// Filename: triangulator.cxx
// Created by:  drose (18Jan07)
//
////////////////////////////////////////////////////////////////////
//
// PANDA 3D SOFTWARE
// Copyright (c) 2001 - 2004, Disney Enterprises, Inc.  All rights reserved
//
// All use of this software is subject to the terms of the Panda 3d
// Software license.  You should have received a copy of this license
// along with this source code; you will also find a current copy of
// the license at http://etc.cmu.edu/panda3d/docs/license/ .
//
// To contact the maintainers of this program write to
// panda3d-general@lists.sourceforge.net .
//
////////////////////////////////////////////////////////////////////

#include "triangulator.h"
#include "randomizer.h"

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::Constructor
//       Access: Published
//  Description: 
////////////////////////////////////////////////////////////////////
Triangulator::
Triangulator() {
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::clear
//       Access: Published
//  Description: Removes all vertices and polygon specifications from
//               the Triangulator, and prepares it to start over.
////////////////////////////////////////////////////////////////////
void Triangulator::
clear() {
  _vertices.clear();
  clear_polygon();
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::add_vertex
//       Access: Published
//  Description: Adds a new vertex to the vertex pool.  Returns the
//               vertex index number.
////////////////////////////////////////////////////////////////////
int Triangulator::
add_vertex(const LPoint2d &point) {
  int index = (int)_vertices.size();
  _vertices.push_back(point);
  return index;
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::clear_polygon
//       Access: Published
//  Description: Removes the current polygon definition (and its set
//               of holes), but does not clear the vertex pool.
////////////////////////////////////////////////////////////////////
void Triangulator::
clear_polygon() {
  _polygon.clear();
  _holes.clear();
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::add_polygon_vertex
//       Access: Published
//  Description: Adds the next consecutive vertex of the polygon.
//               This vertex should index into the vertex pool
//               established by repeated calls to add_vertex().
//
//               The vertices may be listed in either clockwise or
//               counterclockwise order.  Vertices should not be
//               repeated.  In particular, do not repeat the first
//               vertex at the end.
////////////////////////////////////////////////////////////////////
void Triangulator::
add_polygon_vertex(int index) {
  _polygon.push_back(index);
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::begin_hole
//       Access: Published
//  Description: Finishes the previous hole, if any, and prepares to
//               add a new hole.
////////////////////////////////////////////////////////////////////
void Triangulator::
begin_hole() {
  _holes.push_back(vector_int());
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::add_hole_vertex
//       Access: Published
//  Description: Adds the next consecutive vertex of the current hole.
//               This vertex should index into the vertex pool
//               established by repeated calls to add_vertex().
//
//               The vertices may be listed in either clockwise or
//               counterclockwise order.  Vertices should not be
//               repeated.
////////////////////////////////////////////////////////////////////
void Triangulator::
add_hole_vertex(int index) {
  nassertv(!_holes.empty());
  _holes.back().push_back(index);
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::triangulate
//       Access: Published
//  Description: Does the work of triangulating the specified polygon.
//               After this call, you may retrieve the new triangles
//               one at a time by iterating through
//               get_triangle_v0/1/2().
////////////////////////////////////////////////////////////////////
void Triangulator::
triangulate() {
  _result.clear();

  // Set up the list of segments.
  seg.clear();
  seg.push_back(segment_t());  // we don't use the first entry.
  make_segment(_polygon, true);

  Holes::const_iterator hi;
  for (hi = _holes.begin(); hi != _holes.end(); ++hi) {
    make_segment(*hi, false);
  }

  // Shuffle the segment index.
  int num_segments = (int)seg.size() - 1;
  permute.reserve(num_segments);
  int i;
  for (i = 0; i < num_segments; ++i) {
    permute.push_back(i + 1);
  }

  // Actually, I'm not sure why we should shuffle the index.  That
  // makes the result non-deterministic, and isn't one order--for
  // instance, the initial order--as good as any other?
  /*
  Randomizer randomizer;
  for (i = 0; i < num_segments; ++i) {
    int j = randomizer.random_int(num_segments);
    nassertv(j >= 0 && j < num_segments);
    int t = permute[i];
    permute[i] = permute[j];
    permute[j] = t;
  }
  */
  choose_idx = 0;

  //  cerr << "got " << num_segments << " segments\n";
  /*
  for (i = 1; i < (int)seg.size(); ++i) {
    segment_t &s = seg[i];
    printf("  %d. (%g %g), (%g %g)\n", i, s.v0.x, s.v0.y, s.v1.x, s.v1.y);
    printf("    root0 = %d, root1 = %d\n", s.root0, s.root1);
    printf("    next = %d, prev = %d\n", s.next, s.prev);
  }
  */

  construct_trapezoids(num_segments);

  //  cerr << "got " << tr.size() - 1 << " trapezoids\n";
  /*
  for (i = 1; i < (int)tr.size(); ++i) {
    trap_t &t = tr[i];
    cerr << "  " << i << ". state = " << t.state << "\n";
    cerr << "    lseg = " << t.lseg << " rseg = " << t.rseg
         << "\n";
    cerr << "    hi = " << t.hi.x << " " << t.hi.y << " lo = "
         << t.lo.x << " " << t.lo.y << "\n";
  }
  */

  int nmonpoly = monotonate_trapezoids(num_segments);

  //  cerr << "got " << nmonpoly << " monotone polygons\n";

  triangulate_monotone_polygons(num_segments, nmonpoly);

  /*
  Result::iterator ri;
  for (ri = _result.begin(); ri != _result.end(); ++ri) {
    cerr << "tri: " << (*ri)._v0 << " " << (*ri)._v1 << " "
         << (*ri)._v2 << "\n";
  }
  */
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::get_num_triangles
//       Access: Published
//  Description: Returns the number of triangles generated by the
//               previous call to triangulate().
////////////////////////////////////////////////////////////////////
int Triangulator::
get_num_triangles() const {
  return _result.size();
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::get_triangle_v0
//       Access: Published
//  Description: Returns vertex 0 of the nth triangle generated by the
//               previous call to triangulate().
//
//               This is a zero-based index into the vertices added by
//               repeated calls to add_vertex().
////////////////////////////////////////////////////////////////////
int Triangulator::
get_triangle_v0(int n) const {
  nassertr(n >= 0 && n < (int)_result.size(), -1);
  return _result[n]._v0;
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::get_triangle_v1
//       Access: Published
//  Description: Returns vertex 1 of the nth triangle generated by the
//               previous call to triangulate().
//
//               This is a zero-based index into the vertices added by
//               repeated calls to add_vertex().
////////////////////////////////////////////////////////////////////
int Triangulator::
get_triangle_v1(int n) const {
  nassertr(n >= 0 && n < (int)_result.size(), -1);
  return _result[n]._v1;
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::get_triangle_v2
//       Access: Published
//  Description: Returns vertex 2 of the nth triangle generated by the
//               previous call to triangulate().
//
//               This is a zero-based index into the vertices added by
//               repeated calls to add_vertex().
////////////////////////////////////////////////////////////////////
int Triangulator::
get_triangle_v2(int n) const {
  nassertr(n >= 0 && n < (int)_result.size(), -1);
  return _result[n]._v2;
}


// The remainder of the code in this file is adapted more or less from
// the C code published with the referenced paper.

