src/backend/utils/adt/geo_ops.c

/*-------------------------------------------------------------------------
 *
 * geo_ops.c
 *	  2D geometric operations
 *
 * This module implements the geometric functions and operators.  The
 * geometric types are (from simple to more complicated):
 *
 * - point
 * - line
 * - line segment
 * - box
 * - circle
 * - polygon
 *
 * Portions Copyright (c) 1996-2023, PostgreSQL Global Development Group
 * Portions Copyright (c) 1994, Regents of the University of California
 *
 *
 * IDENTIFICATION
 *	  src/backend/utils/adt/geo_ops.c
 *
 *-------------------------------------------------------------------------
 */
#include "postgres.h"

#include <math.h>
#include <limits.h>
#include <float.h>
#include <ctype.h>

#include "libpq/pqformat.h"
#include "miscadmin.h"
#include "nodes/miscnodes.h"
#include "utils/float.h"
#include "utils/fmgrprotos.h"
#include "utils/geo_decls.h"
#include "varatt.h"

/*
 * * Type constructors have this form:
 *   void type_construct(Type *result, ...);
 *
 * * Operators commonly have signatures such as
 *   void type1_operator_type2(Type *result, Type1 *obj1, Type2 *obj2);
 *
 * Common operators are:
 * * Intersection point:
 *   bool type1_interpt_type2(Point *result, Type1 *obj1, Type2 *obj2);
 *		Return whether the two objects intersect. If *result is not NULL,
 *		it is set to the intersection point.
 *
 * * Containment:
 *   bool type1_contain_type2(Type1 *obj1, Type2 *obj2);
 *		Return whether obj1 contains obj2.
 *   bool type1_contain_type2(Type1 *contains_obj, Type1 *contained_obj);
 *		Return whether obj1 contains obj2 (used when types are the same)
 *
 * * Distance of closest point in or on obj1 to obj2:
 *   float8 type1_closept_type2(Point *result, Type1 *obj1, Type2 *obj2);
 *		Returns the shortest distance between two objects.  If *result is not
 *		NULL, it is set to the closest point in or on obj1 to obj2.
 *
 * These functions may be used to implement multiple SQL-level operators.  For
 * example, determining whether two lines are parallel is done by checking
 * whether they don't intersect.
 */

/*
 * Internal routines
 */

enum path_delim
{
	PATH_NONE, PATH_OPEN, PATH_CLOSED
};

/* Routines for points */
static inline void point_construct(Point *result, float8 x, float8 y);
static inline void point_add_point(Point *result, Point *pt1, Point *pt2);
static inline void point_sub_point(Point *result, Point *pt1, Point *pt2);
static inline void point_mul_point(Point *result, Point *pt1, Point *pt2);
static inline void point_div_point(Point *result, Point *pt1, Point *pt2);
static inline bool point_eq_point(Point *pt1, Point *pt2);
static inline float8 point_dt(Point *pt1, Point *pt2);
static inline float8 point_sl(Point *pt1, Point *pt2);
static int	point_inside(Point *p, int npts, Point *plist);

/* Routines for lines */
static inline void line_construct(LINE *result, Point *pt, float8 m);
static inline float8 line_sl(LINE *line);
static inline float8 line_invsl(LINE *line);
static bool line_interpt_line(Point *result, LINE *l1, LINE *l2);
static bool line_contain_point(LINE *line, Point *point);
static float8 line_closept_point(Point *result, LINE *line, Point *point);

/* Routines for line segments */
static inline void statlseg_construct(LSEG *lseg, Point *pt1, Point *pt2);
static inline float8 lseg_sl(LSEG *lseg);
static inline float8 lseg_invsl(LSEG *lseg);
static bool lseg_interpt_line(Point *result, LSEG *lseg, LINE *line);
static bool lseg_interpt_lseg(Point *result, LSEG *l1, LSEG *l2);
static int	lseg_crossing(float8 x, float8 y, float8 prev_x, float8 prev_y);
static bool lseg_contain_point(LSEG *lseg, Point *pt);
static float8 lseg_closept_point(Point *result, LSEG *lseg, Point *pt);
static float8 lseg_closept_line(Point *result, LSEG *lseg, LINE *line);
static float8 lseg_closept_lseg(Point *result, LSEG *on_lseg, LSEG *to_lseg);

/* Routines for boxes */
static inline void box_construct(BOX *result, Point *pt1, Point *pt2);
static void box_cn(Point *center, BOX *box);
static bool box_ov(BOX *box1, BOX *box2);
static float8 box_ar(BOX *box);
static float8 box_ht(BOX *box);
static float8 box_wd(BOX *box);
static bool box_contain_point(BOX *box, Point *point);
static bool box_contain_box(BOX *contains_box, BOX *contained_box);
static bool box_contain_lseg(BOX *box, LSEG *lseg);
static bool box_interpt_lseg(Point *result, BOX *box, LSEG *lseg);
static float8 box_closept_point(Point *result, BOX *box, Point *pt);
static float8 box_closept_lseg(Point *result, BOX *box, LSEG *lseg);

/* Routines for circles */
static float8 circle_ar(CIRCLE *circle);

/* Routines for polygons */
static void make_bound_box(POLYGON *poly);
static void poly_to_circle(CIRCLE *result, POLYGON *poly);
static bool lseg_inside_poly(Point *a, Point *b, POLYGON *poly, int start);
static bool poly_contain_poly(POLYGON *contains_poly, POLYGON *contained_poly);
static bool plist_same(int npts, Point *p1, Point *p2);
static float8 dist_ppoly_internal(Point *pt, POLYGON *poly);

/* Routines for encoding and decoding */
static bool single_decode(char *num, float8 *x, char **endptr_p,
						  const char *type_name, const char *orig_string,
						  Node *escontext);
static void single_encode(float8 x, StringInfo str);
static bool pair_decode(char *str, float8 *x, float8 *y, char **endptr_p,
						const char *type_name, const char *orig_string,
						Node *escontext);
static void pair_encode(float8 x, float8 y, StringInfo str);
static int	pair_count(char *s, char delim);
static bool path_decode(char *str, bool opentype, int npts, Point *p,
						bool *isopen, char **endptr_p,
						const char *type_name, const char *orig_string,
						Node *escontext);
static char *path_encode(enum path_delim path_delim, int npts, Point *pt);


/*
 * Delimiters for input and output strings.
 * LDELIM, RDELIM, and DELIM are left, right, and separator delimiters, respectively.
 * LDELIM_EP, RDELIM_EP are left and right delimiters for paths with endpoints.
 */

#define LDELIM			'('
#define RDELIM			')'
#define DELIM			','
#define LDELIM_EP		'['
#define RDELIM_EP		']'
#define LDELIM_C		'<'
#define RDELIM_C		'>'
#define LDELIM_L		'{'
#define RDELIM_L		'}'


/*
 * Geometric data types are composed of points.
 * This code tries to support a common format throughout the data types,
 *	to allow for more predictable usage and data type conversion.
 * The fundamental unit is the point. Other units are line segments,
 *	open paths, boxes, closed paths, and polygons (which should be considered
 *	non-intersecting closed paths).
 *
 * Data representation is as follows:
 *	point:				(x,y)
 *	line segment:		[(x1,y1),(x2,y2)]
 *	box:				(x1,y1),(x2,y2)
 *	open path:			[(x1,y1),...,(xn,yn)]
 *	closed path:		((x1,y1),...,(xn,yn))
 *	polygon:			((x1,y1),...,(xn,yn))
 *
 * For boxes, the points are opposite corners with the first point at the top right.
 * For closed paths and polygons, the points should be reordered to allow
 *	fast and correct equality comparisons.
 *
 * XXX perhaps points in complex shapes should be reordered internally
 *	to allow faster internal operations, but should keep track of input order
 *	and restore that order for text output - tgl 97/01/16
 */

static bool
single_decode(char *num, float8 *x, char **endptr_p,
			  const char *type_name, const char *orig_string,
			  Node *escontext)
{
	*x = float8in_internal(num, endptr_p, type_name, orig_string, escontext);
	return (!SOFT_ERROR_OCCURRED(escontext));
}								/* single_decode() */

static void
single_encode(float8 x, StringInfo str)
{
	char	   *xstr = float8out_internal(x);

	appendStringInfoString(str, xstr);
	pfree(xstr);
}								/* single_encode() */

static bool
pair_decode(char *str, float8 *x, float8 *y, char **endptr_p,
			const char *type_name, const char *orig_string,
			Node *escontext)
{
	bool		has_delim;

	while (isspace((unsigned char) *str))
		str++;
	if ((has_delim = (*str == LDELIM)))
		str++;

	if (!single_decode(str, x, &str, type_name, orig_string, escontext))
		return false;

	if (*str++ != DELIM)
		goto fail;

	if (!single_decode(str, y, &str, type_name, orig_string, escontext))
		return false;

	if (has_delim)
	{
		if (*str++ != RDELIM)
			goto fail;
		while (isspace((unsigned char) *str))
			str++;
	}

	/* report stopping point if wanted, else complain if not end of string */
	if (endptr_p)
		*endptr_p = str;
	else if (*str != '\0')
		goto fail;
	return true;

fail:
	ereturn(escontext, false,
			(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
			 errmsg("invalid input syntax for type %s: \"%s\"",
					type_name, orig_string)));
}

static void
pair_encode(float8 x, float8 y, StringInfo str)
{
	char	   *xstr = float8out_internal(x);
	char	   *ystr = float8out_internal(y);

	appendStringInfo(str, "%s,%s", xstr, ystr);
	pfree(xstr);
	pfree(ystr);
}

static bool
path_decode(char *str, bool opentype, int npts, Point *p,
			bool *isopen, char **endptr_p,
			const char *type_name, const char *orig_string,
			Node *escontext)
{
	int			depth = 0;
	char	   *cp;
	int			i;

	while (isspace((unsigned char) *str))
		str++;
	if ((*isopen = (*str == LDELIM_EP)))
	{
		/* no open delimiter allowed? */
		if (!opentype)
			goto fail;
		depth++;
		str++;
	}
	else if (*str == LDELIM)
	{
		cp = (str + 1);
		while (isspace((unsigned char) *cp))
			cp++;
		if (*cp == LDELIM)
		{
			depth++;
			str = cp;
		}
		else if (strrchr(str, LDELIM) == str)
		{
			depth++;
			str = cp;
		}
	}

	for (i = 0; i < npts; i++)
	{
		if (!pair_decode(str, &(p->x), &(p->y), &str, type_name, orig_string,
						 escontext))
			return false;
		if (*str == DELIM)
			str++;
		p++;
	}

	while (depth > 0)
	{
		if (*str == RDELIM || (*str == RDELIM_EP && *isopen && depth == 1))
		{
			depth--;
			str++;
			while (isspace((unsigned char) *str))
				str++;
		}
		else
			goto fail;
	}

	/* report stopping point if wanted, else complain if not end of string */
	if (endptr_p)
		*endptr_p = str;
	else if (*str != '\0')
		goto fail;
	return true;

fail:
	ereturn(escontext, false,
			(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
			 errmsg("invalid input syntax for type %s: \"%s\"",
					type_name, orig_string)));
}								/* path_decode() */

static char *
path_encode(enum path_delim path_delim, int npts, Point *pt)
{
	StringInfoData str;
	int			i;

	initStringInfo(&str);

	switch (path_delim)
	{
		case PATH_CLOSED:
			appendStringInfoChar(&str, LDELIM);
			break;
		case PATH_OPEN:
			appendStringInfoChar(&str, LDELIM_EP);
			break;
		case PATH_NONE:
			break;
	}

	for (i = 0; i < npts; i++)
	{
		if (i > 0)
			appendStringInfoChar(&str, DELIM);
		appendStringInfoChar(&str, LDELIM);
		pair_encode(pt->x, pt->y, &str);
		appendStringInfoChar(&str, RDELIM);
		pt++;
	}

	switch (path_delim)
	{
		case PATH_CLOSED:
			appendStringInfoChar(&str, RDELIM);
			break;
		case PATH_OPEN:
			appendStringInfoChar(&str, RDELIM_EP);
			break;
		case PATH_NONE:
			break;
	}

	return str.data;
}								/* path_encode() */

/*-------------------------------------------------------------
 * pair_count - count the number of points
 * allow the following notation:
 * '((1,2),(3,4))'
 * '(1,3,2,4)'
 * require an odd number of delim characters in the string
 *-------------------------------------------------------------*/
static int
pair_count(char *s, char delim)
{
	int			ndelim = 0;

	while ((s = strchr(s, delim)) != NULL)
	{
		ndelim++;
		s++;
	}
	return (ndelim % 2) ? ((ndelim + 1) / 2) : -1;
}


/***********************************************************************
 **
 **		Routines for two-dimensional boxes.
 **
 ***********************************************************************/

/*----------------------------------------------------------
 * Formatting and conversion routines.
 *---------------------------------------------------------*/

/*		box_in	-		convert a string to internal form.
 *
 *		External format: (two corners of box)
 *				"(f8, f8), (f8, f8)"
 *				also supports the older style "(f8, f8, f8, f8)"
 */
Datum
box_in(PG_FUNCTION_ARGS)
{
	char	   *str = PG_GETARG_CSTRING(0);
	Node	   *escontext = fcinfo->context;
	BOX		   *box = (BOX *) palloc(sizeof(BOX));
	bool		isopen;
	float8		x,
				y;

	if (!path_decode(str, false, 2, &(box->high), &isopen, NULL, "box", str,
					 escontext))
		PG_RETURN_NULL();

	/* reorder corners if necessary... */
	if (float8_lt(box->high.x, box->low.x))
	{
		x = box->high.x;
		box->high.x = box->low.x;
		box->low.x = x;
	}
	if (float8_lt(box->high.y, box->low.y))
	{
		y = box->high.y;
		box->high.y = box->low.y;
		box->low.y = y;
	}

	PG_RETURN_BOX_P(box);
}

/*		box_out -		convert a box to external form.
 */
Datum
box_out(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);

	PG_RETURN_CSTRING(path_encode(PATH_NONE, 2, &(box->high)));
}

/*
 *		box_recv			- converts external binary format to box
 */
Datum
box_recv(PG_FUNCTION_ARGS)
{
	StringInfo	buf = (StringInfo) PG_GETARG_POINTER(0);
	BOX		   *box;
	float8		x,
				y;

	box = (BOX *) palloc(sizeof(BOX));

	box->high.x = pq_getmsgfloat8(buf);
	box->high.y = pq_getmsgfloat8(buf);
	box->low.x = pq_getmsgfloat8(buf);
	box->low.y = pq_getmsgfloat8(buf);

	/* reorder corners if necessary... */
	if (float8_lt(box->high.x, box->low.x))
	{
		x = box->high.x;
		box->high.x = box->low.x;
		box->low.x = x;
	}
	if (float8_lt(box->high.y, box->low.y))
	{
		y = box->high.y;
		box->high.y = box->low.y;
		box->low.y = y;
	}

	PG_RETURN_BOX_P(box);
}

/*
 *		box_send			- converts box to binary format
 */
Datum
box_send(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	StringInfoData buf;

	pq_begintypsend(&buf);
	pq_sendfloat8(&buf, box->high.x);
	pq_sendfloat8(&buf, box->high.y);
	pq_sendfloat8(&buf, box->low.x);
	pq_sendfloat8(&buf, box->low.y);
	PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}


/*		box_construct	-		fill in a new box.
 */
static inline void
box_construct(BOX *result, Point *pt1, Point *pt2)
{
	if (float8_gt(pt1->x, pt2->x))
	{
		result->high.x = pt1->x;
		result->low.x = pt2->x;
	}
	else
	{
		result->high.x = pt2->x;
		result->low.x = pt1->x;
	}
	if (float8_gt(pt1->y, pt2->y))
	{
		result->high.y = pt1->y;
		result->low.y = pt2->y;
	}
	else
	{
		result->high.y = pt2->y;
		result->low.y = pt1->y;
	}
}


/*----------------------------------------------------------
 *	Relational operators for BOXes.
 *		<, >, <=, >=, and == are based on box area.
 *---------------------------------------------------------*/

/*		box_same		-		are two boxes identical?
 */
Datum
box_same(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(point_eq_point(&box1->high, &box2->high) &&
				   point_eq_point(&box1->low, &box2->low));
}

