULL;

}

static cairo_status_t

event_queue_insert_stop (cairo_bo_event_queue_t	*event_queue,

			 cairo_bo_edge_t		*edge)

{

    cairo_bo_point32_t point;

    point.y = edge->edge.bottom;

    point.x = _line_compute_intersection_x_for_y (&edge->edge.line,

						  point.y);

    return _cairo_bo_event_queue_insert (event_queue,

					 CAIRO_BO_EVENT_TYPE_STOP,

					 edge, NULL,

					 &point);

}

static void

_cairo_bo_event_queue_fini (cairo_bo_event_queue_t *event_queue)

{

    _pqueue_fini (&event_queue->pqueue);

    _cairo_freepool_fini (&event_queue->pool);

}

static inline cairo_status_t

event_queue_insert_if_intersect_below_current_y (cairo_bo_event_queue_t	*event_queue,

						 cairo_bo_edge_t	*left,

						 cairo_bo_edge_t *right)

{

    cairo_bo_point32_t intersection;

    if (_line_equal (&left->edge.line, &right->edge.line))

	return CAIRO_STATUS_SUCCESS;

    /* The names "left" and "right" here are correct descriptions of

     * the order of the two edges within the active edge list. So if a

     * slope comparison also puts left less than right, then we know

     * that the intersection of these two segments has already

     * occurred before the current sweep line position. */

    if (_slope_compare (left, right) <= 0)

	return CAIRO_STATUS_SUCCESS;

    if (! _cairo_bo_edge_intersect (left, right, &intersection))

	return CAIRO_STATUS_SUCCESS;

    return _cairo_bo_event_queue_insert (event_queue,

					 CAIRO_BO_EVENT_TYPE_INTERSECTION,

					 left, right,

					 &intersection);

}

static void

_cairo_bo_sweep_line_init (cairo_bo_sweep_line_t *sweep_line)

{

    sweep_line->head = NULL;

    sweep_line->current_y = INT32_MIN;

    sweep_line->current_edge = NULL;

}

static cairo_status_t

sweep_line_insert (cairo_bo_sweep_line_t	*sweep_line,

		   cairo_bo_edge_t		*edge)

{

    if (sweep_line->current_edge != NULL) {

	cairo_bo_edge_t *prev, *next;

	int cmp;

	cmp = _cairo_bo_sweep_line_compare_edges (sweep_line,

						  sweep_line->current_edge,

						  edge);

	if (cmp < 0) {

	    prev = sweep_line->current_edge;

	    next = prev->next;

	    while (next != NULL &&

		   _cairo_bo_sweep_line_compare_edges (sweep_line,

						       next, edge) < 0)

	    {

		prev = next, next = prev->next;

	    }

	    prev->next = edge;

	    edge->prev = prev;

	    edge->next = next;

	    if (next != NULL)

		next->prev = edge;

	} else if (cmp > 0) {

	    next = sweep_line->current_edge;

	    prev = next->prev;

	    while (prev != NULL &&

		   _cairo_bo_sweep_line_compare_edges (sweep_line,

						       prev, edge) > 0)

	    {

		next = prev, prev = next->prev;

	    }

	    next->prev = edge;

	    edge->next = next;

	    edge->prev = prev;

	    if (prev != NULL)

		prev->next = edge;

	    else

		sweep_line->head = edge;

	} else {

	    prev = sweep_line->current_edge;

	    edge->prev = prev;

	    edge->next = prev->next;

	    if (prev->next != NULL)

		prev->next->prev = edge;

	    prev->next = edge;

	}

    } else {

	sweep_line->head = edge;

    }

    sweep_line->current_edge = edge;

    return CAIRO_STATUS_SUCCESS;

}

static void

_cairo_bo_sweep_line_delete (cairo_bo_sweep_line_t	*sweep_line,

			     cairo_bo_edge_t	*edge)

{

    if (edge->prev != NULL)

	edge->prev->next = edge->next;

    else

	sweep_line->head = edge->next;

    if (edge->next != NULL)

	edge->next->prev = edge->prev;

    if (sweep_line->current_edge == edge)

	sweep_line->current_edge = edge->prev ? edge->prev : edge->next;

}

static void

_cairo_bo_sweep_line_swap (cairo_bo_sweep_line_t	*sweep_line,

			   cairo_bo_edge_t		*left,

			   cairo_bo_edge_t		*right)

{

    if (left->prev != NULL)

	left->prev->next = right;

    else

	sweep_line->head = right;

    if (right->next != NULL)

	right->next->prev = left;

    left->next = right->next;

    right->next = left;

    right->prev = left->prev;

    left->prev = right;

}

static inline cairo_bool_t

edges_colinear (const cairo_bo_edge_t *a, const cairo_bo_edge_t *b)

{