#define T_X     1
#define T_Y     2
#define T_SINK  3


#define FIRSTPT 1		/* checking whether pt. is inserted */ 
#define LASTPT  2


#define REALLY_BIG 1<<30
#define C_EPS 1.0e-7		/* tolerance value: Used for making */
				/* all decisions about collinearity or */
				/* left/right of segment. Decrease */
				/* this value if the input points are */
				/* spaced very close together */


#define S_LEFT 1		/* for merge-direction */
#define S_RIGHT 2


#define ST_VALID 1		/* for trapezium state */
#define ST_INVALID 2


#define SP_SIMPLE_LRUP 1	/* for splitting trapezoids */
#define SP_SIMPLE_LRDN 2
#define SP_2UP_2DN     3
#define SP_2UP_LEFT    4
#define SP_2UP_RIGHT   5
#define SP_2DN_LEFT    6
#define SP_2DN_RIGHT   7
#define SP_NOSPLIT    -1	

#define TR_FROM_UP 1		/* for traverse-direction */
#define TR_FROM_DN 2

#define TRI_LHS 1
#define TRI_RHS 2


#define CROSS(v0, v1, v2) (((v1).x - (v0).x)*((v2).y - (v0).y) - \
			   ((v1).y - (v0).y)*((v2).x - (v0).x))

#define DOT(v0, v1) ((v0).x * (v1).x + (v0).y * (v1).y)

#define FP_EQUAL(s, t) (fabs(s - t) <= C_EPS)


#define CROSS_SINE(v0, v1) ((v0).x * (v1).y - (v1).x * (v0).y)
#define LENGTH(v0) (sqrt((v0).x * (v0).x + (v0).y * (v0).y))


////////////////////////////////////////////////////////////////////
//     Function: Triangulator::check_left_winding
//       Access: Private
//  Description: Returns true if the list of vertices is
//               counter-clockwise, false if it is clockwise.
////////////////////////////////////////////////////////////////////
bool Triangulator::
check_left_winding(const vector_int &range) const {
  // We do this by computing the polygon's signed area.  If it comes
  // out negative, the polygon is right-winding.

  double area = 0.0;
  size_t j = range.size() - 1;
  for (size_t i = 0; i < range.size(); ++i) {
    const LPoint2d &p0 = _vertices[range[j]];
    const LPoint2d &p1 = _vertices[range[i]];
    area += p0[0] * p1[1] - p0[1] * p1[0];
    j = i;
  }

  return area >= 0.0;
}

////////////////////////////////////////////////////////////////////
//     Function: Triangulator::make_segment
//       Access: Private
//  Description: Converts a linear list of integer vertices to a list
//               of segment_t.  If want_left_winding is true, the list
//               is reversed if necessary to make it left-winding;
//               otherwise, it is reversed to make it right-winding.
////////////////////////////////////////////////////////////////////
void Triangulator::
make_segment(const vector_int &range, bool want_left_winding) {
  int num_points = (int)range.size();
  nassertv(num_points >= 2);

  int first = (int)seg.size();
  int last = first + num_points - 1;

  if (want_left_winding == check_left_winding(range)) {
    // Keep it in its natural order.
    int first = (int)seg.size();
    int last = first + num_points - 1;
    
    seg.push_back(segment_t(this, range[0], range[1],
                            last, first + 1));
    
    for (int i = 1; i < num_points - 1; ++i) {
      seg.push_back(segment_t(this, range[i], range[i + 1],
                              first + i - 1, first + i + 1));
    }
    
    seg.push_back(segment_t(this, range[num_points - 1], range[0],
                            last - 1, first));

  } else {
    // Reverse it.
    seg.push_back(segment_t(this, range[0], range[num_points - 1],
                            last, first + 1));
    
    for (int i = 1; i < num_points - 1; ++i) {
      seg.push_back(segment_t(this, range[num_points - i], range[num_points - i - 1],
                              first + i - 1, first + i + 1));
    }
    
    seg.push_back(segment_t(this, range[1], range[0],
                            last - 1, first));

  }
}
  
/* Return the next segment in the generated random ordering of all the */
/* segments in S */
int Triangulator::
choose_segment() {
  nassertr(choose_idx < (int)permute.size(), 0);
  //  segment_t &s = seg[permute[choose_idx]];
  //  cerr << "choose_segment " << permute[choose_idx] << ": " << s.v0.x << ", " << s.v0.y << " to " << s.v1.x << ", " << s.v1.y << "\n";
  return permute[choose_idx++];
}

double Triangulator::
math_log2(double v) {
  static const double log2 = log(2.0);
  return log(v) / log2;
}

/* Get log*n for given n */
int Triangulator::
math_logstar_n(int n) {
  int i;
  double v;
  
  for (i = 0, v = (double) n; v >= 1; i++)
    v = math_log2(v);
  
  return (i - 1);
}
  

int Triangulator::
math_N(int n, int h) {
  int i;
  double v;

  for (i = 0, v = (int) n; i < h; i++)
    v = math_log2(v);
  
  return (int) ceil((double) 1.0*n/v);
}


/* Return a new node to be added into the query tree */
int Triangulator::newnode() {
  int index = (int)qs.size();
  qs.push_back(node_t());
  //  cerr << "creating new node " << index << "\n";
  return index;
}

/* Return a free trapezoid */
int Triangulator::newtrap() {
  int tr_idx = (int)tr.size();
  tr.push_back(trap_t());
  tr[tr_idx].lseg = -1;
  tr[tr_idx].rseg = -1;
  tr[tr_idx].state = ST_VALID;
  //  cerr << "creating new trapezoid " << tr_idx << "\n";
  return tr_idx;
}


/* Return the maximum of the two points into the yval structure */
int Triangulator::_max(point_t *yval, point_t *v0, point_t *v1) {
  if (v0->y > v1->y + C_EPS)
    *yval = *v0;
  else if (FP_EQUAL(v0->y, v1->y))
    {
      if (v0->x > v1->x + C_EPS)
	*yval = *v0;
      else
	*yval = *v1;
    }
  else
    *yval = *v1;
  
  return 0;
}


/* Return the minimum of the two points into the yval structure */
int Triangulator::_min(point_t *yval, point_t *v0, point_t *v1) {
  if (v0->y < v1->y - C_EPS)
    *yval = *v0;
  else if (FP_EQUAL(v0->y, v1->y))
    {
      if (v0->x < v1->x)
	*yval = *v0;
      else
	*yval = *v1;
    }
  else
    *yval = *v1;
  
  return 0;
}


int Triangulator::
_greater_than(point_t *v0, point_t *v1) {
  if (v0->y > v1->y + C_EPS)
    return true;
  else if (v0->y < v1->y - C_EPS)
    return false;
  else
    return (v0->x > v1->x);
}


int Triangulator::
_equal_to(point_t *v0, point_t *v1) {
  return (FP_EQUAL(v0->y, v1->y) && FP_EQUAL(v0->x, v1->x));
}

int Triangulator::
_greater_than_equal_to(point_t *v0, point_t *v1) {
  if (v0->y > v1->y + C_EPS)
    return true;
  else if (v0->y < v1->y - C_EPS)
    return false;
  else
    return (v0->x >= v1->x);
}

int Triangulator::
_less_than(point_t *v0, point_t *v1) {
  if (v0->y < v1->y - C_EPS)
    return true;
  else if (v0->y > v1->y + C_EPS)
    return false;
  else
    return (v0->x < v1->x);
}


/* Initilialise the query structure (Q) and the trapezoid table (T) 
 * when the first segment is added to start the trapezoidation. The
 * query-tree starts out with 4 trapezoids, one S-node and 2 Y-nodes
 *    
 *                4
 *   -----------------------------------
 *  		  \
 *  	1	   \        2
 *  		    \
 *   -----------------------------------
 *                3
 */

int Triangulator::
init_query_structure(int segnum) {
  int i1, i2, i3, i4, i5, i6, i7, root;
  int t1, t2, t3, t4;
  segment_t *s = &seg[segnum];

  tr.clear();
  qs.clear();

  // We don't use the first elements.
  tr.push_back(trap_t());
  qs.push_back(node_t());

  i1 = newnode();
  qs[i1].nodetype = T_Y;
  _max(&qs[i1].yval, &s->v0, &s->v1); /* root */
  root = i1;

  i2 = newnode();
  qs[i1].right = i2;
  qs[i2].nodetype = T_SINK;
  qs[i2].parent = i1;

  i3 = newnode();
  qs[i1].left = i3;

  qs[i3].nodetype = T_Y;
  _min(&qs[i3].yval, &s->v0, &s->v1); /* root */
  qs[i3].parent = i1;
  
  i4 = newnode();
  qs[i3].left = i4;
  qs[i4].nodetype = T_SINK;
  qs[i4].parent = i3;
  
  i5 = newnode();
  qs[i3].right = i5;
  qs[i5].nodetype = T_X;
  qs[i5].segnum = segnum;
  qs[i5].parent = i3;
  
  i6 = newnode();
  qs[i5].left = i6;
  qs[i6].nodetype = T_SINK;
  qs[i6].parent = i5;

  i7 = newnode();
  qs[i5].right = i7;
  qs[i7].nodetype = T_SINK;
  qs[i7].parent = i5;

  t1 = newtrap();		/* middle left */
  t2 = newtrap();		/* middle right */
  t3 = newtrap();		/* bottom-most */
  t4 = newtrap();		/* topmost */