/*		box_overlap		-		does box1 overlap box2?
 */
Datum
box_overlap(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(box_ov(box1, box2));
}

static bool
box_ov(BOX *box1, BOX *box2)
{
	return (FPle(box1->low.x, box2->high.x) &&
			FPle(box2->low.x, box1->high.x) &&
			FPle(box1->low.y, box2->high.y) &&
			FPle(box2->low.y, box1->high.y));
}

/*		box_left		-		is box1 strictly left of box2?
 */
Datum
box_left(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPlt(box1->high.x, box2->low.x));
}

/*		box_overleft	-		is the right edge of box1 at or left of
 *								the right edge of box2?
 *
 *		This is "less than or equal" for the end of a time range,
 *		when time ranges are stored as rectangles.
 */
Datum
box_overleft(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPle(box1->high.x, box2->high.x));
}

/*		box_right		-		is box1 strictly right of box2?
 */
Datum
box_right(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPgt(box1->low.x, box2->high.x));
}

/*		box_overright	-		is the left edge of box1 at or right of
 *								the left edge of box2?
 *
 *		This is "greater than or equal" for time ranges, when time ranges
 *		are stored as rectangles.
 */
Datum
box_overright(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPge(box1->low.x, box2->low.x));
}

/*		box_below		-		is box1 strictly below box2?
 */
Datum
box_below(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPlt(box1->high.y, box2->low.y));
}

/*		box_overbelow	-		is the upper edge of box1 at or below
 *								the upper edge of box2?
 */
Datum
box_overbelow(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPle(box1->high.y, box2->high.y));
}

/*		box_above		-		is box1 strictly above box2?
 */
Datum
box_above(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPgt(box1->low.y, box2->high.y));
}

/*		box_overabove	-		is the lower edge of box1 at or above
 *								the lower edge of box2?
 */
Datum
box_overabove(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPge(box1->low.y, box2->low.y));
}

/*		box_contained	-		is box1 contained by box2?
 */
Datum
box_contained(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(box_contain_box(box2, box1));
}

/*		box_contain		-		does box1 contain box2?
 */
Datum
box_contain(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(box_contain_box(box1, box2));
}

/*
 * Check whether the second box is in the first box or on its border
 */
static bool
box_contain_box(BOX *contains_box, BOX *contained_box)
{
	return FPge(contains_box->high.x, contained_box->high.x) &&
		FPle(contains_box->low.x, contained_box->low.x) &&
		FPge(contains_box->high.y, contained_box->high.y) &&
		FPle(contains_box->low.y, contained_box->low.y);
}


/*		box_positionop	-
 *				is box1 entirely {above,below} box2?
 *
 * box_below_eq and box_above_eq are obsolete versions that (probably
 * erroneously) accept the equal-boundaries case.  Since these are not
 * in sync with the box_left and box_right code, they are deprecated and
 * not supported in the PG 8.1 rtree operator class extension.
 */
Datum
box_below_eq(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPle(box1->high.y, box2->low.y));
}

Datum
box_above_eq(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPge(box1->low.y, box2->high.y));
}


/*		box_relop		-		is area(box1) relop area(box2), within
 *								our accuracy constraint?
 */
Datum
box_lt(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPlt(box_ar(box1), box_ar(box2)));
}

Datum
box_gt(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPgt(box_ar(box1), box_ar(box2)));
}

Datum
box_eq(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPeq(box_ar(box1), box_ar(box2)));
}

Datum
box_le(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPle(box_ar(box1), box_ar(box2)));
}

Datum
box_ge(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(FPge(box_ar(box1), box_ar(box2)));
}


/*----------------------------------------------------------
 *	"Arithmetic" operators on boxes.
 *---------------------------------------------------------*/

/*		box_area		-		returns the area of the box.
 */
Datum
box_area(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);

	PG_RETURN_FLOAT8(box_ar(box));
}


/*		box_width		-		returns the width of the box
 *								  (horizontal magnitude).
 */
Datum
box_width(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);

	PG_RETURN_FLOAT8(box_wd(box));
}


/*		box_height		-		returns the height of the box
 *								  (vertical magnitude).
 */
Datum
box_height(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);

	PG_RETURN_FLOAT8(box_ht(box));
}


/*		box_distance	-		returns the distance between the
 *								  center points of two boxes.
 */
Datum
box_distance(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);
	Point		a,
				b;

	box_cn(&a, box1);
	box_cn(&b, box2);

	PG_RETURN_FLOAT8(point_dt(&a, &b));
}


/*		box_center		-		returns the center point of the box.
 */
Datum
box_center(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	Point	   *result = (Point *) palloc(sizeof(Point));

	box_cn(result, box);

	PG_RETURN_POINT_P(result);
}


/*		box_ar	-		returns the area of the box.
 */
static float8
box_ar(BOX *box)
{
	return float8_mul(box_wd(box), box_ht(box));
}


/*		box_cn	-		stores the centerpoint of the box into *center.
 */
static void
box_cn(Point *center, BOX *box)
{
	center->x = float8_div(float8_pl(box->high.x, box->low.x), 2.0);
	center->y = float8_div(float8_pl(box->high.y, box->low.y), 2.0);
}


/*		box_wd	-		returns the width (length) of the box
 *								  (horizontal magnitude).
 */
static float8
box_wd(BOX *box)
{
	return float8_mi(box->high.x, box->low.x);
}


/*		box_ht	-		returns the height of the box
 *								  (vertical magnitude).
 */
static float8
box_ht(BOX *box)
{
	return float8_mi(box->high.y, box->low.y);
}


/*----------------------------------------------------------
 *	Funky operations.
 *---------------------------------------------------------*/

/*		box_intersect	-
 *				returns the overlapping portion of two boxes,
 *				  or NULL if they do not intersect.
 */
Datum
box_intersect(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0);
	BOX		   *box2 = PG_GETARG_BOX_P(1);
	BOX		   *result;

	if (!box_ov(box1, box2))
		PG_RETURN_NULL();

	result = (BOX *) palloc(sizeof(BOX));

	result->high.x = float8_min(box1->high.x, box2->high.x);
	result->low.x = float8_max(box1->low.x, box2->low.x);
	result->high.y = float8_min(box1->high.y, box2->high.y);
	result->low.y = float8_max(box1->low.y, box2->low.y);

	PG_RETURN_BOX_P(result);
}


/*		box_diagonal	-
 *				returns a line segment which happens to be the
 *				  positive-slope diagonal of "box".
 */
Datum
box_diagonal(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	LSEG	   *result = (LSEG *) palloc(sizeof(LSEG));

	statlseg_construct(result, &box->high, &box->low);

	PG_RETURN_LSEG_P(result);
}

/***********************************************************************
 **
 **		Routines for 2D lines.
 **
 ***********************************************************************/

static bool
line_decode(char *s, const char *str, LINE *line, Node *escontext)
{
	/* s was already advanced over leading '{' */
	if (!single_decode(s, &line->A, &s, "line", str, escontext))
		return false;
	if (*s++ != DELIM)
		goto fail;
	if (!single_decode(s, &line->B, &s, "line", str, escontext))
		return false;
	if (*s++ != DELIM)
		goto fail;
	if (!single_decode(s, &line->C, &s, "line", str, escontext))
		return false;
	if (*s++ != RDELIM_L)
		goto fail;
	while (isspace((unsigned char) *s))
		s++;
	if (*s != '\0')
		goto fail;
	return true;

fail:
	ereturn(escontext, false,
			(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
			 errmsg("invalid input syntax for type %s: \"%s\"",
					"line", str)));
}

Datum
line_in(PG_FUNCTION_ARGS)
{
	char	   *str = PG_GETARG_CSTRING(0);
	Node	   *escontext = fcinfo->context;
	LINE	   *line = (LINE *) palloc(sizeof(LINE));
	LSEG		lseg;
	bool		isopen;
	char	   *s;

	s = str;
	while (isspace((unsigned char) *s))
		s++;
	if (*s == LDELIM_L)
	{
		if (!line_decode(s + 1, str, line, escontext))
			PG_RETURN_NULL();
		if (FPzero(line->A) && FPzero(line->B))
			ereturn(escontext, (Datum) 0,
					(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
					 errmsg("invalid line specification: A and B cannot both be zero")));
	}
	else
	{
		if (!path_decode(s, true, 2, &lseg.p[0], &isopen, NULL, "line", str,
						 escontext))
			PG_RETURN_NULL();
		if (point_eq_point(&lseg.p[0], &lseg.p[1]))
			ereturn(escontext, (Datum) 0,
					(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
					 errmsg("invalid line specification: must be two distinct points")));

		/*
		 * XXX lseg_sl() and line_construct() can throw overflow/underflow
		 * errors.  Eventually we should allow those to be soft, but the
		 * notational pain seems to outweigh the value for now.
		 */
		line_construct(line, &lseg.p[0], lseg_sl(&lseg));
	}

	PG_RETURN_LINE_P(line);
}


Datum
line_out(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);
	char	   *astr = float8out_internal(line->A);
	char	   *bstr = float8out_internal(line->B);
	char	   *cstr = float8out_internal(line->C);

	PG_RETURN_CSTRING(psprintf("%c%s%c%s%c%s%c", LDELIM_L, astr, DELIM, bstr,
							   DELIM, cstr, RDELIM_L));
}

/*
 *		line_recv			- converts external binary format to line
 */
Datum
line_recv(PG_FUNCTION_ARGS)
{
	StringInfo	buf = (StringInfo) PG_GETARG_POINTER(0);
	LINE	   *line;

	line = (LINE *) palloc(sizeof(LINE));

	line->A = pq_getmsgfloat8(buf);
	line->B = pq_getmsgfloat8(buf);
	line->C = pq_getmsgfloat8(buf);

	if (FPzero(line->A) && FPzero(line->B))
		ereport(ERROR,
				(errcode(ERRCODE_INVALID_BINARY_REPRESENTATION),
				 errmsg("invalid line specification: A and B cannot both be zero")));

	PG_RETURN_LINE_P(line);
}

/*
 *		line_send			- converts line to binary format
 */
Datum
line_send(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);
	StringInfoData buf;

	pq_begintypsend(&buf);
	pq_sendfloat8(&buf, line->A);
	pq_sendfloat8(&buf, line->B);
	pq_sendfloat8(&buf, line->C);
	PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}


/*----------------------------------------------------------
 *	Conversion routines from one line formula to internal.
 *		Internal form:	Ax+By+C=0
 *---------------------------------------------------------*/

/*
 * Fill already-allocated LINE struct from the point and the slope
 */
static inline void
line_construct(LINE *result, Point *pt, float8 m)
{
	if (isinf(m))
	{
		/* vertical - use "x = C" */
		result->A = -1.0;
		result->B = 0.0;
		result->C = pt->x;
	}
	else if (m == 0)
	{
		/* horizontal - use "y = C" */
		result->A = 0.0;
		result->B = -1.0;
		result->C = pt->y;
	}
	else
	{
		/* use "mx - y + yinter = 0" */
		result->A = m;
		result->B = -1.0;
		result->C = float8_mi(pt->y, float8_mul(m, pt->x));
		/* on some platforms, the preceding expression tends to produce -0 */
		if (result->C == 0.0)
			result->C = 0.0;
	}
}

/* line_construct_pp()
 * two points
 */
Datum
line_construct_pp(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);
	LINE	   *result = (LINE *) palloc(sizeof(LINE));

	if (point_eq_point(pt1, pt2))
		ereport(ERROR,
				(errcode(ERRCODE_INVALID_PARAMETER_VALUE),
				 errmsg("invalid line specification: must be two distinct points")));

	line_construct(result, pt1, point_sl(pt1, pt2));

	PG_RETURN_LINE_P(result);
}


/*----------------------------------------------------------
 *	Relative position routines.
 *---------------------------------------------------------*/

Datum
line_intersect(PG_FUNCTION_ARGS)
{
	LINE	   *l1 = PG_GETARG_LINE_P(0);
	LINE	   *l2 = PG_GETARG_LINE_P(1);

	PG_RETURN_BOOL(line_interpt_line(NULL, l1, l2));
}

Datum
line_parallel(PG_FUNCTION_ARGS)
{
	LINE	   *l1 = PG_GETARG_LINE_P(0);
	LINE	   *l2 = PG_GETARG_LINE_P(1);

	PG_RETURN_BOOL(!line_interpt_line(NULL, l1, l2));
}

Datum
line_perp(PG_FUNCTION_ARGS)
{
	LINE	   *l1 = PG_GETARG_LINE_P(0);
	LINE	   *l2 = PG_GETARG_LINE_P(1);

	if (FPzero(l1->A))
		PG_RETURN_BOOL(FPzero(l2->B));
	if (FPzero(l2->A))
		PG_RETURN_BOOL(FPzero(l1->B));
	if (FPzero(l1->B))
		PG_RETURN_BOOL(FPzero(l2->A));
	if (FPzero(l2->B))
		PG_RETURN_BOOL(FPzero(l1->A));

	PG_RETURN_BOOL(FPeq(float8_div(float8_mul(l1->A, l2->A),
								   float8_mul(l1->B, l2->B)), -1.0));
}

Datum
line_vertical(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);

	PG_RETURN_BOOL(FPzero(line->B));
}

Datum
line_horizontal(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);

	PG_RETURN_BOOL(FPzero(line->A));
}


/*
 * Check whether the two lines are the same
 */
Datum
line_eq(PG_FUNCTION_ARGS)
{
	LINE	   *l1 = PG_GETARG_LINE_P(0);
	LINE	   *l2 = PG_GETARG_LINE_P(1);
	float8		ratio;

	/* If any NaNs are involved, insist on exact equality */
	if (unlikely(isnan(l1->A) || isnan(l1->B) || isnan(l1->C) ||
				 isnan(l2->A) || isnan(l2->B) || isnan(l2->C)))
	{
		PG_RETURN_BOOL(float8_eq(l1->A, l2->A) &&
					   float8_eq(l1->B, l2->B) &&
					   float8_eq(l1->C, l2->C));
	}

	/* Otherwise, lines whose parameters are proportional are the same */
	if (!FPzero(l2->A))
		ratio = float8_div(l1->A, l2->A);
	else if (!FPzero(l2->B))
		ratio = float8_div(l1->B, l2->B);
	else if (!FPzero(l2->C))
		ratio = float8_div(l1->C, l2->C);
	else
		ratio = 1.0;

	PG_RETURN_BOOL(FPeq(l1->A, float8_mul(ratio, l2->A)) &&
				   FPeq(l1->B, float8_mul(ratio, l2->B)) &&
				   FPeq(l1->C, float8_mul(ratio, l2->C)));
}


/*----------------------------------------------------------
 *	Line arithmetic routines.
 *---------------------------------------------------------*/

/*
 * Return slope of the line
 */
static inline float8
line_sl(LINE *line)
{
	if (FPzero(line->A))
		return 0.0;
	if (FPzero(line->B))
		return get_float8_infinity();
	return float8_div(line->A, -line->B);
}


/*
 * Return inverse slope of the line
 */
static inline float8
line_invsl(LINE *line)
{
	if (FPzero(line->A))
		return get_float8_infinity();
	if (FPzero(line->B))
		return 0.0;
	return float8_div(line->B, line->A);
}


/* line_distance()
 * Distance between two lines.
 */
Datum
line_distance(PG_FUNCTION_ARGS)
{
	LINE	   *l1 = PG_GETARG_LINE_P(0);
	LINE	   *l2 = PG_GETARG_LINE_P(1);
	float8		ratio;

	if (line_interpt_line(NULL, l1, l2))	/* intersecting? */
		PG_RETURN_FLOAT8(0.0);

	if (!FPzero(l1->A) && !isnan(l1->A) && !FPzero(l2->A) && !isnan(l2->A))
		ratio = float8_div(l1->A, l2->A);
	else if (!FPzero(l1->B) && !isnan(l1->B) && !FPzero(l2->B) && !isnan(l2->B))
		ratio = float8_div(l1->B, l2->B);
	else
		ratio = 1.0;

	PG_RETURN_FLOAT8(float8_div(fabs(float8_mi(l1->C,
											   float8_mul(ratio, l2->C))),
								HYPOT(l1->A, l1->B)));
}