  tr[t4].lo = qs[i1].yval;
  tr[t2].hi = qs[i1].yval;
  tr[t1].hi = qs[i1].yval;
  tr[t3].hi = qs[i3].yval;
  tr[t2].lo = qs[i3].yval;
  tr[t1].lo = qs[i3].yval;
  tr[t4].hi.y = (double) (REALLY_BIG);
  tr[t4].hi.x = (double) (REALLY_BIG);
  tr[t3].lo.y = (double) -1* (REALLY_BIG);
  tr[t3].lo.x = (double) -1* (REALLY_BIG);
  tr[t2].lseg = segnum;
  tr[t1].rseg = segnum;
  tr[t2].u0 = t4;
  tr[t1].u0 = t4;
  tr[t2].d0 = t3;
  tr[t1].d0 = t3;
  tr[t3].u0 = t1;
  tr[t4].d0 = t1;
  tr[t3].u1 = t2;
  tr[t4].d1 = t2;
  
  tr[t1].sink = i6;
  tr[t2].sink = i7;
  tr[t3].sink = i4;
  tr[t4].sink = i2;

  tr[t2].state = ST_VALID;
  tr[t1].state = ST_VALID;
  tr[t4].state = ST_VALID;
  tr[t3].state = ST_VALID;

  qs[i2].trnum = t4;
  qs[i4].trnum = t3;
  qs[i6].trnum = t1;
  qs[i7].trnum = t2;

  s->is_inserted = true;
  return root;
}


/* Retun true if the vertex v is to the left of line segment no.
 * segnum. Takes care of the degenerate cases when both the vertices
 * have the same y--cood, etc.
 */

int Triangulator::
is_left_of(int segnum, point_t *v) {
  segment_t *s = &seg[segnum];
  double area;
  
  if (_greater_than(&s->v1, &s->v0)) /* seg. going upwards */
    {
      if (FP_EQUAL(s->v1.y, v->y))
	{
	  if (v->x < s->v1.x)
	    area = 1.0;
	  else
	    area = -1.0;
	}
      else if (FP_EQUAL(s->v0.y, v->y))
	{
	  if (v->x < s->v0.x)
	    area = 1.0;
	  else
	    area = -1.0;
	}
      else
	area = CROSS(s->v0, s->v1, (*v));
    }
  else				/* v0 > v1 */
    {
      if (FP_EQUAL(s->v1.y, v->y))
	{
	  if (v->x < s->v1.x)
	    area = 1.0;
	  else
	    area = -1.0;
	}
      else if (FP_EQUAL(s->v0.y, v->y))
	{
	  if (v->x < s->v0.x)
	    area = 1.0;
	  else
	    area = -1.0;
	}
      else
	area = CROSS(s->v1, s->v0, (*v));
    }
  
  if (area > 0.0)
    return true;
  else 
    return false;
}



/* Returns true if the corresponding endpoint of the given segment is */
/* already inserted into the segment tree. Use the simple test of */
/* whether the segment which shares this endpoint is already inserted */

int Triangulator::
inserted(int segnum, int whichpt) {
  if (whichpt == FIRSTPT)
    return seg[seg[segnum].prev].is_inserted;
  else
    return seg[seg[segnum].next].is_inserted;
}

/* This is query routine which determines which trapezoid does the 
 * point v lie in. The return value is the trapezoid number. 
 */

int Triangulator::
locate_endpoint(point_t *v, point_t *vo, int r) {
  //  cerr << "locate_endpoint(" << v->x << " " << v->y << ", " << vo->x << " " << vo->y << ", " << r << ")\n";
  node_t *rptr = &qs[r];
  
  switch (rptr->nodetype)
    {
    case T_SINK:
      return rptr->trnum;
      
    case T_Y:
      if (_greater_than(v, &rptr->yval)) /* above */
	return locate_endpoint(v, vo, rptr->right);
      else if (_equal_to(v, &rptr->yval)) /* the point is already */
	{			          /* inserted. */
	  if (_greater_than(vo, &rptr->yval)) /* above */
	    return locate_endpoint(v, vo, rptr->right);
	  else 
	    return locate_endpoint(v, vo, rptr->left); /* below */	    
	}
      else
	return locate_endpoint(v, vo, rptr->left); /* below */

    case T_X:
      if (_equal_to(v, &seg[rptr->segnum].v0) || 
	       _equal_to(v, &seg[rptr->segnum].v1))
	{
	  if (FP_EQUAL(v->y, vo->y)) /* horizontal segment */
	    {
	      if (vo->x < v->x)
		return locate_endpoint(v, vo, rptr->left); /* left */
	      else
		return locate_endpoint(v, vo, rptr->right); /* right */
	    }

	  else if (is_left_of(rptr->segnum, vo))
	    return locate_endpoint(v, vo, rptr->left); /* left */
	  else
	    return locate_endpoint(v, vo, rptr->right); /* right */
	}
      else if (is_left_of(rptr->segnum, v))
	return locate_endpoint(v, vo, rptr->left); /* left */
      else
	return locate_endpoint(v, vo, rptr->right); /* right */	

    default:
      fprintf(stderr, "Haggu !!!!!\n");
      nassertr(false, -1);
      return -1;
    }
}


/* Thread in the segment into the existing trapezoidation. The 
 * limiting trapezoids are given by tfirst and tlast (which are the
 * trapezoids containing the two endpoints of the segment. Merges all
 * possible trapezoids which flank this segment and have been recently
 * divided because of its insertion
 */

int Triangulator::
merge_trapezoids(int segnum, int tfirst, int tlast, int side) {
  int t, tnext, cond;
  int ptnext;

  //  cerr << "merge_trapezoids(" << segnum << ", " << tfirst << ", " << tlast << ", " << side << ")\n";

  /* First merge polys on the LHS */
  t = tfirst;
  while ((t > 0) && _greater_than_equal_to(&tr[t].lo, &tr[tlast].lo))
    {
      if (side == S_LEFT)
	cond = ((((tnext = tr[t].d0) > 0) && (tr[tnext].rseg == segnum)) ||
		(((tnext = tr[t].d1) > 0) && (tr[tnext].rseg == segnum)));
      else
	cond = ((((tnext = tr[t].d0) > 0) && (tr[tnext].lseg == segnum)) ||
		(((tnext = tr[t].d1) > 0) && (tr[tnext].lseg == segnum)));
      
      if (cond)
	{
	  if ((tr[t].lseg == tr[tnext].lseg) &&
	      (tr[t].rseg == tr[tnext].rseg)) /* good neighbours */
	    {			              /* merge them */
	      /* Use the upper node as the new node i.e. t */
	      
	      ptnext = qs[tr[tnext].sink].parent;
	      
	      if (qs[ptnext].left == tr[tnext].sink)
		qs[ptnext].left = tr[t].sink;
	      else
		qs[ptnext].right = tr[t].sink;	/* redirect parent */
	      
	      
	      /* Change the upper neighbours of the lower trapezoids */
	      
	      if ((tr[t].d0 = tr[tnext].d0) > 0)
		if (tr[tr[t].d0].u0 == tnext)
		  tr[tr[t].d0].u0 = t;
		else if (tr[tr[t].d0].u1 == tnext)
		  tr[tr[t].d0].u1 = t;
	      
	      if ((tr[t].d1 = tr[tnext].d1) > 0)
		if (tr[tr[t].d1].u0 == tnext)
		  tr[tr[t].d1].u0 = t;
		else if (tr[tr[t].d1].u1 == tnext)
		  tr[tr[t].d1].u1 = t;
	      
	      tr[t].lo = tr[tnext].lo;
	      tr[tnext].state = ST_INVALID; /* invalidate the lower */
				            /* trapezium */
	    }
	  else		    /* not good neighbours */
	    t = tnext;
	}
      else		    /* do not satisfy the outer if */
	t = tnext;
      
    } /* end-while */
       
  return 0;
}


/* Add in the new segment into the trapezoidation and update Q and T
 * structures. First locate the two endpoints of the segment in the
 * Q-structure. Then start from the topmost trapezoid and go down to
 * the  lower trapezoid dividing all the trapezoids in between .
 */

int Triangulator::
add_segment(int segnum) {
  //  cerr << "add_segment(" << segnum << ")\n";

  segment_t s;
  //  segment_t *so = &seg[segnum];
  int tu, tl, sk, tfirst, tlast; //, tnext;
  int tfirstr = 0, tlastr = 0, tfirstl = 0, tlastl = 0;
  int i1, i2, t, tn; // t1, t2,
  point_t tpt;
  int tritop = 0, tribot = 0, is_swapped = 0;
  int tmptriseg;

  s = seg[segnum];
  if (_greater_than(&s.v1, &s.v0)) /* Get higher vertex in v0 */
    {
      int tmp;
      tpt = s.v0;
      s.v0 = s.v1;
      s.v1 = tpt;
      tmp = s.root0;
      s.root0 = s.root1;
      s.root1 = tmp;
      is_swapped = true;
    }

  if ((is_swapped) ? !inserted(segnum, LASTPT) :
       !inserted(segnum, FIRSTPT))     /* insert v0 in the tree */
    {
      int tmp_d;

      tu = locate_endpoint(&s.v0, &s.v1, s.root0);
      tl = newtrap();		/* tl is the new lower trapezoid */
      tr[tl].state = ST_VALID;
      tr[tl] = tr[tu];
      tr[tl].hi.y = s.v0.y;
      tr[tu].lo.y = s.v0.y;
      tr[tl].hi.x = s.v0.x;
      tr[tu].lo.x = s.v0.x;
      tr[tu].d0 = tl;      
      tr[tu].d1 = 0;
      tr[tl].u0 = tu;
      tr[tl].u1 = 0;

      if (((tmp_d = tr[tl].d0) > 0) && (tr[tmp_d].u0 == tu))
	tr[tmp_d].u0 = tl;
      if (((tmp_d = tr[tl].d0) > 0) && (tr[tmp_d].u1 == tu))
	tr[tmp_d].u1 = tl;

      if (((tmp_d = tr[tl].d1) > 0) && (tr[tmp_d].u0 == tu))
	tr[tmp_d].u0 = tl;
      if (((tmp_d = tr[tl].d1) > 0) && (tr[tmp_d].u1 == tu))
	tr[tmp_d].u1 = tl;

      /* Now update the query structure and obtain the sinks for the */
      /* two trapezoids */ 
      
      i1 = newnode();		/* Upper trapezoid sink */
      i2 = newnode();		/* Lower trapezoid sink */
      sk = tr[tu].sink;
      
      qs[sk].nodetype = T_Y;
      qs[sk].yval = s.v0;
      qs[sk].segnum = segnum;	/* not really reqd ... maybe later */
      qs[sk].left = i2;
      qs[sk].right = i1;

      qs[i1].nodetype = T_SINK;
      qs[i1].trnum = tu;
      qs[i1].parent = sk;

      qs[i2].nodetype = T_SINK;
      qs[i2].trnum = tl;
      qs[i2].parent = sk;

      tr[tu].sink = i1;
      tr[tl].sink = i2;
      tfirst = tl;
    }
  else				/* v0 already present */
    {       /* Get the topmost intersecting trapezoid */
      tfirst = locate_endpoint(&s.v0, &s.v1, s.root0);
      tritop = 1;
    }


  if ((is_swapped) ? !inserted(segnum, FIRSTPT) :
       !inserted(segnum, LASTPT))     /* insert v1 in the tree */
    {
      int tmp_d;

      tu = locate_endpoint(&s.v1, &s.v0, s.root1);

      tl = newtrap();		/* tl is the new lower trapezoid */
      tr[tl].state = ST_VALID;
      tr[tl] = tr[tu];
      tr[tl].hi.y = s.v1.y;
      tr[tu].lo.y = s.v1.y;
      tr[tl].hi.x = s.v1.x;
      tr[tu].lo.x = s.v1.x;
      tr[tu].d0 = tl;      
      tr[tu].d1 = 0;
      tr[tl].u0 = tu;
      tr[tl].u1 = 0;

      if (((tmp_d = tr[tl].d0) > 0) && (tr[tmp_d].u0 == tu))
	tr[tmp_d].u0 = tl;
      if (((tmp_d = tr[tl].d0) > 0) && (tr[tmp_d].u1 == tu))
	tr[tmp_d].u1 = tl;

      if (((tmp_d = tr[tl].d1) > 0) && (tr[tmp_d].u0 == tu))
	tr[tmp_d].u0 = tl;
      if (((tmp_d = tr[tl].d1) > 0) && (tr[tmp_d].u1 == tu))
	tr[tmp_d].u1 = tl;
      
      /* Now update the query structure and obtain the sinks for the */
      /* two trapezoids */ 
      
      i1 = newnode();		/* Upper trapezoid sink */
      i2 = newnode();		/* Lower trapezoid sink */
      sk = tr[tu].sink;
      
      qs[sk].nodetype = T_Y;
      qs[sk].yval = s.v1;
      qs[sk].segnum = segnum;	/* not really reqd ... maybe later */
      qs[sk].left = i2;
      qs[sk].right = i1;

      qs[i1].nodetype = T_SINK;
      qs[i1].trnum = tu;
      qs[i1].parent = sk;

      qs[i2].nodetype = T_SINK;
      qs[i2].trnum = tl;
      qs[i2].parent = sk;

      tr[tu].sink = i1;
      tr[tl].sink = i2;
      tlast = tu;
    }
  else				/* v1 already present */
    {       /* Get the lowermost intersecting trapezoid */
      tlast = locate_endpoint(&s.v1, &s.v0, s.root1);
      tribot = 1;
    }
  
  /* Thread the segment into the query tree creating a new X-node */
  /* First, split all the trapezoids which are intersected by s into */
  /* two */

  t = tfirst;			/* topmost trapezoid */
  
  while ((t > 0) && 
	 _greater_than_equal_to(&tr[t].lo, &tr[tlast].lo))
				/* traverse from top to bot */
    {
      int t_sav, tn_sav;
      sk = tr[t].sink;
      i1 = newnode();		/* left trapezoid sink */
      i2 = newnode();		/* right trapezoid sink */
      
      qs[sk].nodetype = T_X;
      qs[sk].segnum = segnum;
      qs[sk].left = i1;
      qs[sk].right = i2;

      qs[i1].nodetype = T_SINK;	/* left trapezoid (use existing one) */
      qs[i1].trnum = t;
      qs[i1].parent = sk;

      qs[i2].nodetype = T_SINK;	/* right trapezoid (allocate new) */
      tn = newtrap();
      qs[i2].trnum = tn;
      tr[tn].state = ST_VALID;
      qs[i2].parent = sk;

      if (t == tfirst)
	tfirstr = tn;
      if (_equal_to(&tr[t].lo, &tr[tlast].lo))
	tlastr = tn;

      tr[tn] = tr[t];
      tr[t].sink = i1;
      tr[tn].sink = i2;
      t_sav = t;
      tn_sav = tn;

      /* error */

      if ((tr[t].d0 <= 0) && (tr[t].d1 <= 0)) /* case cannot arise */
	{
	  fprintf(stderr, "add_segment: error\n");
	  break;
	}
      
      /* only one trapezoid below. partition t into two and make the */
      /* two resulting trapezoids t and tn as the upper neighbours of */
      /* the sole lower trapezoid */
      
      else if ((tr[t].d0 > 0) && (tr[t].d1 <= 0))
	{			/* Only one trapezoid below */
	  if ((tr[t].u0 > 0) && (tr[t].u1 > 0))
	    {			/* continuation of a chain from abv. */
	      if (tr[t].usave > 0) /* three upper neighbours */
		{
		  if (tr[t].uside == S_LEFT)
		    {
		      tr[tn].u0 = tr[t].u1;
		      tr[t].u1 = -1;
		      tr[tn].u1 = tr[t].usave;
		      
		      tr[tr[t].u0].d0 = t;
		      tr[tr[tn].u0].d0 = tn;
		      tr[tr[tn].u1].d0 = tn;
		    }
		  else		/* intersects in the right */
		    {
		      tr[tn].u1 = -1;
		      tr[tn].u0 = tr[t].u1;
		      tr[t].u1 = tr[t].u0;
		      tr[t].u0 = tr[t].usave;

		      tr[tr[t].u0].d0 = t;
		      tr[tr[t].u1].d0 = t;
		      tr[tr[tn].u0].d0 = tn;		      
		    }
		  tr[tn].usave = 0;
		  tr[t].usave = 0;
		}
	      else		/* No usave.... simple case */
		{
		  tr[tn].u0 = tr[t].u1;
                  tr[tn].u1 = -1;
		  tr[t].u1 = -1;
		  tr[tr[tn].u0].d0 = tn;
		}
	    }
	  else 
	    {			/* fresh seg. or upward cusp */
	      int tmp_u = tr[t].u0;
	      int td0, td1;
	      if (((td0 = tr[tmp_u].d0) > 0) && 
		  ((td1 = tr[tmp_u].d1) > 0))
		{		/* upward cusp */
		  if ((tr[td0].rseg > 0) &&
		      !is_left_of(tr[td0].rseg, &s.v1))
		    {
                      tr[tn].u1 = -1;
                      tr[t].u1 = -1;
		      tr[t].u0 = -1;
		      tr[tr[tn].u0].d1 = tn;
		    }
		  else		/* cusp going leftwards */
		    { 
                      tr[t].u1 = -1;
                      tr[tn].u1 = -1;
		      tr[tn].u0 = -1;
		      tr[tr[t].u0].d0 = t;
		    }
		}
	      else		/* fresh segment */
		{
		  tr[tr[t].u0].d0 = t;
		  tr[tr[t].u0].d1 = tn;
		}	      
	    }
	  