/* line_interpt()
 * Point where two lines l1, l2 intersect (if any)
 */
Datum
line_interpt(PG_FUNCTION_ARGS)
{
	LINE	   *l1 = PG_GETARG_LINE_P(0);
	LINE	   *l2 = PG_GETARG_LINE_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	if (!line_interpt_line(result, l1, l2))
		PG_RETURN_NULL();
	PG_RETURN_POINT_P(result);
}

/*
 * Internal version of line_interpt
 *
 * Return whether two lines intersect. If *result is not NULL, it is set to
 * the intersection point.
 *
 * NOTE: If the lines are identical then we will find they are parallel
 * and report "no intersection".  This is a little weird, but since
 * there's no *unique* intersection, maybe it's appropriate behavior.
 *
 * If the lines have NaN constants, we will return true, and the intersection
 * point would have NaN coordinates.  We shouldn't return false in this case
 * because that would mean the lines are parallel.
 */
static bool
line_interpt_line(Point *result, LINE *l1, LINE *l2)
{
	float8		x,
				y;

	if (!FPzero(l1->B))
	{
		if (FPeq(l2->A, float8_mul(l1->A, float8_div(l2->B, l1->B))))
			return false;

		x = float8_div(float8_mi(float8_mul(l1->B, l2->C),
								 float8_mul(l2->B, l1->C)),
					   float8_mi(float8_mul(l1->A, l2->B),
								 float8_mul(l2->A, l1->B)));
		y = float8_div(-float8_pl(float8_mul(l1->A, x), l1->C), l1->B);
	}
	else if (!FPzero(l2->B))
	{
		if (FPeq(l1->A, float8_mul(l2->A, float8_div(l1->B, l2->B))))
			return false;

		x = float8_div(float8_mi(float8_mul(l2->B, l1->C),
								 float8_mul(l1->B, l2->C)),
					   float8_mi(float8_mul(l2->A, l1->B),
								 float8_mul(l1->A, l2->B)));
		y = float8_div(-float8_pl(float8_mul(l2->A, x), l2->C), l2->B);
	}
	else
		return false;

	/* On some platforms, the preceding expressions tend to produce -0. */
	if (x == 0.0)
		x = 0.0;
	if (y == 0.0)
		y = 0.0;

	if (result != NULL)
		point_construct(result, x, y);

	return true;
}


/***********************************************************************
 **
 **		Routines for 2D paths (sequences of line segments, also
 **				called `polylines').
 **
 **				This is not a general package for geometric paths,
 **				which of course include polygons; the emphasis here
 **				is on (for example) usefulness in wire layout.
 **
 ***********************************************************************/

/*----------------------------------------------------------
 *	String to path / path to string conversion.
 *		External format:
 *				"((xcoord, ycoord),... )"
 *				"[(xcoord, ycoord),... ]"
 *				"(xcoord, ycoord),... "
 *				"[xcoord, ycoord,... ]"
 *		Also support older format:
 *				"(closed, npts, xcoord, ycoord,... )"
 *---------------------------------------------------------*/

Datum
path_area(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);
	float8		area = 0.0;
	int			i,
				j;

	if (!path->closed)
		PG_RETURN_NULL();

	for (i = 0; i < path->npts; i++)
	{
		j = (i + 1) % path->npts;
		area = float8_pl(area, float8_mul(path->p[i].x, path->p[j].y));
		area = float8_mi(area, float8_mul(path->p[i].y, path->p[j].x));
	}

	PG_RETURN_FLOAT8(float8_div(fabs(area), 2.0));
}


Datum
path_in(PG_FUNCTION_ARGS)
{
	char	   *str = PG_GETARG_CSTRING(0);
	Node	   *escontext = fcinfo->context;
	PATH	   *path;
	bool		isopen;
	char	   *s;
	int			npts;
	int			size;
	int			base_size;
	int			depth = 0;

	if ((npts = pair_count(str, ',')) <= 0)
		ereturn(escontext, (Datum) 0,
				(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
				 errmsg("invalid input syntax for type %s: \"%s\"",
						"path", str)));

	s = str;
	while (isspace((unsigned char) *s))
		s++;

	/* skip single leading paren */
	if ((*s == LDELIM) && (strrchr(s, LDELIM) == s))
	{
		s++;
		depth++;
	}

	base_size = sizeof(path->p[0]) * npts;
	size = offsetof(PATH, p) + base_size;

	/* Check for integer overflow */
	if (base_size / npts != sizeof(path->p[0]) || size <= base_size)
		ereturn(escontext, (Datum) 0,
				(errcode(ERRCODE_PROGRAM_LIMIT_EXCEEDED),
				 errmsg("too many points requested")));

	path = (PATH *) palloc(size);

	SET_VARSIZE(path, size);
	path->npts = npts;

	if (!path_decode(s, true, npts, &(path->p[0]), &isopen, &s, "path", str,
					 escontext))
		PG_RETURN_NULL();

	if (depth >= 1)
	{
		if (*s++ != RDELIM)
			ereturn(escontext, (Datum) 0,
					(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
					 errmsg("invalid input syntax for type %s: \"%s\"",
							"path", str)));
		while (isspace((unsigned char) *s))
			s++;
	}
	if (*s != '\0')
		ereturn(escontext, (Datum) 0,
				(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
				 errmsg("invalid input syntax for type %s: \"%s\"",
						"path", str)));

	path->closed = (!isopen);
	/* prevent instability in unused pad bytes */
	path->dummy = 0;

	PG_RETURN_PATH_P(path);
}


Datum
path_out(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);

	PG_RETURN_CSTRING(path_encode(path->closed ? PATH_CLOSED : PATH_OPEN, path->npts, path->p));
}

/*
 *		path_recv			- converts external binary format to path
 *
 * External representation is closed flag (a boolean byte), int32 number
 * of points, and the points.
 */
Datum
path_recv(PG_FUNCTION_ARGS)
{
	StringInfo	buf = (StringInfo) PG_GETARG_POINTER(0);
	PATH	   *path;
	int			closed;
	int32		npts;
	int32		i;
	int			size;

	closed = pq_getmsgbyte(buf);
	npts = pq_getmsgint(buf, sizeof(int32));
	if (npts <= 0 || npts >= (int32) ((INT_MAX - offsetof(PATH, p)) / sizeof(Point)))
		ereport(ERROR,
				(errcode(ERRCODE_INVALID_BINARY_REPRESENTATION),
				 errmsg("invalid number of points in external \"path\" value")));

	size = offsetof(PATH, p) + sizeof(path->p[0]) * npts;
	path = (PATH *) palloc(size);

	SET_VARSIZE(path, size);
	path->npts = npts;
	path->closed = (closed ? 1 : 0);
	/* prevent instability in unused pad bytes */
	path->dummy = 0;

	for (i = 0; i < npts; i++)
	{
		path->p[i].x = pq_getmsgfloat8(buf);
		path->p[i].y = pq_getmsgfloat8(buf);
	}

	PG_RETURN_PATH_P(path);
}

/*
 *		path_send			- converts path to binary format
 */
Datum
path_send(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);
	StringInfoData buf;
	int32		i;

	pq_begintypsend(&buf);
	pq_sendbyte(&buf, path->closed ? 1 : 0);
	pq_sendint32(&buf, path->npts);
	for (i = 0; i < path->npts; i++)
	{
		pq_sendfloat8(&buf, path->p[i].x);
		pq_sendfloat8(&buf, path->p[i].y);
	}
	PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}


/*----------------------------------------------------------
 *	Relational operators.
 *		These are based on the path cardinality,
 *		as stupid as that sounds.
 *
 *		Better relops and access methods coming soon.
 *---------------------------------------------------------*/

Datum
path_n_lt(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);

	PG_RETURN_BOOL(p1->npts < p2->npts);
}

Datum
path_n_gt(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);

	PG_RETURN_BOOL(p1->npts > p2->npts);
}

Datum
path_n_eq(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);

	PG_RETURN_BOOL(p1->npts == p2->npts);
}

Datum
path_n_le(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);

	PG_RETURN_BOOL(p1->npts <= p2->npts);
}

Datum
path_n_ge(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);

	PG_RETURN_BOOL(p1->npts >= p2->npts);
}

/*----------------------------------------------------------
 * Conversion operators.
 *---------------------------------------------------------*/

Datum
path_isclosed(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);

	PG_RETURN_BOOL(path->closed);
}

Datum
path_isopen(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);

	PG_RETURN_BOOL(!path->closed);
}

Datum
path_npoints(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);

	PG_RETURN_INT32(path->npts);
}


Datum
path_close(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P_COPY(0);

	path->closed = true;

	PG_RETURN_PATH_P(path);
}

Datum
path_open(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P_COPY(0);

	path->closed = false;

	PG_RETURN_PATH_P(path);
}


/* path_inter -
 *		Does p1 intersect p2 at any point?
 *		Use bounding boxes for a quick (O(n)) check, then do a
 *		O(n^2) iterative edge check.
 */
Datum
path_inter(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);
	BOX			b1,
				b2;
	int			i,
				j;
	LSEG		seg1,
				seg2;

	Assert(p1->npts > 0 && p2->npts > 0);

	b1.high.x = b1.low.x = p1->p[0].x;
	b1.high.y = b1.low.y = p1->p[0].y;
	for (i = 1; i < p1->npts; i++)
	{
		b1.high.x = float8_max(p1->p[i].x, b1.high.x);
		b1.high.y = float8_max(p1->p[i].y, b1.high.y);
		b1.low.x = float8_min(p1->p[i].x, b1.low.x);
		b1.low.y = float8_min(p1->p[i].y, b1.low.y);
	}
	b2.high.x = b2.low.x = p2->p[0].x;
	b2.high.y = b2.low.y = p2->p[0].y;
	for (i = 1; i < p2->npts; i++)
	{
		b2.high.x = float8_max(p2->p[i].x, b2.high.x);
		b2.high.y = float8_max(p2->p[i].y, b2.high.y);
		b2.low.x = float8_min(p2->p[i].x, b2.low.x);
		b2.low.y = float8_min(p2->p[i].y, b2.low.y);
	}
	if (!box_ov(&b1, &b2))
		PG_RETURN_BOOL(false);

	/* pairwise check lseg intersections */
	for (i = 0; i < p1->npts; i++)
	{
		int			iprev;

		if (i > 0)
			iprev = i - 1;
		else
		{
			if (!p1->closed)
				continue;
			iprev = p1->npts - 1;	/* include the closure segment */
		}

		for (j = 0; j < p2->npts; j++)
		{
			int			jprev;

			if (j > 0)
				jprev = j - 1;
			else
			{
				if (!p2->closed)
					continue;
				jprev = p2->npts - 1;	/* include the closure segment */
			}

			statlseg_construct(&seg1, &p1->p[iprev], &p1->p[i]);
			statlseg_construct(&seg2, &p2->p[jprev], &p2->p[j]);
			if (lseg_interpt_lseg(NULL, &seg1, &seg2))
				PG_RETURN_BOOL(true);
		}
	}

	/* if we dropped through, no two segs intersected */
	PG_RETURN_BOOL(false);
}

/* path_distance()
 * This essentially does a cartesian product of the lsegs in the
 *	two paths, and finds the min distance between any two lsegs
 */
Datum
path_distance(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);
	float8		min = 0.0;		/* initialize to keep compiler quiet */
	bool		have_min = false;
	float8		tmp;
	int			i,
				j;
	LSEG		seg1,
				seg2;

	for (i = 0; i < p1->npts; i++)
	{
		int			iprev;

		if (i > 0)
			iprev = i - 1;
		else
		{
			if (!p1->closed)
				continue;
			iprev = p1->npts - 1;	/* include the closure segment */
		}

		for (j = 0; j < p2->npts; j++)
		{
			int			jprev;

			if (j > 0)
				jprev = j - 1;
			else
			{
				if (!p2->closed)
					continue;
				jprev = p2->npts - 1;	/* include the closure segment */
			}

			statlseg_construct(&seg1, &p1->p[iprev], &p1->p[i]);
			statlseg_construct(&seg2, &p2->p[jprev], &p2->p[j]);

			tmp = lseg_closept_lseg(NULL, &seg1, &seg2);
			if (!have_min || float8_lt(tmp, min))
			{
				min = tmp;
				have_min = true;
			}
		}
	}

	if (!have_min)
		PG_RETURN_NULL();

	PG_RETURN_FLOAT8(min);
}


/*----------------------------------------------------------
 *	"Arithmetic" operations.
 *---------------------------------------------------------*/

Datum
path_length(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);
	float8		result = 0.0;
	int			i;

	for (i = 0; i < path->npts; i++)
	{
		int			iprev;

		if (i > 0)
			iprev = i - 1;
		else
		{
			if (!path->closed)
				continue;
			iprev = path->npts - 1; /* include the closure segment */
		}

		result = float8_pl(result, point_dt(&path->p[iprev], &path->p[i]));
	}

	PG_RETURN_FLOAT8(result);
}

/***********************************************************************
 **
 **		Routines for 2D points.
 **
 ***********************************************************************/

/*----------------------------------------------------------
 *	String to point, point to string conversion.
 *		External format:
 *				"(x,y)"
 *				"x,y"
 *---------------------------------------------------------*/

Datum
point_in(PG_FUNCTION_ARGS)
{
	char	   *str = PG_GETARG_CSTRING(0);
	Point	   *point = (Point *) palloc(sizeof(Point));

	/* Ignore failure from pair_decode, since our return value won't matter */
	pair_decode(str, &point->x, &point->y, NULL, "point", str, fcinfo->context);
	PG_RETURN_POINT_P(point);
}

Datum
point_out(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);

	PG_RETURN_CSTRING(path_encode(PATH_NONE, 1, pt));
}

/*
 *		point_recv			- converts external binary format to point
 */
Datum
point_recv(PG_FUNCTION_ARGS)
{
	StringInfo	buf = (StringInfo) PG_GETARG_POINTER(0);
	Point	   *point;

	point = (Point *) palloc(sizeof(Point));
	point->x = pq_getmsgfloat8(buf);
	point->y = pq_getmsgfloat8(buf);
	PG_RETURN_POINT_P(point);
}

/*
 *		point_send			- converts point to binary format
 */
Datum
point_send(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	StringInfoData buf;

	pq_begintypsend(&buf);
	pq_sendfloat8(&buf, pt->x);
	pq_sendfloat8(&buf, pt->y);
	PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}


/*
 * Initialize a point
 */
static inline void
point_construct(Point *result, float8 x, float8 y)
{
	result->x = x;
	result->y = y;
}


/*----------------------------------------------------------
 *	Relational operators for Points.
 *		Since we do have a sense of coordinates being
 *		"equal" to a given accuracy (point_vert, point_horiz),
 *		the other ops must preserve that sense.  This means
 *		that results may, strictly speaking, be a lie (unless
 *		EPSILON = 0.0).
 *---------------------------------------------------------*/

Datum
point_left(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(FPlt(pt1->x, pt2->x));
}

Datum
point_right(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(FPgt(pt1->x, pt2->x));
}

Datum
point_above(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(FPgt(pt1->y, pt2->y));
}

Datum
point_below(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(FPlt(pt1->y, pt2->y));
}

Datum
point_vert(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(FPeq(pt1->x, pt2->x));
}

Datum
point_horiz(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(FPeq(pt1->y, pt2->y));
}

Datum
point_eq(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(point_eq_point(pt1, pt2));
}

Datum
point_ne(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(!point_eq_point(pt1, pt2));
}


/*
 * Check whether the two points are the same
 */
static inline bool
point_eq_point(Point *pt1, Point *pt2)
{
	/* If any NaNs are involved, insist on exact equality */
	if (unlikely(isnan(pt1->x) || isnan(pt1->y) ||
				 isnan(pt2->x) || isnan(pt2->y)))
		return (float8_eq(pt1->x, pt2->x) && float8_eq(pt1->y, pt2->y));

	return (FPeq(pt1->x, pt2->x) && FPeq(pt1->y, pt2->y));
}


/*----------------------------------------------------------
 *	"Arithmetic" operators on points.
 *---------------------------------------------------------*/

Datum
point_distance(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_FLOAT8(point_dt(pt1, pt2));
}

static inline float8
point_dt(Point *pt1, Point *pt2)
{
	return HYPOT(float8_mi(pt1->x, pt2->x), float8_mi(pt1->y, pt2->y));
}

Datum
point_slope(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);

	PG_RETURN_FLOAT8(point_sl(pt1, pt2));
}


/*
 * Return slope of two points
 *
 * Note that this function returns Inf when the points are the same.
 */
static inline float8
point_sl(Point *pt1, Point *pt2)
{
	if (FPeq(pt1->x, pt2->x))
		return get_float8_infinity();
	if (FPeq(pt1->y, pt2->y))
		return 0.0;
	return float8_div(float8_mi(pt1->y, pt2->y), float8_mi(pt1->x, pt2->x));
}


/*
 * Return inverse slope of two points
 *
 * Note that this function returns 0.0 when the points are the same.
 */
static inline float8
point_invsl(Point *pt1, Point *pt2)
{
	if (FPeq(pt1->x, pt2->x))
		return 0.0;
	if (FPeq(pt1->y, pt2->y))
		return get_float8_infinity();
	return float8_div(float8_mi(pt1->x, pt2->x), float8_mi(pt2->y, pt1->y));
}


/***********************************************************************
 **
 **		Routines for 2D line segments.
 **
 ***********************************************************************/

/*----------------------------------------------------------
 *	String to lseg, lseg to string conversion.
 *		External forms: "[(x1, y1), (x2, y2)]"
 *						"(x1, y1), (x2, y2)"
 *						"x1, y1, x2, y2"
 *		closed form ok	"((x1, y1), (x2, y2))"
 *		(old form)		"(x1, y1, x2, y2)"
 *---------------------------------------------------------*/

Datum
lseg_in(PG_FUNCTION_ARGS)
{
	char	   *str = PG_GETARG_CSTRING(0);
	Node	   *escontext = fcinfo->context;
	LSEG	   *lseg = (LSEG *) palloc(sizeof(LSEG));
	bool		isopen;

	if (!path_decode(str, true, 2, &lseg->p[0], &isopen, NULL, "lseg", str,
					 escontext))
		PG_RETURN_NULL();

	PG_RETURN_LSEG_P(lseg);
}


Datum
lseg_out(PG_FUNCTION_ARGS)
{
	LSEG	   *ls = PG_GETARG_LSEG_P(0);

	PG_RETURN_CSTRING(path_encode(PATH_OPEN, 2, &ls->p[0]));
}

/*
 *		lseg_recv			- converts external binary format to lseg
 */
Datum
lseg_recv(PG_FUNCTION_ARGS)
{
	StringInfo	buf = (StringInfo) PG_GETARG_POINTER(0);
	LSEG	   *lseg;

	lseg = (LSEG *) palloc(sizeof(LSEG));

	lseg->p[0].x = pq_getmsgfloat8(buf);
	lseg->p[0].y = pq_getmsgfloat8(buf);
	lseg->p[1].x = pq_getmsgfloat8(buf);
	lseg->p[1].y = pq_getmsgfloat8(buf);

	PG_RETURN_LSEG_P(lseg);
}

/*
 *		lseg_send			- converts lseg to binary format
 */
Datum
lseg_send(PG_FUNCTION_ARGS)
{
	LSEG	   *ls = PG_GETARG_LSEG_P(0);
	StringInfoData buf;

	pq_begintypsend(&buf);
	pq_sendfloat8(&buf, ls->p[0].x);
	pq_sendfloat8(&buf, ls->p[0].y);
	pq_sendfloat8(&buf, ls->p[1].x);
	pq_sendfloat8(&buf, ls->p[1].y);
	PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}


/* lseg_construct -
 *		form a LSEG from two Points.
 */
Datum
lseg_construct(PG_FUNCTION_ARGS)
{
	Point	   *pt1 = PG_GETARG_POINT_P(0);
	Point	   *pt2 = PG_GETARG_POINT_P(1);
	LSEG	   *result = (LSEG *) palloc(sizeof(LSEG));

	statlseg_construct(result, pt1, pt2);

	PG_RETURN_LSEG_P(result);
}

/* like lseg_construct, but assume space already allocated */
static inline void
statlseg_construct(LSEG *lseg, Point *pt1, Point *pt2)
{
	lseg->p[0].x = pt1->x;
	lseg->p[0].y = pt1->y;
	lseg->p[1].x = pt2->x;
	lseg->p[1].y = pt2->y;
}


/*
 * Return slope of the line segment
 */
static inline float8
lseg_sl(LSEG *lseg)
{
	return point_sl(&lseg->p[0], &lseg->p[1]);
}


/*
 * Return inverse slope of the line segment
 */
static inline float8
lseg_invsl(LSEG *lseg)
{
	return point_invsl(&lseg->p[0], &lseg->p[1]);
}


Datum
lseg_length(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);

	PG_RETURN_FLOAT8(point_dt(&lseg->p[0], &lseg->p[1]));
}

/*----------------------------------------------------------
 *	Relative position routines.
 *---------------------------------------------------------*/

/*
 **  find intersection of the two lines, and see if it falls on
 **  both segments.
 */
Datum
lseg_intersect(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(lseg_interpt_lseg(NULL, l1, l2));
}


Datum
lseg_parallel(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(FPeq(lseg_sl(l1), lseg_sl(l2)));
}

/*
 * Determine if two line segments are perpendicular.
 */
Datum
lseg_perp(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(FPeq(lseg_sl(l1), lseg_invsl(l2)));
}

Datum
lseg_vertical(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);

	PG_RETURN_BOOL(FPeq(lseg->p[0].x, lseg->p[1].x));
}

Datum
lseg_horizontal(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);

	PG_RETURN_BOOL(FPeq(lseg->p[0].y, lseg->p[1].y));
}


Datum
lseg_eq(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(point_eq_point(&l1->p[0], &l2->p[0]) &&
				   point_eq_point(&l1->p[1], &l2->p[1]));
}

Datum
lseg_ne(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(!point_eq_point(&l1->p[0], &l2->p[0]) ||
				   !point_eq_point(&l1->p[1], &l2->p[1]));
}

Datum
lseg_lt(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(FPlt(point_dt(&l1->p[0], &l1->p[1]),
						point_dt(&l2->p[0], &l2->p[1])));
}

Datum
lseg_le(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(FPle(point_dt(&l1->p[0], &l1->p[1]),
						point_dt(&l2->p[0], &l2->p[1])));
}

Datum
lseg_gt(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(FPgt(point_dt(&l1->p[0], &l1->p[1]),
						point_dt(&l2->p[0], &l2->p[1])));
}

Datum
lseg_ge(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(FPge(point_dt(&l1->p[0], &l1->p[1]),
						point_dt(&l2->p[0], &l2->p[1])));
}


/*----------------------------------------------------------
 *	Line arithmetic routines.
 *---------------------------------------------------------*/

/* lseg_distance -
 *		If two segments don't intersect, then the closest
 *		point will be from one of the endpoints to the other
 *		segment.
 */
Datum
lseg_distance(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);

	PG_RETURN_FLOAT8(lseg_closept_lseg(NULL, l1, l2));
}


Datum
lseg_center(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	result->x = float8_div(float8_pl(lseg->p[0].x, lseg->p[1].x), 2.0);
	result->y = float8_div(float8_pl(lseg->p[0].y, lseg->p[1].y), 2.0);

	PG_RETURN_POINT_P(result);
}


/*
 * Return whether the two segments intersect. If *result is not NULL,
 * it is set to the intersection point.
 *
 * This function is almost perfectly symmetric, even though it doesn't look
 * like it.  See lseg_interpt_line() for the other half of it.
 */
static bool
lseg_interpt_lseg(Point *result, LSEG *l1, LSEG *l2)
{
	Point		interpt;
	LINE		tmp;

	line_construct(&tmp, &l2->p[0], lseg_sl(l2));
	if (!lseg_interpt_line(&interpt, l1, &tmp))
		return false;

	/*
	 * If the line intersection point isn't within l2, there is no valid
	 * segment intersection point at all.
	 */
	if (!lseg_contain_point(l2, &interpt))
		return false;

	if (result != NULL)
		*result = interpt;

	return true;
}

Datum
lseg_interpt(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	if (!lseg_interpt_lseg(result, l1, l2))
		PG_RETURN_NULL();
	PG_RETURN_POINT_P(result);
}

/***********************************************************************
 **
 **		Routines for position comparisons of differently-typed
 **				2D objects.
 **
 ***********************************************************************/

/*---------------------------------------------------------------------
 *		dist_
 *				Minimum distance from one object to another.
 *-------------------------------------------------------------------*/

/*
 * Distance from a point to a line
 */
Datum
dist_pl(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	LINE	   *line = PG_GETARG_LINE_P(1);

	PG_RETURN_FLOAT8(line_closept_point(NULL, line, pt));
}

/*
 * Distance from a line to a point
 */
Datum
dist_lp(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);
	Point	   *pt = PG_GETARG_POINT_P(1);

	PG_RETURN_FLOAT8(line_closept_point(NULL, line, pt));
}

/*
 * Distance from a point to a lseg
 */
Datum
dist_ps(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	LSEG	   *lseg = PG_GETARG_LSEG_P(1);

	PG_RETURN_FLOAT8(lseg_closept_point(NULL, lseg, pt));
}

/*
 * Distance from a lseg to a point
 */
Datum
dist_sp(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	Point	   *pt = PG_GETARG_POINT_P(1);

	PG_RETURN_FLOAT8(lseg_closept_point(NULL, lseg, pt));
}

static float8
dist_ppath_internal(Point *pt, PATH *path)
{
	float8		result = 0.0;	/* keep compiler quiet */
	bool		have_min = false;
	float8		tmp;
	int			i;
	LSEG		lseg;

	Assert(path->npts > 0);

	/*
	 * The distance from a point to a path is the smallest distance from the
	 * point to any of its constituent segments.
	 */
	for (i = 0; i < path->npts; i++)
	{
		int			iprev;

		if (i > 0)
			iprev = i - 1;
		else
		{
			if (!path->closed)
				continue;
			iprev = path->npts - 1; /* Include the closure segment */
		}

		statlseg_construct(&lseg, &path->p[iprev], &path->p[i]);
		tmp = lseg_closept_point(NULL, &lseg, pt);
		if (!have_min || float8_lt(tmp, result))
		{
			result = tmp;
			have_min = true;
		}
	}

	return result;
}

/*
 * Distance from a point to a path
 */
Datum
dist_ppath(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	PATH	   *path = PG_GETARG_PATH_P(1);

	PG_RETURN_FLOAT8(dist_ppath_internal(pt, path));
}

/*
 * Distance from a path to a point
 */
Datum
dist_pathp(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);
	Point	   *pt = PG_GETARG_POINT_P(1);

	PG_RETURN_FLOAT8(dist_ppath_internal(pt, path));
}

/*
 * Distance from a point to a box
 */
Datum
dist_pb(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);

	PG_RETURN_FLOAT8(box_closept_point(NULL, box, pt));
}

/*
 * Distance from a box to a point
 */
Datum
dist_bp(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	Point	   *pt = PG_GETARG_POINT_P(1);

	PG_RETURN_FLOAT8(box_closept_point(NULL, box, pt));
}

/*
 * Distance from a lseg to a line
 */
Datum
dist_sl(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	LINE	   *line = PG_GETARG_LINE_P(1);

	PG_RETURN_FLOAT8(lseg_closept_line(NULL, lseg, line));
}

/*
 * Distance from a line to a lseg
 */
Datum
dist_ls(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);
	LSEG	   *lseg = PG_GETARG_LSEG_P(1);

	PG_RETURN_FLOAT8(lseg_closept_line(NULL, lseg, line));
}

/*
 * Distance from a lseg to a box
 */
Datum
dist_sb(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);

	PG_RETURN_FLOAT8(box_closept_lseg(NULL, box, lseg));
}

/*
 * Distance from a box to a lseg
 */
Datum
dist_bs(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	LSEG	   *lseg = PG_GETARG_LSEG_P(1);

	PG_RETURN_FLOAT8(box_closept_lseg(NULL, box, lseg));
}

static float8
dist_cpoly_internal(CIRCLE *circle, POLYGON *poly)
{
	float8		result;

	/* calculate distance to center, and subtract radius */
	result = float8_mi(dist_ppoly_internal(&circle->center, poly),
					   circle->radius);
	if (result < 0.0)
		result = 0.0;

	return result;
}

/*
 * Distance from a circle to a polygon
 */
Datum
dist_cpoly(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	POLYGON    *poly = PG_GETARG_POLYGON_P(1);

	PG_RETURN_FLOAT8(dist_cpoly_internal(circle, poly));
}

/*
 * Distance from a polygon to a circle
 */
Datum
dist_polyc(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_FLOAT8(dist_cpoly_internal(circle, poly));
}

/*
 * Distance from a point to a polygon
 */
Datum
dist_ppoly(PG_FUNCTION_ARGS)
{
	Point	   *point = PG_GETARG_POINT_P(0);
	POLYGON    *poly = PG_GETARG_POLYGON_P(1);

	PG_RETURN_FLOAT8(dist_ppoly_internal(point, poly));
}

Datum
dist_polyp(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	Point	   *point = PG_GETARG_POINT_P(1);

	PG_RETURN_FLOAT8(dist_ppoly_internal(point, poly));
}

static float8
dist_ppoly_internal(Point *pt, POLYGON *poly)
{
	float8		result;
	float8		d;
	int			i;
	LSEG		seg;

	if (point_inside(pt, poly->npts, poly->p) != 0)
		return 0.0;

	/* initialize distance with segment between first and last points */
	seg.p[0].x = poly->p[0].x;
	seg.p[0].y = poly->p[0].y;
	seg.p[1].x = poly->p[poly->npts - 1].x;
	seg.p[1].y = poly->p[poly->npts - 1].y;
	result = lseg_closept_point(NULL, &seg, pt);

	/* check distances for other segments */
	for (i = 0; i < poly->npts - 1; i++)
	{
		seg.p[0].x = poly->p[i].x;
		seg.p[0].y = poly->p[i].y;
		seg.p[1].x = poly->p[i + 1].x;
		seg.p[1].y = poly->p[i + 1].y;
		d = lseg_closept_point(NULL, &seg, pt);
		if (float8_lt(d, result))
			result = d;
	}

	return result;
}


/*---------------------------------------------------------------------
 *		interpt_
 *				Intersection point of objects.
 *				We choose to ignore the "point" of intersection between
 *				  lines and boxes, since there are typically two.
 *-------------------------------------------------------------------*/

/*
 * Return whether the line segment intersect with the line. If *result is not
 * NULL, it is set to the intersection point.
 */
static bool
lseg_interpt_line(Point *result, LSEG *lseg, LINE *line)
{
	Point		interpt;
	LINE		tmp;

	/*
	 * First, we promote the line segment to a line, because we know how to
	 * find the intersection point of two lines.  If they don't have an
	 * intersection point, we are done.
	 */
	line_construct(&tmp, &lseg->p[0], lseg_sl(lseg));
	if (!line_interpt_line(&interpt, &tmp, line))
		return false;

	/*
	 * Then, we check whether the intersection point is actually on the line
	 * segment.
	 */
	if (!lseg_contain_point(lseg, &interpt))
		return false;
	if (result != NULL)
	{
		/*
		 * If there is an intersection, then check explicitly for matching
		 * endpoints since there may be rounding effects with annoying LSB
		 * residue.
		 */
		if (point_eq_point(&lseg->p[0], &interpt))
			*result = lseg->p[0];
		else if (point_eq_point(&lseg->p[1], &interpt))
			*result = lseg->p[1];
		else
			*result = interpt;
	}

	return true;
}

/*---------------------------------------------------------------------
 *		close_
 *				Point of closest proximity between objects.
 *-------------------------------------------------------------------*/

/*
 * If *result is not NULL, it is set to the intersection point of a
 * perpendicular of the line through the point.  Returns the distance
 * of those two points.
 */
static float8
line_closept_point(Point *result, LINE *line, Point *point)
{
	Point		closept;
	LINE		tmp;