	  if (FP_EQUAL(tr[t].lo.y, tr[tlast].lo.y) && 
	      FP_EQUAL(tr[t].lo.x, tr[tlast].lo.x) && tribot)
	    {		/* bottom forms a triangle */

	      if (is_swapped)	
		tmptriseg = seg[segnum].prev;
	      else
		tmptriseg = seg[segnum].next;
	      
	      if ((tmptriseg > 0) && is_left_of(tmptriseg, &s.v0))
		{
				/* L-R downward cusp */
		  tr[tr[t].d0].u0 = t;
                  tr[tn].d1 = -1;
		  tr[tn].d0 = -1;
		}
	      else
		{
				/* R-L downward cusp */
		  tr[tr[tn].d0].u1 = tn;
                  tr[t].d1 = -1;
		  tr[t].d0 = -1;
		}
	    }
	  else
	    {
	      if ((tr[tr[t].d0].u0 > 0) && (tr[tr[t].d0].u1 > 0))
		{
		  if (tr[tr[t].d0].u0 == t) /* passes thru LHS */
		    {
		      tr[tr[t].d0].usave = tr[tr[t].d0].u1;
		      tr[tr[t].d0].uside = S_LEFT;
		    }
		  else
		    {
		      tr[tr[t].d0].usave = tr[tr[t].d0].u0;
		      tr[tr[t].d0].uside = S_RIGHT;
		    }		    
		}
	      tr[tr[t].d0].u0 = t;
	      tr[tr[t].d0].u1 = tn;
	    }
	  
	  t = tr[t].d0;
	}


      else if ((tr[t].d0 <= 0) && (tr[t].d1 > 0))
	{			/* Only one trapezoid below */
	  if ((tr[t].u0 > 0) && (tr[t].u1 > 0))
	    {			/* continuation of a chain from abv. */
	      if (tr[t].usave > 0) /* three upper neighbours */
		{
		  if (tr[t].uside == S_LEFT)
		    {
		      tr[tn].u0 = tr[t].u1;
		      tr[t].u1 = -1;
		      tr[tn].u1 = tr[t].usave;
		      
		      tr[tr[t].u0].d0 = t;
		      tr[tr[tn].u0].d0 = tn;
		      tr[tr[tn].u1].d0 = tn;
		    }
		  else		/* intersects in the right */
		    {
		      tr[tn].u1 = -1;
		      tr[tn].u0 = tr[t].u1;
		      tr[t].u1 = tr[t].u0;
		      tr[t].u0 = tr[t].usave;

		      tr[tr[t].u0].d0 = t;
		      tr[tr[t].u1].d0 = t;
		      tr[tr[tn].u0].d0 = tn;		      
		    }
		  
                  tr[tn].usave = 0;
		  tr[t].usave = 0;
		}
	      else		/* No usave.... simple case */
		{
		  tr[tn].u0 = tr[t].u1;
                  tr[tn].u1 = -1;
		  tr[t].u1 = -1;
		  tr[tr[tn].u0].d0 = tn;
		}
	    }
	  else 
	    {			/* fresh seg. or upward cusp */
	      int tmp_u = tr[t].u0;
	      int td0, td1;
	      if (((td0 = tr[tmp_u].d0) > 0) && 
		  ((td1 = tr[tmp_u].d1) > 0))
		{		/* upward cusp */
		  if ((tr[td0].rseg > 0) &&
		      !is_left_of(tr[td0].rseg, &s.v1))
		    {
                      tr[tn].u1 = -1;
                      tr[t].u1 = -1;
		      tr[t].u0 = -1;
		      tr[tr[tn].u0].d1 = tn;
		    }
		  else 
		    {
                      tr[t].u1 = -1;
                      tr[tn].u1 = -1;
		      tr[tn].u0 = -1;
		      tr[tr[t].u0].d0 = t;
		    }
		}
	      else		/* fresh segment */
		{
		  tr[tr[t].u0].d0 = t;
		  tr[tr[t].u0].d1 = tn;
		}
	    }
	  
	  if (FP_EQUAL(tr[t].lo.y, tr[tlast].lo.y) && 
	      FP_EQUAL(tr[t].lo.x, tr[tlast].lo.x) && tribot)
	    {		/* bottom forms a triangle */
	      if (is_swapped)	
		tmptriseg = seg[segnum].prev;
	      else
		tmptriseg = seg[segnum].next;

	      if ((tmptriseg > 0) && is_left_of(tmptriseg, &s.v0))
		{
		  /* L-R downward cusp */
		  tr[tr[t].d1].u0 = t;
                  tr[tn].d1 = -1;
		  tr[tn].d0 = -1;
		}
	      else
		{
		  /* R-L downward cusp */
		  tr[tr[tn].d1].u1 = tn;
                  tr[t].d1 = -1;
		  tr[t].d0 = -1;
		}
	    }		
	  else
	    {
	      if ((tr[tr[t].d1].u0 > 0) && (tr[tr[t].d1].u1 > 0))
		{
		  if (tr[tr[t].d1].u0 == t) /* passes thru LHS */
		    {
		      tr[tr[t].d1].usave = tr[tr[t].d1].u1;
		      tr[tr[t].d1].uside = S_LEFT;
		    }
		  else
		    {
		      tr[tr[t].d1].usave = tr[tr[t].d1].u0;
		      tr[tr[t].d1].uside = S_RIGHT;
		    }		    
		}
	      tr[tr[t].d1].u0 = t;
	      tr[tr[t].d1].u1 = tn;
	    }
	  
	  t = tr[t].d1;
	}

      /* two trapezoids below. Find out which one is intersected by */
      /* this segment and proceed down that one */
      
      else
	{
          //	  int tmpseg = tr[tr[t].d0].rseg;
	  double y0, yt;
	  point_t tmppt;
	  int tnext, i_d0, i_d1;

          i_d1 = false;
	  i_d0 = false;
	  if (FP_EQUAL(tr[t].lo.y, s.v0.y))
	    {
	      if (tr[t].lo.x > s.v0.x)
		i_d0 = true;
	      else
		i_d1 = true;
	    }
	  else
	    {
              y0 = tr[t].lo.y;
	      tmppt.y = y0;
	      yt = (y0 - s.v0.y)/(s.v1.y - s.v0.y);
	      tmppt.x = s.v0.x + yt * (s.v1.x - s.v0.x);
	      
	      if (_less_than(&tmppt, &tr[t].lo))
		i_d0 = true;
	      else
		i_d1 = true;
	    }
	  
	  /* check continuity from the top so that the lower-neighbour */
	  /* values are properly filled for the upper trapezoid */

	  if ((tr[t].u0 > 0) && (tr[t].u1 > 0))
	    {			/* continuation of a chain from abv. */
	      if (tr[t].usave > 0) /* three upper neighbours */
		{
		  if (tr[t].uside == S_LEFT)
		    {
		      tr[tn].u0 = tr[t].u1;
		      tr[t].u1 = -1;
		      tr[tn].u1 = tr[t].usave;
		      
		      tr[tr[t].u0].d0 = t;
		      tr[tr[tn].u0].d0 = tn;
		      tr[tr[tn].u1].d0 = tn;
		    }
		  else		/* intersects in the right */
		    {
		      tr[tn].u1 = -1;
		      tr[tn].u0 = tr[t].u1;
		      tr[t].u1 = tr[t].u0;
		      tr[t].u0 = tr[t].usave;

		      tr[tr[t].u0].d0 = t;
		      tr[tr[t].u1].d0 = t;
		      tr[tr[tn].u0].d0 = tn;		      
		    }
		  