	/*
	 * We drop a perpendicular to find the intersection point.  Ordinarily we
	 * should always find it, but that can fail in the presence of NaN
	 * coordinates, and perhaps even from simple roundoff issues.
	 */
	line_construct(&tmp, point, line_invsl(line));
	if (!line_interpt_line(&closept, &tmp, line))
	{
		if (result != NULL)
			*result = *point;

		return get_float8_nan();
	}

	if (result != NULL)
		*result = closept;

	return point_dt(&closept, point);
}

Datum
close_pl(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	LINE	   *line = PG_GETARG_LINE_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	if (isnan(line_closept_point(result, line, pt)))
		PG_RETURN_NULL();

	PG_RETURN_POINT_P(result);
}


/*
 * Closest point on line segment to specified point.
 *
 * If *result is not NULL, set it to the closest point on the line segment
 * to the point.  Returns the distance of the two points.
 */
static float8
lseg_closept_point(Point *result, LSEG *lseg, Point *pt)
{
	Point		closept;
	LINE		tmp;

	/*
	 * To find the closest point, we draw a perpendicular line from the point
	 * to the line segment.
	 */
	line_construct(&tmp, pt, point_invsl(&lseg->p[0], &lseg->p[1]));
	lseg_closept_line(&closept, lseg, &tmp);

	if (result != NULL)
		*result = closept;

	return point_dt(&closept, pt);
}

Datum
close_ps(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	LSEG	   *lseg = PG_GETARG_LSEG_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	if (isnan(lseg_closept_point(result, lseg, pt)))
		PG_RETURN_NULL();

	PG_RETURN_POINT_P(result);
}


/*
 * Closest point on line segment to line segment
 */
static float8
lseg_closept_lseg(Point *result, LSEG *on_lseg, LSEG *to_lseg)
{
	Point		point;
	float8		dist,
				d;

	/* First, we handle the case when the line segments are intersecting. */
	if (lseg_interpt_lseg(result, on_lseg, to_lseg))
		return 0.0;

	/*
	 * Then, we find the closest points from the endpoints of the second line
	 * segment, and keep the closest one.
	 */
	dist = lseg_closept_point(result, on_lseg, &to_lseg->p[0]);
	d = lseg_closept_point(&point, on_lseg, &to_lseg->p[1]);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = point;
	}

	/* The closest point can still be one of the endpoints, so we test them. */
	d = lseg_closept_point(NULL, to_lseg, &on_lseg->p[0]);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = on_lseg->p[0];
	}
	d = lseg_closept_point(NULL, to_lseg, &on_lseg->p[1]);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = on_lseg->p[1];
	}

	return dist;
}

Datum
close_lseg(PG_FUNCTION_ARGS)
{
	LSEG	   *l1 = PG_GETARG_LSEG_P(0);
	LSEG	   *l2 = PG_GETARG_LSEG_P(1);
	Point	   *result;

	if (lseg_sl(l1) == lseg_sl(l2))
		PG_RETURN_NULL();

	result = (Point *) palloc(sizeof(Point));

	if (isnan(lseg_closept_lseg(result, l2, l1)))
		PG_RETURN_NULL();

	PG_RETURN_POINT_P(result);
}


/*
 * Closest point on or in box to specified point.
 *
 * If *result is not NULL, set it to the closest point on the box to the
 * given point, and return the distance of the two points.
 */
static float8
box_closept_point(Point *result, BOX *box, Point *pt)
{
	float8		dist,
				d;
	Point		point,
				closept;
	LSEG		lseg;

	if (box_contain_point(box, pt))
	{
		if (result != NULL)
			*result = *pt;

		return 0.0;
	}

	/* pairwise check lseg distances */
	point.x = box->low.x;
	point.y = box->high.y;
	statlseg_construct(&lseg, &box->low, &point);
	dist = lseg_closept_point(result, &lseg, pt);

	statlseg_construct(&lseg, &box->high, &point);
	d = lseg_closept_point(&closept, &lseg, pt);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = closept;
	}

	point.x = box->high.x;
	point.y = box->low.y;
	statlseg_construct(&lseg, &box->low, &point);
	d = lseg_closept_point(&closept, &lseg, pt);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = closept;
	}

	statlseg_construct(&lseg, &box->high, &point);
	d = lseg_closept_point(&closept, &lseg, pt);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = closept;
	}

	return dist;
}

Datum
close_pb(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	if (isnan(box_closept_point(result, box, pt)))
		PG_RETURN_NULL();

	PG_RETURN_POINT_P(result);
}

/*
 * Closest point on line segment to line.
 *
 * Return the distance between the line and the closest point of the line
 * segment to the line.  If *result is not NULL, set it to that point.
 *
 * NOTE: When the lines are parallel, endpoints of one of the line segment
 * are FPeq(), in presence of NaN or Infinite coordinates, or perhaps =
 * even because of simple roundoff issues, there may not be a single closest
 * point.  We are likely to set the result to the second endpoint in these
 * cases.
 */
static float8
lseg_closept_line(Point *result, LSEG *lseg, LINE *line)
{
	float8		dist1,
				dist2;

	if (lseg_interpt_line(result, lseg, line))
		return 0.0;

	dist1 = line_closept_point(NULL, line, &lseg->p[0]);
	dist2 = line_closept_point(NULL, line, &lseg->p[1]);

	if (dist1 < dist2)
	{
		if (result != NULL)
			*result = lseg->p[0];

		return dist1;
	}
	else
	{
		if (result != NULL)
			*result = lseg->p[1];

		return dist2;
	}
}

Datum
close_ls(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);
	LSEG	   *lseg = PG_GETARG_LSEG_P(1);
	Point	   *result;

	if (lseg_sl(lseg) == line_sl(line))
		PG_RETURN_NULL();

	result = (Point *) palloc(sizeof(Point));

	if (isnan(lseg_closept_line(result, lseg, line)))
		PG_RETURN_NULL();

	PG_RETURN_POINT_P(result);
}


/*
 * Closest point on or in box to line segment.
 *
 * Returns the distance between the closest point on or in the box to
 * the line segment.  If *result is not NULL, it is set to that point.
 */
static float8
box_closept_lseg(Point *result, BOX *box, LSEG *lseg)
{
	float8		dist,
				d;
	Point		point,
				closept;
	LSEG		bseg;

	if (box_interpt_lseg(result, box, lseg))
		return 0.0;

	/* pairwise check lseg distances */
	point.x = box->low.x;
	point.y = box->high.y;
	statlseg_construct(&bseg, &box->low, &point);
	dist = lseg_closept_lseg(result, &bseg, lseg);

	statlseg_construct(&bseg, &box->high, &point);
	d = lseg_closept_lseg(&closept, &bseg, lseg);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = closept;
	}

	point.x = box->high.x;
	point.y = box->low.y;
	statlseg_construct(&bseg, &box->low, &point);
	d = lseg_closept_lseg(&closept, &bseg, lseg);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = closept;
	}

	statlseg_construct(&bseg, &box->high, &point);
	d = lseg_closept_lseg(&closept, &bseg, lseg);
	if (float8_lt(d, dist))
	{
		dist = d;
		if (result != NULL)
			*result = closept;
	}

	return dist;
}

Datum
close_sb(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	if (isnan(box_closept_lseg(result, box, lseg)))
		PG_RETURN_NULL();

	PG_RETURN_POINT_P(result);
}


/*---------------------------------------------------------------------
 *		on_
 *				Whether one object lies completely within another.
 *-------------------------------------------------------------------*/

/*
 *		Does the point satisfy the equation?
 */
static bool
line_contain_point(LINE *line, Point *point)
{
	return FPzero(float8_pl(float8_pl(float8_mul(line->A, point->x),
									  float8_mul(line->B, point->y)),
							line->C));
}

Datum
on_pl(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	LINE	   *line = PG_GETARG_LINE_P(1);

	PG_RETURN_BOOL(line_contain_point(line, pt));
}


/*
 *		Determine colinearity by detecting a triangle inequality.
 * This algorithm seems to behave nicely even with lsb residues - tgl 1997-07-09
 */
static bool
lseg_contain_point(LSEG *lseg, Point *pt)
{
	return FPeq(point_dt(pt, &lseg->p[0]) +
				point_dt(pt, &lseg->p[1]),
				point_dt(&lseg->p[0], &lseg->p[1]));
}

Datum
on_ps(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	LSEG	   *lseg = PG_GETARG_LSEG_P(1);

	PG_RETURN_BOOL(lseg_contain_point(lseg, pt));
}


/*
 * Check whether the point is in the box or on its border
 */
static bool
box_contain_point(BOX *box, Point *point)
{
	return box->high.x >= point->x && box->low.x <= point->x &&
		box->high.y >= point->y && box->low.y <= point->y;
}

Datum
on_pb(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(box_contain_point(box, pt));
}

Datum
box_contain_pt(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	Point	   *pt = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(box_contain_point(box, pt));
}

/* on_ppath -
 *		Whether a point lies within (on) a polyline.
 *		If open, we have to (groan) check each segment.
 * (uses same algorithm as for point intersecting segment - tgl 1997-07-09)
 *		If closed, we use the old O(n) ray method for point-in-polygon.
 *				The ray is horizontal, from pt out to the right.
 *				Each segment that crosses the ray counts as an
 *				intersection; note that an endpoint or edge may touch
 *				but not cross.
 *				(we can do p-in-p in lg(n), but it takes preprocessing)
 */
Datum
on_ppath(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	PATH	   *path = PG_GETARG_PATH_P(1);
	int			i,
				n;
	float8		a,
				b;

	/*-- OPEN --*/
	if (!path->closed)
	{
		n = path->npts - 1;
		a = point_dt(pt, &path->p[0]);
		for (i = 0; i < n; i++)
		{
			b = point_dt(pt, &path->p[i + 1]);
			if (FPeq(float8_pl(a, b), point_dt(&path->p[i], &path->p[i + 1])))
				PG_RETURN_BOOL(true);
			a = b;
		}
		PG_RETURN_BOOL(false);
	}

	/*-- CLOSED --*/
	PG_RETURN_BOOL(point_inside(pt, path->npts, path->p) != 0);
}


/*
 * Check whether the line segment is on the line or close enough
 *
 * It is, if both of its points are on the line or close enough.
 */
Datum
on_sl(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	LINE	   *line = PG_GETARG_LINE_P(1);

	PG_RETURN_BOOL(line_contain_point(line, &lseg->p[0]) &&
				   line_contain_point(line, &lseg->p[1]));
}


/*
 * Check whether the line segment is in the box or on its border
 *
 * It is, if both of its points are in the box or on its border.
 */
static bool
box_contain_lseg(BOX *box, LSEG *lseg)
{
	return box_contain_point(box, &lseg->p[0]) &&
		box_contain_point(box, &lseg->p[1]);
}

Datum
on_sb(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(box_contain_lseg(box, lseg));
}

/*---------------------------------------------------------------------
 *		inter_
 *				Whether one object intersects another.
 *-------------------------------------------------------------------*/

Datum
inter_sl(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	LINE	   *line = PG_GETARG_LINE_P(1);

	PG_RETURN_BOOL(lseg_interpt_line(NULL, lseg, line));
}


/*
 * Do line segment and box intersect?
 *
 * Segment completely inside box counts as intersection.
 * If you want only segments crossing box boundaries,
 *	try converting box to path first.
 *
 * This function also sets the *result to the closest point on the line
 * segment to the center of the box when they overlap and the result is
 * not NULL.  It is somewhat arbitrary, but maybe the best we can do as
 * there are typically two points they intersect.
 *
 * Optimize for non-intersection by checking for box intersection first.
 * - thomas 1998-01-30
 */
static bool
box_interpt_lseg(Point *result, BOX *box, LSEG *lseg)
{
	BOX			lbox;
	LSEG		bseg;
	Point		point;

	lbox.low.x = float8_min(lseg->p[0].x, lseg->p[1].x);
	lbox.low.y = float8_min(lseg->p[0].y, lseg->p[1].y);
	lbox.high.x = float8_max(lseg->p[0].x, lseg->p[1].x);
	lbox.high.y = float8_max(lseg->p[0].y, lseg->p[1].y);

	/* nothing close to overlap? then not going to intersect */
	if (!box_ov(&lbox, box))
		return false;

	if (result != NULL)
	{
		box_cn(&point, box);
		lseg_closept_point(result, lseg, &point);
	}

	/* an endpoint of segment is inside box? then clearly intersects */
	if (box_contain_point(box, &lseg->p[0]) ||
		box_contain_point(box, &lseg->p[1]))
		return true;

	/* pairwise check lseg intersections */
	point.x = box->low.x;
	point.y = box->high.y;
	statlseg_construct(&bseg, &box->low, &point);
	if (lseg_interpt_lseg(NULL, &bseg, lseg))
		return true;

	statlseg_construct(&bseg, &box->high, &point);
	if (lseg_interpt_lseg(NULL, &bseg, lseg))
		return true;

	point.x = box->high.x;
	point.y = box->low.y;
	statlseg_construct(&bseg, &box->low, &point);
	if (lseg_interpt_lseg(NULL, &bseg, lseg))
		return true;

	statlseg_construct(&bseg, &box->high, &point);
	if (lseg_interpt_lseg(NULL, &bseg, lseg))
		return true;

	/* if we dropped through, no two segs intersected */
	return false;
}

Datum
inter_sb(PG_FUNCTION_ARGS)
{
	LSEG	   *lseg = PG_GETARG_LSEG_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);

	PG_RETURN_BOOL(box_interpt_lseg(NULL, box, lseg));
}


/* inter_lb()
 * Do line and box intersect?
 */
Datum
inter_lb(PG_FUNCTION_ARGS)
{
	LINE	   *line = PG_GETARG_LINE_P(0);
	BOX		   *box = PG_GETARG_BOX_P(1);
	LSEG		bseg;
	Point		p1,
				p2;

	/* pairwise check lseg intersections */
	p1.x = box->low.x;
	p1.y = box->low.y;
	p2.x = box->low.x;
	p2.y = box->high.y;
	statlseg_construct(&bseg, &p1, &p2);
	if (lseg_interpt_line(NULL, &bseg, line))
		PG_RETURN_BOOL(true);
	p1.x = box->high.x;
	p1.y = box->high.y;
	statlseg_construct(&bseg, &p1, &p2);
	if (lseg_interpt_line(NULL, &bseg, line))
		PG_RETURN_BOOL(true);
	p2.x = box->high.x;
	p2.y = box->low.y;
	statlseg_construct(&bseg, &p1, &p2);
	if (lseg_interpt_line(NULL, &bseg, line))
		PG_RETURN_BOOL(true);
	p1.x = box->low.x;
	p1.y = box->low.y;
	statlseg_construct(&bseg, &p1, &p2);
	if (lseg_interpt_line(NULL, &bseg, line))
		PG_RETURN_BOOL(true);

	/* if we dropped through, no intersection */
	PG_RETURN_BOOL(false);
}

/*------------------------------------------------------------------
 * The following routines define a data type and operator class for
 * POLYGONS .... Part of which (the polygon's bounding box) is built on
 * top of the BOX data type.
 *
 * make_bound_box - create the bounding box for the input polygon
 *------------------------------------------------------------------*/

/*---------------------------------------------------------------------
 * Make the smallest bounding box for the given polygon.
 *---------------------------------------------------------------------*/
static void
make_bound_box(POLYGON *poly)
{
	int			i;
	float8		x1,
				y1,
				x2,
				y2;

	Assert(poly->npts > 0);

	x1 = x2 = poly->p[0].x;
	y2 = y1 = poly->p[0].y;
	for (i = 1; i < poly->npts; i++)
	{
		if (float8_lt(poly->p[i].x, x1))
			x1 = poly->p[i].x;
		if (float8_gt(poly->p[i].x, x2))
			x2 = poly->p[i].x;
		if (float8_lt(poly->p[i].y, y1))
			y1 = poly->p[i].y;
		if (float8_gt(poly->p[i].y, y2))
			y2 = poly->p[i].y;
	}

	poly->boundbox.low.x = x1;
	poly->boundbox.high.x = x2;
	poly->boundbox.low.y = y1;
	poly->boundbox.high.y = y2;
}

/*------------------------------------------------------------------
 * poly_in - read in the polygon from a string specification
 *
 *		External format:
 *				"((x0,y0),...,(xn,yn))"
 *				"x0,y0,...,xn,yn"
 *				also supports the older style "(x1,...,xn,y1,...yn)"
 *------------------------------------------------------------------*/
Datum
poly_in(PG_FUNCTION_ARGS)
{
	char	   *str = PG_GETARG_CSTRING(0);
	Node	   *escontext = fcinfo->context;
	POLYGON    *poly;
	int			npts;
	int			size;
	int			base_size;
	bool		isopen;

	if ((npts = pair_count(str, ',')) <= 0)
		ereturn(escontext, (Datum) 0,
				(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
				 errmsg("invalid input syntax for type %s: \"%s\"",
						"polygon", str)));

	base_size = sizeof(poly->p[0]) * npts;
	size = offsetof(POLYGON, p) + base_size;

	/* Check for integer overflow */
	if (base_size / npts != sizeof(poly->p[0]) || size <= base_size)
		ereturn(escontext, (Datum) 0,
				(errcode(ERRCODE_PROGRAM_LIMIT_EXCEEDED),
				 errmsg("too many points requested")));

	poly = (POLYGON *) palloc0(size);	/* zero any holes */

	SET_VARSIZE(poly, size);
	poly->npts = npts;

	if (!path_decode(str, false, npts, &(poly->p[0]), &isopen, NULL, "polygon",
					 str, escontext))
		PG_RETURN_NULL();

	make_bound_box(poly);

	PG_RETURN_POLYGON_P(poly);
}

/*---------------------------------------------------------------
 * poly_out - convert internal POLYGON representation to the
 *			  character string format "((f8,f8),...,(f8,f8))"
 *---------------------------------------------------------------*/
Datum
poly_out(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);

	PG_RETURN_CSTRING(path_encode(PATH_CLOSED, poly->npts, poly->p));
}

/*
 *		poly_recv			- converts external binary format to polygon
 *
 * External representation is int32 number of points, and the points.
 * We recompute the bounding box on read, instead of trusting it to
 * be valid.  (Checking it would take just as long, so may as well
 * omit it from external representation.)
 */
Datum
poly_recv(PG_FUNCTION_ARGS)
{
	StringInfo	buf = (StringInfo) PG_GETARG_POINTER(0);
	POLYGON    *poly;
	int32		npts;
	int32		i;
	int			size;

	npts = pq_getmsgint(buf, sizeof(int32));
	if (npts <= 0 || npts >= (int32) ((INT_MAX - offsetof(POLYGON, p)) / sizeof(Point)))
		ereport(ERROR,
				(errcode(ERRCODE_INVALID_BINARY_REPRESENTATION),
				 errmsg("invalid number of points in external \"polygon\" value")));

	size = offsetof(POLYGON, p) + sizeof(poly->p[0]) * npts;
	poly = (POLYGON *) palloc0(size);	/* zero any holes */

	SET_VARSIZE(poly, size);
	poly->npts = npts;

	for (i = 0; i < npts; i++)
	{
		poly->p[i].x = pq_getmsgfloat8(buf);
		poly->p[i].y = pq_getmsgfloat8(buf);
	}

	make_bound_box(poly);

	PG_RETURN_POLYGON_P(poly);
}

/*
 *		poly_send			- converts polygon to binary format
 */
Datum
poly_send(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	StringInfoData buf;
	int32		i;

	pq_begintypsend(&buf);
	pq_sendint32(&buf, poly->npts);
	for (i = 0; i < poly->npts; i++)
	{
		pq_sendfloat8(&buf, poly->p[i].x);
		pq_sendfloat8(&buf, poly->p[i].y);
	}
	PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}


/*-------------------------------------------------------
 * Is polygon A strictly left of polygon B? i.e. is
 * the right most point of A left of the left most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_left(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.high.x < polyb->boundbox.low.x;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-------------------------------------------------------
 * Is polygon A overlapping or left of polygon B? i.e. is
 * the right most point of A at or left of the right most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_overleft(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.high.x <= polyb->boundbox.high.x;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-------------------------------------------------------
 * Is polygon A strictly right of polygon B? i.e. is
 * the left most point of A right of the right most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_right(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.low.x > polyb->boundbox.high.x;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-------------------------------------------------------
 * Is polygon A overlapping or right of polygon B? i.e. is
 * the left most point of A at or right of the left most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_overright(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.low.x >= polyb->boundbox.low.x;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-------------------------------------------------------
 * Is polygon A strictly below polygon B? i.e. is
 * the upper most point of A below the lower most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_below(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.high.y < polyb->boundbox.low.y;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-------------------------------------------------------
 * Is polygon A overlapping or below polygon B? i.e. is
 * the upper most point of A at or below the upper most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_overbelow(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.high.y <= polyb->boundbox.high.y;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-------------------------------------------------------
 * Is polygon A strictly above polygon B? i.e. is
 * the lower most point of A above the upper most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_above(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.low.y > polyb->boundbox.high.y;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-------------------------------------------------------
 * Is polygon A overlapping or above polygon B? i.e. is
 * the lower most point of A at or above the lower most point
 * of B?
 *-------------------------------------------------------*/
Datum
poly_overabove(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = polya->boundbox.low.y >= polyb->boundbox.low.y;

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}


/*-------------------------------------------------------
 * Is polygon A the same as polygon B? i.e. are all the
 * points the same?
 * Check all points for matches in both forward and reverse
 *	direction since polygons are non-directional and are
 *	closed shapes.
 *-------------------------------------------------------*/
Datum
poly_same(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	if (polya->npts != polyb->npts)
		result = false;
	else
		result = plist_same(polya->npts, polya->p, polyb->p);

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*-----------------------------------------------------------------
 * Determine if polygon A overlaps polygon B
 *-----------------------------------------------------------------*/
static bool
poly_overlap_internal(POLYGON *polya, POLYGON *polyb)
{
	bool		result;

	Assert(polya->npts > 0 && polyb->npts > 0);

	/* Quick check by bounding box */
	result = box_ov(&polya->boundbox, &polyb->boundbox);

	/*
	 * Brute-force algorithm - try to find intersected edges, if so then
	 * polygons are overlapped else check is one polygon inside other or not
	 * by testing single point of them.
	 */
	if (result)
	{
		int			ia,
					ib;
		LSEG		sa,
					sb;

		/* Init first of polya's edge with last point */
		sa.p[0] = polya->p[polya->npts - 1];
		result = false;

		for (ia = 0; ia < polya->npts && !result; ia++)
		{
			/* Second point of polya's edge is a current one */
			sa.p[1] = polya->p[ia];

			/* Init first of polyb's edge with last point */
			sb.p[0] = polyb->p[polyb->npts - 1];

			for (ib = 0; ib < polyb->npts && !result; ib++)
			{
				sb.p[1] = polyb->p[ib];
				result = lseg_interpt_lseg(NULL, &sa, &sb);
				sb.p[0] = sb.p[1];
			}

			/*
			 * move current endpoint to the first point of next edge
			 */
			sa.p[0] = sa.p[1];
		}

		if (!result)
		{
			result = (point_inside(polya->p, polyb->npts, polyb->p) ||
					  point_inside(polyb->p, polya->npts, polya->p));
		}
	}

	return result;
}

Datum
poly_overlap(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = poly_overlap_internal(polya, polyb);

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}

/*
 * Tests special kind of segment for in/out of polygon.
 * Special kind means:
 *	- point a should be on segment s
 *	- segment (a,b) should not be contained by s
 * Returns true if:
 *	- segment (a,b) is collinear to s and (a,b) is in polygon
 *	- segment (a,b) s not collinear to s. Note: that doesn't
 *	  mean that segment is in polygon!
 */

static bool
touched_lseg_inside_poly(Point *a, Point *b, LSEG *s, POLYGON *poly, int start)
{
	/* point a is on s, b is not */
	LSEG		t;

	t.p[0] = *a;
	t.p[1] = *b;

	if (point_eq_point(a, s->p))
	{
		if (lseg_contain_point(&t, s->p + 1))
			return lseg_inside_poly(b, s->p + 1, poly, start);
	}
	else if (point_eq_point(a, s->p + 1))
	{
		if (lseg_contain_point(&t, s->p))
			return lseg_inside_poly(b, s->p, poly, start);
	}
	else if (lseg_contain_point(&t, s->p))
	{
		return lseg_inside_poly(b, s->p, poly, start);
	}
	else if (lseg_contain_point(&t, s->p + 1))
	{
		return lseg_inside_poly(b, s->p + 1, poly, start);
	}

	return true;				/* may be not true, but that will check later */
}

/*
 * Returns true if segment (a,b) is in polygon, option
 * start is used for optimization - function checks
 * polygon's edges starting from start
 */
static bool
lseg_inside_poly(Point *a, Point *b, POLYGON *poly, int start)
{
	LSEG		s,
				t;
	int			i;
	bool		res = true,
				intersection = false;

	/* since this function recurses, it could be driven to stack overflow */
	check_stack_depth();

	t.p[0] = *a;
	t.p[1] = *b;
	s.p[0] = poly->p[(start == 0) ? (poly->npts - 1) : (start - 1)];

	for (i = start; i < poly->npts && res; i++)
	{
		Point		interpt;

		CHECK_FOR_INTERRUPTS();

		s.p[1] = poly->p[i];

		if (lseg_contain_point(&s, t.p))
		{
			if (lseg_contain_point(&s, t.p + 1))
				return true;	/* t is contained by s */

			/* Y-cross */
			res = touched_lseg_inside_poly(t.p, t.p + 1, &s, poly, i + 1);
		}
		else if (lseg_contain_point(&s, t.p + 1))
		{
			/* Y-cross */
			res = touched_lseg_inside_poly(t.p + 1, t.p, &s, poly, i + 1);
		}
		else if (lseg_interpt_lseg(&interpt, &t, &s))
		{
			/*
			 * segments are X-crossing, go to check each subsegment
			 */

			intersection = true;
			res = lseg_inside_poly(t.p, &interpt, poly, i + 1);
			if (res)
				res = lseg_inside_poly(t.p + 1, &interpt, poly, i + 1);
		}

		s.p[0] = s.p[1];
	}

	if (res && !intersection)
	{
		Point		p;

		/*
		 * if X-intersection wasn't found, then check central point of tested
		 * segment. In opposite case we already check all subsegments
		 */
		p.x = float8_div(float8_pl(t.p[0].x, t.p[1].x), 2.0);
		p.y = float8_div(float8_pl(t.p[0].y, t.p[1].y), 2.0);

		res = point_inside(&p, poly->npts, poly->p);
	}

	return res;
}

/*
 * Check whether the first polygon contains the second
 */
static bool
poly_contain_poly(POLYGON *contains_poly, POLYGON *contained_poly)
{
	int			i;
	LSEG		s;

	Assert(contains_poly->npts > 0 && contained_poly->npts > 0);

	/*
	 * Quick check to see if contained's bounding box is contained in
	 * contains' bb.
	 */
	if (!box_contain_box(&contains_poly->boundbox, &contained_poly->boundbox))
		return false;

	s.p[0] = contained_poly->p[contained_poly->npts - 1];

	for (i = 0; i < contained_poly->npts; i++)
	{
		s.p[1] = contained_poly->p[i];
		if (!lseg_inside_poly(s.p, s.p + 1, contains_poly, 0))
			return false;
		s.p[0] = s.p[1];
	}

	return true;
}

Datum
poly_contain(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	result = poly_contain_poly(polya, polyb);

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}


/*-----------------------------------------------------------------
 * Determine if polygon A is contained by polygon B
 *-----------------------------------------------------------------*/
Datum
poly_contained(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	bool		result;

	/* Just switch the arguments and pass it off to poly_contain */
	result = poly_contain_poly(polyb, polya);

	/*
	 * Avoid leaking memory for toasted inputs ... needed for rtree indexes
	 */
	PG_FREE_IF_COPY(polya, 0);
	PG_FREE_IF_COPY(polyb, 1);

	PG_RETURN_BOOL(result);
}


Datum
poly_contain_pt(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	Point	   *p = PG_GETARG_POINT_P(1);

	PG_RETURN_BOOL(point_inside(p, poly->npts, poly->p) != 0);
}

Datum
pt_contained_poly(PG_FUNCTION_ARGS)
{
	Point	   *p = PG_GETARG_POINT_P(0);
	POLYGON    *poly = PG_GETARG_POLYGON_P(1);

	PG_RETURN_BOOL(point_inside(p, poly->npts, poly->p) != 0);
}


Datum
poly_distance(PG_FUNCTION_ARGS)
{
	POLYGON    *polya = PG_GETARG_POLYGON_P(0);
	POLYGON    *polyb = PG_GETARG_POLYGON_P(1);
	float8		min = 0.0;		/* initialize to keep compiler quiet */
	bool		have_min = false;
	float8		tmp;
	int			i,
				j;
	LSEG		seg1,
				seg2;

	/*
	 * Distance is zero if polygons overlap.  We must check this because the
	 * path distance will not give the right answer if one poly is entirely
	 * within the other.
	 */
	if (poly_overlap_internal(polya, polyb))
		PG_RETURN_FLOAT8(0.0);

	/*
	 * When they don't overlap, the distance calculation is identical to that
	 * for closed paths (i.e., we needn't care about the fact that polygons
	 * include their contained areas).  See path_distance().
	 */
	for (i = 0; i < polya->npts; i++)
	{
		int			iprev;

		if (i > 0)
			iprev = i - 1;
		else
			iprev = polya->npts - 1;

		for (j = 0; j < polyb->npts; j++)
		{
			int			jprev;

			if (j > 0)
				jprev = j - 1;
			else
				jprev = polyb->npts - 1;

			statlseg_construct(&seg1, &polya->p[iprev], &polya->p[i]);
			statlseg_construct(&seg2, &polyb->p[jprev], &polyb->p[j]);

			tmp = lseg_closept_lseg(NULL, &seg1, &seg2);
			if (!have_min || float8_lt(tmp, min))
			{
				min = tmp;
				have_min = true;
			}
		}
	}

	if (!have_min)
		PG_RETURN_NULL();

	PG_RETURN_FLOAT8(min);
}


/***********************************************************************
 **
 **		Routines for 2D points.
 **
 ***********************************************************************/

Datum
construct_point(PG_FUNCTION_ARGS)
{
	float8		x = PG_GETARG_FLOAT8(0);
	float8		y = PG_GETARG_FLOAT8(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	point_construct(result, x, y);

	PG_RETURN_POINT_P(result);
}


static inline void
point_add_point(Point *result, Point *pt1, Point *pt2)
{
	point_construct(result,
					float8_pl(pt1->x, pt2->x),
					float8_pl(pt1->y, pt2->y));
}

Datum
point_add(PG_FUNCTION_ARGS)
{
	Point	   *p1 = PG_GETARG_POINT_P(0);
	Point	   *p2 = PG_GETARG_POINT_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	point_add_point(result, p1, p2);

	PG_RETURN_POINT_P(result);
}


static inline void
point_sub_point(Point *result, Point *pt1, Point *pt2)
{
	point_construct(result,
					float8_mi(pt1->x, pt2->x),
					float8_mi(pt1->y, pt2->y));
}

Datum
point_sub(PG_FUNCTION_ARGS)
{
	Point	   *p1 = PG_GETARG_POINT_P(0);
	Point	   *p2 = PG_GETARG_POINT_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	point_sub_point(result, p1, p2);

	PG_RETURN_POINT_P(result);
}


static inline void
point_mul_point(Point *result, Point *pt1, Point *pt2)
{
	point_construct(result,
					float8_mi(float8_mul(pt1->x, pt2->x),
							  float8_mul(pt1->y, pt2->y)),
					float8_pl(float8_mul(pt1->x, pt2->y),
							  float8_mul(pt1->y, pt2->x)));
}

Datum
point_mul(PG_FUNCTION_ARGS)
{
	Point	   *p1 = PG_GETARG_POINT_P(0);
	Point	   *p2 = PG_GETARG_POINT_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	point_mul_point(result, p1, p2);

	PG_RETURN_POINT_P(result);
}


static inline void
point_div_point(Point *result, Point *pt1, Point *pt2)
{
	float8		div;

	div = float8_pl(float8_mul(pt2->x, pt2->x), float8_mul(pt2->y, pt2->y));

	point_construct(result,
					float8_div(float8_pl(float8_mul(pt1->x, pt2->x),
										 float8_mul(pt1->y, pt2->y)), div),
					float8_div(float8_mi(float8_mul(pt1->y, pt2->x),
										 float8_mul(pt1->x, pt2->y)), div));
}