                  tr[tn].usave = 0;
		  tr[t].usave = 0;
		}
	      else		/* No usave.... simple case */
		{
		  tr[tn].u0 = tr[t].u1;
		  tr[tn].u1 = -1;
		  tr[t].u1 = -1;
		  tr[tr[tn].u0].d0 = tn;
		}
	    }
	  else 
	    {			/* fresh seg. or upward cusp */
	      int tmp_u = tr[t].u0;
	      int td0, td1;
	      if (((td0 = tr[tmp_u].d0) > 0) && 
		  ((td1 = tr[tmp_u].d1) > 0))
		{		/* upward cusp */
		  if ((tr[td0].rseg > 0) &&
		      !is_left_of(tr[td0].rseg, &s.v1))
		    {
                      tr[tn].u1 = -1;
                      tr[t].u1 = -1;
		      tr[t].u0 = -1;
		      tr[tr[tn].u0].d1 = tn;
		    }
		  else 
		    {
                      tr[t].u1 = -1;
                      tr[tn].u1 = -1;
		      tr[tn].u0 = -1;
		      tr[tr[t].u0].d0 = t;
		    }
		}
	      else		/* fresh segment */
		{
		  tr[tr[t].u0].d0 = t;
		  tr[tr[t].u0].d1 = tn;
		}
	    }
	  
	  if (FP_EQUAL(tr[t].lo.y, tr[tlast].lo.y) && 
	      FP_EQUAL(tr[t].lo.x, tr[tlast].lo.x) && tribot)
	    {
	      /* this case arises only at the lowest trapezoid.. i.e.
		 tlast, if the lower endpoint of the segment is
		 already inserted in the structure */
	      
	      tr[tr[t].d0].u0 = t;
	      tr[tr[t].d0].u1 = -1;
	      tr[tr[t].d1].u0 = tn;
	      tr[tr[t].d1].u1 = -1;

	      tr[tn].d0 = tr[t].d1;
              tr[tn].d1 = -1;
	      tr[t].d1 = -1;
	      
	      tnext = tr[t].d1;	      
	    }
	  else if (i_d0)
				/* intersecting d0 */
	    {
	      tr[tr[t].d0].u0 = t;
	      tr[tr[t].d0].u1 = tn;
	      tr[tr[t].d1].u0 = tn;
	      tr[tr[t].d1].u1 = -1;
	      
	      /* new code to determine the bottom neighbours of the */
	      /* newly partitioned trapezoid */
	      
	      tr[t].d1 = -1;

	      tnext = tr[t].d0;
	    }
	  else			/* intersecting d1 */
	    {
	      tr[tr[t].d0].u0 = t;
	      tr[tr[t].d0].u1 = -1;
	      tr[tr[t].d1].u0 = t;
	      tr[tr[t].d1].u1 = tn;

	      /* new code to determine the bottom neighbours of the */
	      /* newly partitioned trapezoid */
	      
	      tr[tn].d0 = tr[t].d1;
	      tr[tn].d1 = -1;
	      
	      tnext = tr[t].d1;
	    }	    
	  
	  t = tnext;
	}
      
      tr[tn_sav].lseg  = segnum;
      tr[t_sav].rseg = segnum;
    } /* end-while */
  
  /* Now combine those trapezoids which share common segments. We can */
  /* use the pointers to the parent to connect these together. This */
  /* works only because all these new trapezoids have been formed */
  /* due to splitting by the segment, and hence have only one parent */

  tfirstl = tfirst; 
  tlastl = tlast;
  merge_trapezoids(segnum, tfirstl, tlastl, S_LEFT);
  merge_trapezoids(segnum, tfirstr, tlastr, S_RIGHT);

  seg[segnum].is_inserted = true;
  return 0;
}


/* Update the roots stored for each of the endpoints of the segment.
 * This is done to speed up the location-query for the endpoint when
 * the segment is inserted into the trapezoidation subsequently
 */
int Triangulator::
find_new_roots(int segnum) {
  //  cerr << "find_new_roots(" << segnum << ")\n";
  segment_t *s = &seg[segnum];
  
  if (s->is_inserted)
    return 0;

  s->root0 = locate_endpoint(&s->v0, &s->v1, s->root0);
  s->root0 = tr[s->root0].sink;

  s->root1 = locate_endpoint(&s->v1, &s->v0, s->root1);
  s->root1 = tr[s->root1].sink;  
  return 0;
}


/* Main routine to perform trapezoidation */
int Triangulator::
construct_trapezoids(int nseg) {
  //  cerr << "construct_trapezoids(" << nseg << ")\n";
  int i;
  int root, h;
  
  /* Add the first segment and get the query structure and trapezoid */
  /* list initialised */

  root = init_query_structure(choose_segment());

  for (i = 1; i <= nseg; i++) {
    seg[i].root1 = root;
    seg[i].root0 = root;
  }
  
  for (h = 1; h <= math_logstar_n(nseg); h++)
    {
      for (i = math_N(nseg, h -1) + 1; i <= math_N(nseg, h); i++) {
	add_segment(choose_segment());
      }
      
      /* Find a new root for each of the segment endpoints */
      for (i = 1; i <= nseg; i++)
	find_new_roots(i);
    }
  
  for (i = math_N(nseg, math_logstar_n(nseg)) + 1; i <= nseg; i++)
    add_segment(choose_segment());

  return 0;
}



/* Function returns true if the trapezoid lies inside the polygon */
int Triangulator::
inside_polygon(trap_t *t) {
  int rseg = t->rseg;

  if (t->state == ST_INVALID)
    return 0;

  if ((t->lseg <= 0) || (t->rseg <= 0))
    return 0;
  
  if (((t->u0 <= 0) && (t->u1 <= 0)) || 
      ((t->d0 <= 0) && (t->d1 <= 0))) /* triangle */
    return (_greater_than(&seg[rseg].v1, &seg[rseg].v0));
  
  return 0;
}


/* return a new mon structure from the table */
int Triangulator::
newmon() {
  int index = (int)mon.size();
  mon.push_back(0);
  //  cerr << "newmon " << index << "\n";
  return index;
}


/* return a new chain element from the table */
int Triangulator::
new_chain_element() {
  int index = (int)mchain.size();
  mchain.push_back(monchain_t());
  //  cerr << "new_chain_element " << index << "\n";
  return index;
}


double Triangulator::
get_angle(point_t *vp0, point_t *vpnext, point_t *vp1) {
  point_t v0, v1;
  
  v0.x = vpnext->x - vp0->x;
  v0.y = vpnext->y - vp0->y;

  v1.x = vp1->x - vp0->x;
  v1.y = vp1->y - vp0->y;

  if (CROSS_SINE(v0, v1) >= 0)	/* sine is positive */
    return DOT(v0, v1)/LENGTH(v0)/LENGTH(v1);
  else
    return (-1.0 * DOT(v0, v1)/LENGTH(v0)/LENGTH(v1) - 2);
}


/* (v0, v1) is the new diagonal to be added to the polygon. Find which */
/* chain to use and return the positions of v0 and v1 in p and q */ 
int Triangulator::
get_vertex_positions(int v0, int v1, int *ip, int *iq) {
  vertexchain_t *vp0, *vp1;
  int i;
  double angle, temp;
  int tp = 0, tq = 0;

  vp0 = &vert[v0];
  vp1 = &vert[v1];
  
  /* p is identified as follows. Scan from (v0, v1) rightwards till */
  /* you hit the first segment starting from v0. That chain is the */
  /* chain of our interest */
  
  angle = -4.0;
  for (i = 0; i < 4; i++)
    {
      if (vp0->vnext[i] <= 0)
	continue;
      if ((temp = get_angle(&vp0->pt, &(vert[vp0->vnext[i]].pt), 
			    &vp1->pt)) > angle)
	{
	  angle = temp;
	  tp = i;
	}
    }

  *ip = tp;

  /* Do similar actions for q */

  angle = -4.0;
  for (i = 0; i < 4; i++)
    {
      if (vp1->vnext[i] <= 0)
	continue;      
      if ((temp = get_angle(&vp1->pt, &(vert[vp1->vnext[i]].pt), 
			    &vp0->pt)) > angle)
	{
	  angle = temp;
	  tq = i;
	}
    }

  *iq = tq;

  return 0;
}

  
/* v0 and v1 are specified in anti-clockwise order with respect to 
 * the current monotone polygon mcur. Split the current polygon into 
 * two polygons using the diagonal (v0, v1) 
 */
int Triangulator::
make_new_monotone_poly(int mcur, int v0, int v1) {
  int p, q, ip, iq;
  int mnew = newmon();
  int i, j, nf0, nf1;
  vertexchain_t *vp0, *vp1;

  if (v0 <= 0 || v1 <= 0) {
    return -1;
  }
  
  vp0 = &vert[v0];
  vp1 = &vert[v1];

  get_vertex_positions(v0, v1, &ip, &iq);

  p = vp0->vpos[ip];
  q = vp1->vpos[iq];