Datum
point_div(PG_FUNCTION_ARGS)
{
	Point	   *p1 = PG_GETARG_POINT_P(0);
	Point	   *p2 = PG_GETARG_POINT_P(1);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));

	point_div_point(result, p1, p2);

	PG_RETURN_POINT_P(result);
}


/***********************************************************************
 **
 **		Routines for 2D boxes.
 **
 ***********************************************************************/

Datum
points_box(PG_FUNCTION_ARGS)
{
	Point	   *p1 = PG_GETARG_POINT_P(0);
	Point	   *p2 = PG_GETARG_POINT_P(1);
	BOX		   *result;

	result = (BOX *) palloc(sizeof(BOX));

	box_construct(result, p1, p2);

	PG_RETURN_BOX_P(result);
}

Datum
box_add(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	Point	   *p = PG_GETARG_POINT_P(1);
	BOX		   *result;

	result = (BOX *) palloc(sizeof(BOX));

	point_add_point(&result->high, &box->high, p);
	point_add_point(&result->low, &box->low, p);

	PG_RETURN_BOX_P(result);
}

Datum
box_sub(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	Point	   *p = PG_GETARG_POINT_P(1);
	BOX		   *result;

	result = (BOX *) palloc(sizeof(BOX));

	point_sub_point(&result->high, &box->high, p);
	point_sub_point(&result->low, &box->low, p);

	PG_RETURN_BOX_P(result);
}

Datum
box_mul(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	Point	   *p = PG_GETARG_POINT_P(1);
	BOX		   *result;
	Point		high,
				low;

	result = (BOX *) palloc(sizeof(BOX));

	point_mul_point(&high, &box->high, p);
	point_mul_point(&low, &box->low, p);

	box_construct(result, &high, &low);

	PG_RETURN_BOX_P(result);
}

Datum
box_div(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	Point	   *p = PG_GETARG_POINT_P(1);
	BOX		   *result;
	Point		high,
				low;

	result = (BOX *) palloc(sizeof(BOX));

	point_div_point(&high, &box->high, p);
	point_div_point(&low, &box->low, p);

	box_construct(result, &high, &low);

	PG_RETURN_BOX_P(result);
}

/*
 * Convert point to empty box
 */
Datum
point_box(PG_FUNCTION_ARGS)
{
	Point	   *pt = PG_GETARG_POINT_P(0);
	BOX		   *box;

	box = (BOX *) palloc(sizeof(BOX));

	box->high.x = pt->x;
	box->low.x = pt->x;
	box->high.y = pt->y;
	box->low.y = pt->y;

	PG_RETURN_BOX_P(box);
}

/*
 * Smallest bounding box that includes both of the given boxes
 */
Datum
boxes_bound_box(PG_FUNCTION_ARGS)
{
	BOX		   *box1 = PG_GETARG_BOX_P(0),
			   *box2 = PG_GETARG_BOX_P(1),
			   *container;

	container = (BOX *) palloc(sizeof(BOX));

	container->high.x = float8_max(box1->high.x, box2->high.x);
	container->low.x = float8_min(box1->low.x, box2->low.x);
	container->high.y = float8_max(box1->high.y, box2->high.y);
	container->low.y = float8_min(box1->low.y, box2->low.y);

	PG_RETURN_BOX_P(container);
}


/***********************************************************************
 **
 **		Routines for 2D paths.
 **
 ***********************************************************************/

/* path_add()
 * Concatenate two paths (only if they are both open).
 */
Datum
path_add(PG_FUNCTION_ARGS)
{
	PATH	   *p1 = PG_GETARG_PATH_P(0);
	PATH	   *p2 = PG_GETARG_PATH_P(1);
	PATH	   *result;
	int			size,
				base_size;
	int			i;

	if (p1->closed || p2->closed)
		PG_RETURN_NULL();

	base_size = sizeof(p1->p[0]) * (p1->npts + p2->npts);
	size = offsetof(PATH, p) + base_size;

	/* Check for integer overflow */
	if (base_size / sizeof(p1->p[0]) != (p1->npts + p2->npts) ||
		size <= base_size)
		ereport(ERROR,
				(errcode(ERRCODE_PROGRAM_LIMIT_EXCEEDED),
				 errmsg("too many points requested")));

	result = (PATH *) palloc(size);

	SET_VARSIZE(result, size);
	result->npts = (p1->npts + p2->npts);
	result->closed = p1->closed;
	/* prevent instability in unused pad bytes */
	result->dummy = 0;

	for (i = 0; i < p1->npts; i++)
	{
		result->p[i].x = p1->p[i].x;
		result->p[i].y = p1->p[i].y;
	}
	for (i = 0; i < p2->npts; i++)
	{
		result->p[i + p1->npts].x = p2->p[i].x;
		result->p[i + p1->npts].y = p2->p[i].y;
	}

	PG_RETURN_PATH_P(result);
}

/* path_add_pt()
 * Translation operators.
 */
Datum
path_add_pt(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P_COPY(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	int			i;

	for (i = 0; i < path->npts; i++)
		point_add_point(&path->p[i], &path->p[i], point);

	PG_RETURN_PATH_P(path);
}

Datum
path_sub_pt(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P_COPY(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	int			i;

	for (i = 0; i < path->npts; i++)
		point_sub_point(&path->p[i], &path->p[i], point);

	PG_RETURN_PATH_P(path);
}

/* path_mul_pt()
 * Rotation and scaling operators.
 */
Datum
path_mul_pt(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P_COPY(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	int			i;

	for (i = 0; i < path->npts; i++)
		point_mul_point(&path->p[i], &path->p[i], point);

	PG_RETURN_PATH_P(path);
}

Datum
path_div_pt(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P_COPY(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	int			i;

	for (i = 0; i < path->npts; i++)
		point_div_point(&path->p[i], &path->p[i], point);

	PG_RETURN_PATH_P(path);
}


Datum
path_poly(PG_FUNCTION_ARGS)
{
	PATH	   *path = PG_GETARG_PATH_P(0);
	POLYGON    *poly;
	int			size;
	int			i;

	/* This is not very consistent --- other similar cases return NULL ... */
	if (!path->closed)
		ereport(ERROR,
				(errcode(ERRCODE_INVALID_PARAMETER_VALUE),
				 errmsg("open path cannot be converted to polygon")));

	/*
	 * Never overflows: the old size fit in MaxAllocSize, and the new size is
	 * just a small constant larger.
	 */
	size = offsetof(POLYGON, p) + sizeof(poly->p[0]) * path->npts;
	poly = (POLYGON *) palloc(size);

	SET_VARSIZE(poly, size);
	poly->npts = path->npts;

	for (i = 0; i < path->npts; i++)
	{
		poly->p[i].x = path->p[i].x;
		poly->p[i].y = path->p[i].y;
	}

	make_bound_box(poly);

	PG_RETURN_POLYGON_P(poly);
}


/***********************************************************************
 **
 **		Routines for 2D polygons.
 **
 ***********************************************************************/

Datum
poly_npoints(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);

	PG_RETURN_INT32(poly->npts);
}


Datum
poly_center(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	Point	   *result;
	CIRCLE		circle;

	result = (Point *) palloc(sizeof(Point));

	poly_to_circle(&circle, poly);
	*result = circle.center;

	PG_RETURN_POINT_P(result);
}


Datum
poly_box(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	BOX		   *box;

	box = (BOX *) palloc(sizeof(BOX));
	*box = poly->boundbox;

	PG_RETURN_BOX_P(box);
}


/* box_poly()
 * Convert a box to a polygon.
 */
Datum
box_poly(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	POLYGON    *poly;
	int			size;

	/* map four corners of the box to a polygon */
	size = offsetof(POLYGON, p) + sizeof(poly->p[0]) * 4;
	poly = (POLYGON *) palloc(size);

	SET_VARSIZE(poly, size);
	poly->npts = 4;

	poly->p[0].x = box->low.x;
	poly->p[0].y = box->low.y;
	poly->p[1].x = box->low.x;
	poly->p[1].y = box->high.y;
	poly->p[2].x = box->high.x;
	poly->p[2].y = box->high.y;
	poly->p[3].x = box->high.x;
	poly->p[3].y = box->low.y;

	box_construct(&poly->boundbox, &box->high, &box->low);

	PG_RETURN_POLYGON_P(poly);
}


Datum
poly_path(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	PATH	   *path;
	int			size;
	int			i;

	/*
	 * Never overflows: the old size fit in MaxAllocSize, and the new size is
	 * smaller by a small constant.
	 */
	size = offsetof(PATH, p) + sizeof(path->p[0]) * poly->npts;
	path = (PATH *) palloc(size);

	SET_VARSIZE(path, size);
	path->npts = poly->npts;
	path->closed = true;
	/* prevent instability in unused pad bytes */
	path->dummy = 0;

	for (i = 0; i < poly->npts; i++)
	{
		path->p[i].x = poly->p[i].x;
		path->p[i].y = poly->p[i].y;
	}

	PG_RETURN_PATH_P(path);
}


/***********************************************************************
 **
 **		Routines for circles.
 **
 ***********************************************************************/

/*----------------------------------------------------------
 * Formatting and conversion routines.
 *---------------------------------------------------------*/

/*		circle_in		-		convert a string to internal form.
 *
 *		External format: (center and radius of circle)
 *				"<(f8,f8),f8>"
 *				also supports quick entry style "f8,f8,f8"
 */
Datum
circle_in(PG_FUNCTION_ARGS)
{
	char	   *str = PG_GETARG_CSTRING(0);
	Node	   *escontext = fcinfo->context;
	CIRCLE	   *circle = (CIRCLE *) palloc(sizeof(CIRCLE));
	char	   *s,
			   *cp;
	int			depth = 0;

	s = str;
	while (isspace((unsigned char) *s))
		s++;
	if (*s == LDELIM_C)
		depth++, s++;
	else if (*s == LDELIM)
	{
		/* If there are two left parens, consume the first one */
		cp = (s + 1);
		while (isspace((unsigned char) *cp))
			cp++;
		if (*cp == LDELIM)
			depth++, s = cp;
	}

	/* pair_decode will consume parens around the pair, if any */
	if (!pair_decode(s, &circle->center.x, &circle->center.y, &s, "circle", str,
					 escontext))
		PG_RETURN_NULL();

	if (*s == DELIM)
		s++;

	if (!single_decode(s, &circle->radius, &s, "circle", str, escontext))
		PG_RETURN_NULL();

	/* We have to accept NaN. */
	if (circle->radius < 0.0)
		ereturn(escontext, (Datum) 0,
				(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
				 errmsg("invalid input syntax for type %s: \"%s\"",
						"circle", str)));

	while (depth > 0)
	{
		if ((*s == RDELIM) || ((*s == RDELIM_C) && (depth == 1)))
		{
			depth--;
			s++;
			while (isspace((unsigned char) *s))
				s++;
		}
		else
			ereturn(escontext, (Datum) 0,
					(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
					 errmsg("invalid input syntax for type %s: \"%s\"",
							"circle", str)));
	}

	if (*s != '\0')
		ereturn(escontext, (Datum) 0,
				(errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
				 errmsg("invalid input syntax for type %s: \"%s\"",
						"circle", str)));

	PG_RETURN_CIRCLE_P(circle);
}

/*		circle_out		-		convert a circle to external form.
 */
Datum
circle_out(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	StringInfoData str;

	initStringInfo(&str);

	appendStringInfoChar(&str, LDELIM_C);
	appendStringInfoChar(&str, LDELIM);
	pair_encode(circle->center.x, circle->center.y, &str);
	appendStringInfoChar(&str, RDELIM);
	appendStringInfoChar(&str, DELIM);
	single_encode(circle->radius, &str);
	appendStringInfoChar(&str, RDELIM_C);

	PG_RETURN_CSTRING(str.data);
}

/*
 *		circle_recv			- converts external binary format to circle
 */
Datum
circle_recv(PG_FUNCTION_ARGS)
{
	StringInfo	buf = (StringInfo) PG_GETARG_POINTER(0);
	CIRCLE	   *circle;

	circle = (CIRCLE *) palloc(sizeof(CIRCLE));

	circle->center.x = pq_getmsgfloat8(buf);
	circle->center.y = pq_getmsgfloat8(buf);
	circle->radius = pq_getmsgfloat8(buf);

	/* We have to accept NaN. */
	if (circle->radius < 0.0)
		ereport(ERROR,
				(errcode(ERRCODE_INVALID_BINARY_REPRESENTATION),
				 errmsg("invalid radius in external \"circle\" value")));

	PG_RETURN_CIRCLE_P(circle);
}

/*
 *		circle_send			- converts circle to binary format
 */
Datum
circle_send(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	StringInfoData buf;

	pq_begintypsend(&buf);
	pq_sendfloat8(&buf, circle->center.x);
	pq_sendfloat8(&buf, circle->center.y);
	pq_sendfloat8(&buf, circle->radius);
	PG_RETURN_BYTEA_P(pq_endtypsend(&buf));
}


/*----------------------------------------------------------
 *	Relational operators for CIRCLEs.
 *		<, >, <=, >=, and == are based on circle area.
 *---------------------------------------------------------*/

/*		circles identical?
 *
 * We consider NaNs values to be equal to each other to let those circles
 * to be found.
 */
Datum
circle_same(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(((isnan(circle1->radius) && isnan(circle2->radius)) ||
					FPeq(circle1->radius, circle2->radius)) &&
				   point_eq_point(&circle1->center, &circle2->center));
}

/*		circle_overlap	-		does circle1 overlap circle2?
 */
Datum
circle_overlap(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPle(point_dt(&circle1->center, &circle2->center),
						float8_pl(circle1->radius, circle2->radius)));
}

/*		circle_overleft -		is the right edge of circle1 at or left of
 *								the right edge of circle2?
 */
Datum
circle_overleft(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPle(float8_pl(circle1->center.x, circle1->radius),
						float8_pl(circle2->center.x, circle2->radius)));
}

/*		circle_left		-		is circle1 strictly left of circle2?
 */
Datum
circle_left(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPlt(float8_pl(circle1->center.x, circle1->radius),
						float8_mi(circle2->center.x, circle2->radius)));
}

/*		circle_right	-		is circle1 strictly right of circle2?
 */
Datum
circle_right(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPgt(float8_mi(circle1->center.x, circle1->radius),
						float8_pl(circle2->center.x, circle2->radius)));
}

/*		circle_overright	-	is the left edge of circle1 at or right of
 *								the left edge of circle2?
 */
Datum
circle_overright(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPge(float8_mi(circle1->center.x, circle1->radius),
						float8_mi(circle2->center.x, circle2->radius)));
}

/*		circle_contained		-		is circle1 contained by circle2?
 */
Datum
circle_contained(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPle(point_dt(&circle1->center, &circle2->center),
						float8_mi(circle2->radius, circle1->radius)));
}

/*		circle_contain	-		does circle1 contain circle2?
 */
Datum
circle_contain(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPle(point_dt(&circle1->center, &circle2->center),
						float8_mi(circle1->radius, circle2->radius)));
}


/*		circle_below		-		is circle1 strictly below circle2?
 */
Datum
circle_below(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPlt(float8_pl(circle1->center.y, circle1->radius),
						float8_mi(circle2->center.y, circle2->radius)));
}

/*		circle_above	-		is circle1 strictly above circle2?
 */
Datum
circle_above(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPgt(float8_mi(circle1->center.y, circle1->radius),
						float8_pl(circle2->center.y, circle2->radius)));
}

/*		circle_overbelow -		is the upper edge of circle1 at or below
 *								the upper edge of circle2?
 */
Datum
circle_overbelow(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPle(float8_pl(circle1->center.y, circle1->radius),
						float8_pl(circle2->center.y, circle2->radius)));
}

/*		circle_overabove	-	is the lower edge of circle1 at or above
 *								the lower edge of circle2?
 */
Datum
circle_overabove(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPge(float8_mi(circle1->center.y, circle1->radius),
						float8_mi(circle2->center.y, circle2->radius)));
}


/*		circle_relop	-		is area(circle1) relop area(circle2), within
 *								our accuracy constraint?
 */
Datum
circle_eq(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPeq(circle_ar(circle1), circle_ar(circle2)));
}

Datum
circle_ne(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPne(circle_ar(circle1), circle_ar(circle2)));
}

Datum
circle_lt(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPlt(circle_ar(circle1), circle_ar(circle2)));
}

Datum
circle_gt(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPgt(circle_ar(circle1), circle_ar(circle2)));
}

Datum
circle_le(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPle(circle_ar(circle1), circle_ar(circle2)));
}

Datum
circle_ge(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);

	PG_RETURN_BOOL(FPge(circle_ar(circle1), circle_ar(circle2)));
}


/*----------------------------------------------------------
 *	"Arithmetic" operators on circles.
 *---------------------------------------------------------*/

/* circle_add_pt()
 * Translation operator.
 */
Datum
circle_add_pt(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	CIRCLE	   *result;

	result = (CIRCLE *) palloc(sizeof(CIRCLE));

	point_add_point(&result->center, &circle->center, point);
	result->radius = circle->radius;

	PG_RETURN_CIRCLE_P(result);
}

Datum
circle_sub_pt(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	CIRCLE	   *result;

	result = (CIRCLE *) palloc(sizeof(CIRCLE));

	point_sub_point(&result->center, &circle->center, point);
	result->radius = circle->radius;

	PG_RETURN_CIRCLE_P(result);
}


/* circle_mul_pt()
 * Rotation and scaling operators.
 */
Datum
circle_mul_pt(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	CIRCLE	   *result;

	result = (CIRCLE *) palloc(sizeof(CIRCLE));

	point_mul_point(&result->center, &circle->center, point);
	result->radius = float8_mul(circle->radius, HYPOT(point->x, point->y));

	PG_RETURN_CIRCLE_P(result);
}

Datum
circle_div_pt(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	CIRCLE	   *result;

	result = (CIRCLE *) palloc(sizeof(CIRCLE));

	point_div_point(&result->center, &circle->center, point);
	result->radius = float8_div(circle->radius, HYPOT(point->x, point->y));

	PG_RETURN_CIRCLE_P(result);
}


/*		circle_area		-		returns the area of the circle.
 */
Datum
circle_area(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);

	PG_RETURN_FLOAT8(circle_ar(circle));
}


/*		circle_diameter -		returns the diameter of the circle.
 */
Datum
circle_diameter(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);

	PG_RETURN_FLOAT8(float8_mul(circle->radius, 2.0));
}


/*		circle_radius	-		returns the radius of the circle.
 */
Datum
circle_radius(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);

	PG_RETURN_FLOAT8(circle->radius);
}


/*		circle_distance -		returns the distance between
 *								  two circles.
 */
Datum
circle_distance(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle1 = PG_GETARG_CIRCLE_P(0);
	CIRCLE	   *circle2 = PG_GETARG_CIRCLE_P(1);
	float8		result;

	result = float8_mi(point_dt(&circle1->center, &circle2->center),
					   float8_pl(circle1->radius, circle2->radius));
	if (result < 0.0)
		result = 0.0;

	PG_RETURN_FLOAT8(result);
}


Datum
circle_contain_pt(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	float8		d;

	d = point_dt(&circle->center, point);
	PG_RETURN_BOOL(d <= circle->radius);
}


Datum
pt_contained_circle(PG_FUNCTION_ARGS)
{
	Point	   *point = PG_GETARG_POINT_P(0);
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(1);
	float8		d;

	d = point_dt(&circle->center, point);
	PG_RETURN_BOOL(d <= circle->radius);
}


/*		dist_pc -		returns the distance between
 *						  a point and a circle.
 */
Datum
dist_pc(PG_FUNCTION_ARGS)
{
	Point	   *point = PG_GETARG_POINT_P(0);
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(1);
	float8		result;

	result = float8_mi(point_dt(point, &circle->center),
					   circle->radius);
	if (result < 0.0)
		result = 0.0;

	PG_RETURN_FLOAT8(result);
}

/*
 * Distance from a circle to a point
 */
Datum
dist_cpoint(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	Point	   *point = PG_GETARG_POINT_P(1);
	float8		result;

	result = float8_mi(point_dt(point, &circle->center), circle->radius);
	if (result < 0.0)
		result = 0.0;

	PG_RETURN_FLOAT8(result);
}

/*		circle_center	-		returns the center point of the circle.
 */
Datum
circle_center(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	Point	   *result;

	result = (Point *) palloc(sizeof(Point));
	result->x = circle->center.x;
	result->y = circle->center.y;

	PG_RETURN_POINT_P(result);
}


/*		circle_ar		-		returns the area of the circle.
 */
static float8
circle_ar(CIRCLE *circle)
{
	return float8_mul(float8_mul(circle->radius, circle->radius), M_PI);
}


/*----------------------------------------------------------
 *	Conversion operators.
 *---------------------------------------------------------*/

Datum
cr_circle(PG_FUNCTION_ARGS)
{
	Point	   *center = PG_GETARG_POINT_P(0);
	float8		radius = PG_GETARG_FLOAT8(1);
	CIRCLE	   *result;

	result = (CIRCLE *) palloc(sizeof(CIRCLE));

	result->center.x = center->x;
	result->center.y = center->y;
	result->radius = radius;

	PG_RETURN_CIRCLE_P(result);
}

Datum
circle_box(PG_FUNCTION_ARGS)
{
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(0);
	BOX		   *box;
	float8		delta;

	box = (BOX *) palloc(sizeof(BOX));

	delta = float8_div(circle->radius, sqrt(2.0));

	box->high.x = float8_pl(circle->center.x, delta);
	box->low.x = float8_mi(circle->center.x, delta);
	box->high.y = float8_pl(circle->center.y, delta);
	box->low.y = float8_mi(circle->center.y, delta);

	PG_RETURN_BOX_P(box);
}

/* box_circle()
 * Convert a box to a circle.
 */
Datum
box_circle(PG_FUNCTION_ARGS)
{
	BOX		   *box = PG_GETARG_BOX_P(0);
	CIRCLE	   *circle;

	circle = (CIRCLE *) palloc(sizeof(CIRCLE));

	circle->center.x = float8_div(float8_pl(box->high.x, box->low.x), 2.0);
	circle->center.y = float8_div(float8_pl(box->high.y, box->low.y), 2.0);

	circle->radius = point_dt(&circle->center, &box->high);

	PG_RETURN_CIRCLE_P(circle);
}


Datum
circle_poly(PG_FUNCTION_ARGS)
{
	int32		npts = PG_GETARG_INT32(0);
	CIRCLE	   *circle = PG_GETARG_CIRCLE_P(1);
	POLYGON    *poly;
	int			base_size,
				size;
	int			i;
	float8		angle;
	float8		anglestep;

	if (FPzero(circle->radius))
		ereport(ERROR,
				(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
				 errmsg("cannot convert circle with radius zero to polygon")));

	if (npts < 2)
		ereport(ERROR,
				(errcode(ERRCODE_INVALID_PARAMETER_VALUE),
				 errmsg("must request at least 2 points")));

	base_size = sizeof(poly->p[0]) * npts;
	size = offsetof(POLYGON, p) + base_size;

	/* Check for integer overflow */
	if (base_size / npts != sizeof(poly->p[0]) || size <= base_size)
		ereport(ERROR,
				(errcode(ERRCODE_PROGRAM_LIMIT_EXCEEDED),
				 errmsg("too many points requested")));

	poly = (POLYGON *) palloc0(size);	/* zero any holes */
	SET_VARSIZE(poly, size);
	poly->npts = npts;

	anglestep = float8_div(2.0 * M_PI, npts);

	for (i = 0; i < npts; i++)
	{
		angle = float8_mul(anglestep, i);

		poly->p[i].x = float8_mi(circle->center.x,
								 float8_mul(circle->radius, cos(angle)));
		poly->p[i].y = float8_pl(circle->center.y,
								 float8_mul(circle->radius, sin(angle)));
	}

	make_bound_box(poly);

	PG_RETURN_POLYGON_P(poly);
}

/*
 * Convert polygon to circle
 *
 * The result must be preallocated.
 *
 * XXX This algorithm should use weighted means of line segments
 *	rather than straight average values of points - tgl 97/01/21.
 */
static void
poly_to_circle(CIRCLE *result, POLYGON *poly)
{
	int			i;

	Assert(poly->npts > 0);

	result->center.x = 0;
	result->center.y = 0;
	result->radius = 0;

	for (i = 0; i < poly->npts; i++)
		point_add_point(&result->center, &result->center, &poly->p[i]);
	result->center.x = float8_div(result->center.x, poly->npts);
	result->center.y = float8_div(result->center.y, poly->npts);

	for (i = 0; i < poly->npts; i++)
		result->radius = float8_pl(result->radius,
								   point_dt(&poly->p[i], &result->center));
	result->radius = float8_div(result->radius, poly->npts);
}

Datum
poly_circle(PG_FUNCTION_ARGS)
{
	POLYGON    *poly = PG_GETARG_POLYGON_P(0);
	CIRCLE	   *result;

	result = (CIRCLE *) palloc(sizeof(CIRCLE));

	poly_to_circle(result, poly);

	PG_RETURN_CIRCLE_P(result);
}


/***********************************************************************
 **
 **		Private routines for multiple types.
 **
 ***********************************************************************/

/*
 *	Test to see if the point is inside the polygon, returns 1/0, or 2 if
 *	the point is on the polygon.
 *	Code adapted but not copied from integer-based routines in WN: A
 *	Server for the HTTP
 *	version 1.15.1, file wn/image.c
 *	http://hopf.math.northwestern.edu/index.html
 *	Description of algorithm:  http://www.linuxjournal.com/article/2197
 *							   http://www.linuxjournal.com/article/2029
 */

#define POINT_ON_POLYGON INT_MAX

static int
point_inside(Point *p, int npts, Point *plist)
{
	float8		x0,
				y0;
	float8		prev_x,
				prev_y;
	int			i = 0;
	float8		x,
				y;
	int			cross,
				total_cross = 0;

	Assert(npts > 0);

	/* compute first polygon point relative to single point */
	x0 = float8_mi(plist[0].x, p->x);
	y0 = float8_mi(plist[0].y, p->y);

	prev_x = x0;
	prev_y = y0;
	/* loop over polygon points and aggregate total_cross */
	for (i = 1; i < npts; i++)
	{
		/* compute next polygon point relative to single point */
		x = float8_mi(plist[i].x, p->x);
		y = float8_mi(plist[i].y, p->y);

		/* compute previous to current point crossing */
		if ((cross = lseg_crossing(x, y, prev_x, prev_y)) == POINT_ON_POLYGON)
			return 2;
		total_cross += cross;

		prev_x = x;
		prev_y = y;
	}

	/* now do the first point */
	if ((cross = lseg_crossing(x0, y0, prev_x, prev_y)) == POINT_ON_POLYGON)
		return 2;
	total_cross += cross;

	if (total_cross != 0)
		return 1;
	return 0;
}


/* lseg_crossing()
 * Returns +/-2 if line segment crosses the positive X-axis in a +/- direction.
 * Returns +/-1 if one point is on the positive X-axis.
 * Returns 0 if both points are on the positive X-axis, or there is no crossing.
 * Returns POINT_ON_POLYGON if the segment contains (0,0).
 * Wow, that is one confusing API, but it is used above, and when summed,
 * can tell is if a point is in a polygon.
 */

static int
lseg_crossing(float8 x, float8 y, float8 prev_x, float8 prev_y)
{
	float8		z;
	int			y_sign;

	if (FPzero(y))
	{							/* y == 0, on X axis */
		if (FPzero(x))			/* (x,y) is (0,0)? */
			return POINT_ON_POLYGON;
		else if (FPgt(x, 0))
		{						/* x > 0 */
			if (FPzero(prev_y)) /* y and prev_y are zero */
				/* prev_x > 0? */
				return FPgt(prev_x, 0.0) ? 0 : POINT_ON_POLYGON;
			return FPlt(prev_y, 0.0) ? 1 : -1;
		}
		else
		{						/* x < 0, x not on positive X axis */
			if (FPzero(prev_y))
				/* prev_x < 0? */
				return FPlt(prev_x, 0.0) ? 0 : POINT_ON_POLYGON;
			return 0;
		}
	}
	else
	{							/* y != 0 */
		/* compute y crossing direction from previous point */
		y_sign = FPgt(y, 0.0) ? 1 : -1;

		if (FPzero(prev_y))
			/* previous point was on X axis, so new point is either off or on */
			return FPlt(prev_x, 0.0) ? 0 : y_sign;
		else if ((y_sign < 0 && FPlt(prev_y, 0.0)) ||
				 (y_sign > 0 && FPgt(prev_y, 0.0)))
			/* both above or below X axis */
			return 0;			/* same sign */
		else
		{						/* y and prev_y cross X-axis */
			if (FPge(x, 0.0) && FPgt(prev_x, 0.0))
				/* both non-negative so cross positive X-axis */
				return 2 * y_sign;
			if (FPlt(x, 0.0) && FPle(prev_x, 0.0))
				/* both non-positive so do not cross positive X-axis */
				return 0;

			/* x and y cross axes, see URL above point_inside() */
			z = float8_mi(float8_mul(float8_mi(x, prev_x), y),
						  float8_mul(float8_mi(y, prev_y), x));
			if (FPzero(z))
				return POINT_ON_POLYGON;
			if ((y_sign < 0 && FPlt(z, 0.0)) ||
				(y_sign > 0 && FPgt(z, 0.0)))
				return 0;
			return 2 * y_sign;
		}
	}
}


static bool
plist_same(int npts, Point *p1, Point *p2)
{
	int			i,
				ii,
				j;

	/* find match for first point */
	for (i = 0; i < npts; i++)
	{
		if (point_eq_point(&p2[i], &p1[0]))
		{

			/* match found? then look forward through remaining points */
			for (ii = 1, j = i + 1; ii < npts; ii++, j++)
			{
				if (j >= npts)
					j = 0;
				if (!point_eq_point(&p2[j], &p1[ii]))
					break;
			}
			if (ii == npts)
				return true;

			/* match not found forwards? then look backwards */
			for (ii = 1, j = i - 1; ii < npts; ii++, j--)
			{
				if (j < 0)
					j = (npts - 1);
				if (!point_eq_point(&p2[j], &p1[ii]))
					break;
			}
			if (ii == npts)
				return true;
		}
	}

	return false;
}


/*-------------------------------------------------------------------------
 * Determine the hypotenuse.
 *
 * If required, x and y are swapped to make x the larger number. The
 * traditional formula of x^2+y^2 is rearranged to factor x outside the
 * sqrt. This allows computation of the hypotenuse for significantly
 * larger values, and with a higher precision than when using the naive
 * formula.  In particular, this cannot overflow unless the final result
 * would be out-of-range.
 *
 * sqrt( x^2 + y^2 ) = sqrt( x^2( 1 + y^2/x^2) )
 *					 = x * sqrt( 1 + y^2/x^2 )
 *					 = x * sqrt( 1 + y/x * y/x )
 *
 * It is expected that this routine will eventually be replaced with the
 * C99 hypot() function.
 *
 * This implementation conforms to IEEE Std 1003.1 and GLIBC, in that the
 * case of hypot(inf,nan) results in INF, and not NAN.
 *-----------------------------------------------------------------------
 */
float8
pg_hypot(float8 x, float8 y)
{
	float8		yx,
				result;

	/* Handle INF and NaN properly */
	if (isinf(x) || isinf(y))
		return get_float8_infinity();

	if (isnan(x) || isnan(y))
		return get_float8_nan();

	/* Else, drop any minus signs */
	x = fabs(x);
	y = fabs(y);

	/* Swap x and y if needed to make x the larger one */
	if (x < y)
	{
		float8		temp = x;

		x = y;
		y = temp;
	}

	/*
	 * If y is zero, the hypotenuse is x.  This test saves a few cycles in
	 * such cases, but more importantly it also protects against
	 * divide-by-zero errors, since now x >= y.
	 */
	if (y == 0.0)
		return x;

	/* Determine the hypotenuse */
	yx = y / x;
	result = x * sqrt(1.0 + (yx * yx));

	if (unlikely(isinf(result)))
		float_overflow_error();
	if (unlikely(result == 0.0))
		float_underflow_error();

	return result;
}