  /* At this stage, we have got the positions of v0 and v1 in the */
  /* desired chain. Now modify the linked lists */

  i = new_chain_element();	/* for the new list */
  j = new_chain_element();

  mchain[i].vnum = v0;
  mchain[j].vnum = v1;

  mchain[i].next = mchain[p].next;
  mchain[mchain[p].next].prev = i;
  mchain[i].prev = j;
  mchain[j].next = i;
  mchain[j].prev = mchain[q].prev;
  mchain[mchain[q].prev].next = j;

  mchain[p].next = q;
  mchain[q].prev = p;

  nf0 = vp0->nextfree;
  nf1 = vp1->nextfree;

  vp0->vnext[ip] = v1;

  vp0->vpos[nf0] = i;
  vp0->vnext[nf0] = mchain[mchain[i].next].vnum;
  vp1->vpos[nf1] = j;
  vp1->vnext[nf1] = v0;

  vp0->nextfree++;
  vp1->nextfree++;

#ifdef DEBUG
  fprintf(stderr, "make_poly: mcur = %d, (v0, v1) = (%d, %d)\n", 
	  mcur, v0, v1);
  fprintf(stderr, "next posns = (p, q) = (%d, %d)\n", p, q);
#endif

  mon[mcur] = p;
  mon[mnew] = i;
  return mnew;
}

/* Main routine to get monotone polygons from the trapezoidation of 
 * the polygon.
 */

int Triangulator::
monotonate_trapezoids(int n) {
  int i;
  int tr_start;

  vert.clear();
  visited.clear();
  mchain.clear();
  mon.clear();

  vert.insert(vert.begin(), n + 1, vertexchain_t());
  mchain.insert(mchain.begin(), n + 1, monchain_t());

  visited.insert(visited.begin(), tr.size(), 0);
  
  /* First locate a trapezoid which lies inside the polygon */
  /* and which is triangular */
  for (i = 1; i < (int)tr.size(); i++)
    if (inside_polygon(&tr[i]))
      break;
  if (i >= (int)tr.size()) {
    // No valid trapezoids.
    return 0;
  }
  //  printf("start = %d\n", i);
  tr_start = i;
  
  /* Initialise the mon data-structure and start spanning all the */
  /* trapezoids within the polygon */

#if 0
  for (i = 1; i <= n; i++)
    {
      mchain[i].prev = i - 1;
      mchain[i].next = i + 1;
      mchain[i].vnum = i;
      vert[i].pt = seg[i].v0;
      vert[i].vnext[0] = i + 1;	/* next vertex */
      vert[i].vpos[0] = i;	/* locn. of next vertex */
      vert[i].nextfree = 1;
      vert[i].user_i = seg[i].v0_i;
    }
  mchain[1].prev = n;
  mchain[n].next = 1;
  vert[n].vnext[0] = 1;
  vert[n].vpos[0] = n;
  mon.push_back(1); /* position of any vertex in the first */
                    /* chain  */

#else

  for (i = 1; i <= n; i++)
    {
      mchain[i].prev = seg[i].prev;
      mchain[i].next = seg[i].next;
      mchain[i].vnum = i;
      vert[i].pt = seg[i].v0;
      vert[i].vnext[0] = seg[i].next; /* next vertex */
      vert[i].vpos[0] = i;	/* locn. of next vertex */
      vert[i].nextfree = 1;
      vert[i].user_i = seg[i].v0_i;
    }

  mon.push_back(1); /* position of any vertex in the first */
                    /* chain  */

#endif
  
  /* traverse the polygon */
  if (tr[tr_start].u0 > 0)
    traverse_polygon(0, tr_start, tr[tr_start].u0, TR_FROM_UP);
  else if (tr[tr_start].d0 > 0)
    traverse_polygon(0, tr_start, tr[tr_start].d0, TR_FROM_DN);
  
  /* return the number of polygons created */
  return newmon();
}


/* recursively visit all the trapezoids */
int Triangulator::
traverse_polygon(int mcur, int trnum, int from, int dir) {
  //  printf("traverse_polygon(%d, %d, %d, %d)\n", mcur, trnum, from, dir);

  trap_t *t = &tr[trnum];
  //  int howsplit;
  int mnew;
  int v0, v1;  //, v0next, v1next;
  int retval = 0;  //, tmp;
  int do_switch = false;

  if (mcur < 0 || trnum <= 0)
    return 0;

  //  printf("visited size = %d, visited[trnum] = %d\n", visited.size(), visited[trnum]);

  if (visited[trnum])
    return 0;

  visited[trnum] = true;
  
  /* We have much more information available here. */
  /* rseg: goes upwards   */
  /* lseg: goes downwards */

  /* Initially assume that dir = TR_FROM_DN (from the left) */
  /* Switch v0 and v1 if necessary afterwards */


  /* special cases for triangles with cusps at the opposite ends. */
  /* take care of this first */
  if ((t->u0 <= 0) && (t->u1 <= 0))
    {
      if ((t->d0 > 0) && (t->d1 > 0)) /* downward opening triangle */
	{
	  v0 = tr[t->d1].lseg;
	  v1 = t->lseg;
	  if (from == t->d1)
	    {
	      do_switch = true;
	      mnew = make_new_monotone_poly(mcur, v1, v0);
	      traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
	      traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);	    
	    }
	  else
	    {
	      mnew = make_new_monotone_poly(mcur, v0, v1);
	      traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
	      traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);
	    }
	}
      else
	{
	  retval = SP_NOSPLIT;	/* Just traverse all neighbours */
	  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
	  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
	  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
	  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
	}
    }
  
  else if ((t->d0 <= 0) && (t->d1 <= 0))
    {
      if ((t->u0 > 0) && (t->u1 > 0)) /* upward opening triangle */
	{
	  v0 = t->rseg;
	  v1 = tr[t->u0].rseg;
	  if (from == t->u1)
	    {
	      do_switch = true;
	      mnew = make_new_monotone_poly(mcur, v1, v0);
	      traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
	      traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);	    
	    }
	  else
	    {
	      mnew = make_new_monotone_poly(mcur, v0, v1);
	      traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
	      traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
	    }
	}
      else
	{
	  retval = SP_NOSPLIT;	/* Just traverse all neighbours */
	  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
	  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
	  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
	  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
	}
    }
  
  else if ((t->u0 > 0) && (t->u1 > 0)) 
    {
      if ((t->d0 > 0) && (t->d1 > 0)) /* downward + upward cusps */
	{
	  v0 = tr[t->d1].lseg;
	  v1 = tr[t->u0].rseg;
	  retval = SP_2UP_2DN;
	  if (((dir == TR_FROM_DN) && (t->d1 == from)) ||
	      ((dir == TR_FROM_UP) && (t->u1 == from)))
	    {
	      do_switch = true;
	      mnew = make_new_monotone_poly(mcur, v1, v0);
	      traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
	      traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
	      traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);
	      traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);
	    }
	  else
	    {
	      mnew = make_new_monotone_poly(mcur, v0, v1);
	      traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
	      traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
	      traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
	      traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);	      
	    }
	}
      else			/* only downward cusp */
	{
	  if (_equal_to(&t->lo, &seg[t->lseg].v1))
	    {
	      v0 = tr[t->u0].rseg;
	      v1 = seg[t->lseg].next;

	      retval = SP_2UP_LEFT;
	      if ((dir == TR_FROM_UP) && (t->u0 == from))
		{
		  do_switch = true;
		  mnew = make_new_monotone_poly(mcur, v1, v0);
		  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);
		}
	      else
		{
		  mnew = make_new_monotone_poly(mcur, v0, v1);
		  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);
		}
	    }
	  else
	    {
	      v0 = t->rseg;
	      v1 = tr[t->u0].rseg;	
	      retval = SP_2UP_RIGHT;
	      if ((dir == TR_FROM_UP) && (t->u1 == from))
		{
		  do_switch = true;
		  mnew = make_new_monotone_poly(mcur, v1, v0);
		  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);
		}
	      else
		{
		  mnew = make_new_monotone_poly(mcur, v0, v1);
		  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
		}
	    }
	}
    }
  else if ((t->u0 > 0) || (t->u1 > 0)) /* no downward cusp */
    {
      if ((t->d0 > 0) && (t->d1 > 0)) /* only upward cusp */
	{
	  if (_equal_to(&t->hi, &seg[t->lseg].v0))
	    {
	      v0 = tr[t->d1].lseg;
	      v1 = t->lseg;
	      retval = SP_2DN_LEFT;
	      if (!((dir == TR_FROM_DN) && (t->d0 == from)))
		{
		  do_switch = true;
		  mnew = make_new_monotone_poly(mcur, v1, v0);
		  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);
		}
	      else
		{
		  mnew = make_new_monotone_poly(mcur, v0, v1);
		  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);	      
		}
	    }
	  else
	    {
	      v0 = tr[t->d1].lseg;
	      v1 = seg[t->rseg].next;

	      retval = SP_2DN_RIGHT;	    
	      if ((dir == TR_FROM_DN) && (t->d1 == from))
		{
		  do_switch = true;
		  mnew = make_new_monotone_poly(mcur, v1, v0);
		  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);
		}
	      else
		{
		  mnew = make_new_monotone_poly(mcur, v0, v1);
		  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);
		}
	    }
	}
      else			/* no cusp */
	{
	  if (_equal_to(&t->hi, &seg[t->lseg].v0) &&
	      _equal_to(&t->lo, &seg[t->rseg].v0))
	    {
	      v0 = t->rseg;
	      v1 = t->lseg;
	      retval = SP_SIMPLE_LRDN;
	      if (dir == TR_FROM_UP)
		{
		  do_switch = true;
		  mnew = make_new_monotone_poly(mcur, v1, v0);
		  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);
		}
	      else
		{
		  mnew = make_new_monotone_poly(mcur, v0, v1);
		  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
		}
	    }
	  else if (_equal_to(&t->hi, &seg[t->rseg].v1) &&
		   _equal_to(&t->lo, &seg[t->lseg].v1))
	    {
	      v0 = seg[t->rseg].next;
	      v1 = seg[t->lseg].next;

	      retval = SP_SIMPLE_LRUP;
	      if (dir == TR_FROM_UP)
		{
		  do_switch = true;
		  mnew = make_new_monotone_poly(mcur, v1, v0);
		  traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->d0, trnum, TR_FROM_UP);
		}
	      else
		{
		  mnew = make_new_monotone_poly(mcur, v0, v1);
		  traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);
		  traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
		  traverse_polygon(mnew, t->u0, trnum, TR_FROM_DN);
		  traverse_polygon(mnew, t->u1, trnum, TR_FROM_DN);
		}
	    }
	  else			/* no split possible */
	    {
	      retval = SP_NOSPLIT;
	      traverse_polygon(mcur, t->u0, trnum, TR_FROM_DN);
	      traverse_polygon(mcur, t->d0, trnum, TR_FROM_UP);
	      traverse_polygon(mcur, t->u1, trnum, TR_FROM_DN);
	      traverse_polygon(mcur, t->d1, trnum, TR_FROM_UP);	      	      
	    }
	}
    }

  return retval;
}


/* For each monotone polygon, find the ymax and ymin (to determine the */
/* two y-monotone chains) and pass on this monotone polygon for greedy */
/* triangulation. */
/* Take care not to triangulate duplicate monotone polygons */

void Triangulator::
triangulate_monotone_polygons(int nvert, int nmonpoly) {
  int i;
  point_t ymax, ymin;
  int p, vfirst, posmax, posmin, v;
  int vcount, processed;

  #ifdef DEBUG
  for (i = 0; i < nmonpoly; i++)
    {
      fprintf(stderr, "\n\nPolygon %d: ", i);
      vfirst = mchain[mon[i]].vnum;
      p = mchain[mon[i]].next;
      fprintf (stderr, "%d ", mchain[mon[i]].vnum);
      while (mchain[p].vnum != vfirst)
	{
	  fprintf(stderr, "%d ", mchain[p].vnum);
          if (mchain[p].vnum == -1) {
            fprintf(stderr, " xx");
            break;
          }
	  p = mchain[p].next;
	}
    }
  fprintf(stderr, "\n");
  #endif

  for (i = 0; i < nmonpoly; i++)
    {
      vcount = 1;
      processed = false;
      vfirst = mchain[mon[i]].vnum;
      if (vfirst <= 0) {
        return;
      }
      ymin = vert[vfirst].pt;
      ymax = ymin;
      posmin = mon[i];
      posmax = posmin;
      mchain[mon[i]].marked = true;
      p = mchain[mon[i]].next;
      while ((v = mchain[p].vnum) != vfirst)
	{
          if (v <= 0) {
            return;
          }
          if (mchain[p].marked)
	   {
	     processed = true;
	     break;		/* break from while */
	   }
          else
            mchain[p].marked = true;
          
	  if (_greater_than(&vert[v].pt, &ymax))
	    {
	      ymax = vert[v].pt;
	      posmax = p;
	    }
	  if (_less_than(&vert[v].pt, &ymin))
	    {
	      ymin = vert[v].pt;
	      posmin = p;
	    }
	  p = mchain[p].next;
	  vcount++;
       }

      if (processed)		/* Go to next polygon */
	continue;
      
      if (vcount == 3)		/* already a triangle */
	{
          _result.push_back(Triangle(this, mchain[p].vnum,
                                     mchain[mchain[p].next].vnum,
                                     mchain[mchain[p].prev].vnum));
	}
      else			/* triangulate the polygon */
	{
	  v = mchain[mchain[posmax].next].vnum;
	  if (_equal_to(&vert[v].pt, &ymin))
	    {			/* LHS is a single line */
	      triangulate_single_polygon(nvert, posmax, TRI_LHS);
	    }
	  else
	    triangulate_single_polygon(nvert, posmax, TRI_RHS);
	}
    }
}


/* A greedy corner-cutting algorithm to triangulate a y-monotone 
 * polygon in O(n) time.
 * Joseph O-Rourke, Computational Geometry in C.
 */
void Triangulator::
triangulate_single_polygon(int nvert, int posmax, int side) {
  int v;
  vector_int rc;	/* reflex chain */
  int ri;
  int endv, tmp, vpos;

  //  cerr << "triangulate_single_polygon(" << nvert << ", " << posmax << ", " << side << ")\n";

  if (side == TRI_RHS)		/* RHS segment is a single segment */
    {
      rc.push_back(mchain[posmax].vnum);
      tmp = mchain[posmax].next;
      rc.push_back(mchain[tmp].vnum);
      ri = 1;
      
      vpos = mchain[tmp].next;
      v = mchain[vpos].vnum;
      
      if ((endv = mchain[mchain[posmax].prev].vnum) == 0)
	endv = nvert;
    }
  else				/* LHS is a single segment */
    {
      tmp = mchain[posmax].next;
      rc.push_back(mchain[tmp].vnum);
      tmp = mchain[tmp].next;
      rc.push_back(mchain[tmp].vnum);
      ri = 1;

      vpos = mchain[tmp].next;
      v = mchain[vpos].vnum;

      endv = mchain[posmax].vnum;
    }
  
  while ((v != endv) || (ri > 1))
    {
      if (v <= 0) {
        // Something went wrong.
        return;
      }
      if (ri > 0)		/* reflex chain is non-empty */
	{
	  if (CROSS(vert[v].pt, vert[rc[ri - 1]].pt, 
		    vert[rc[ri]].pt) > 0)
	    {			/* convex corner: cut if off */
              _result.push_back(Triangle(this, rc[ri - 1], rc[ri], v));
	      ri--;
              rc.pop_back();
              nassertv(ri + 1 == (int)rc.size());
	    }
	  else		/* non-convex */
	    {		/* add v to the chain */
	      ri++;
              rc.push_back(v);
              nassertv(ri + 1 == (int)rc.size());
	      vpos = mchain[vpos].next;
	      v = mchain[vpos].vnum;
	    }
	}
      else			/* reflex-chain empty: add v to the */
	{			/* reflex chain and advance it  */
          ri++;
          rc.push_back(v);
          nassertv(ri + 1 == (int)rc.size());
	  vpos = mchain[vpos].next;
	  v = mchain[vpos].vnum;
	}
    } /* end-while */
  
  /* reached the bottom vertex. Add in the triangle formed */
  _result.push_back(Triangle(this, rc[ri - 1], rc[ri], v));  
  ri--;
}


