svn merge ^/branches/released . --username marha

1 files changed, 2427 insertions, 0 deletions

diff --git a/mesalib/src/glu/sgi/libnurbs/nurbtess/sampleMonoPoly.cc b/mesalib/src/glu/sgi/libnurbs/nurbtess/sampleMonoPoly.cc
new file mode 100644
index 000000000..051f24108
--- /dev/null
+++ b/mesalib/src/glu/sgi/libnurbs/nurbtess/sampleMonoPoly.cc
@@ -0,0 +1,2427 @@
+/*
+** License Applicability. Except to the extent portions of this file are
+** made subject to an alternative license as permitted in the SGI Free
+** Software License B, Version 1.1 (the "License"), the contents of this
+** file are subject only to the provisions of the License. You may not use
+** this file except in compliance with the License. You may obtain a copy
+** of the License at Silicon Graphics, Inc., attn: Legal Services, 1600
+** Amphitheatre Parkway, Mountain View, CA 94043-1351, or at:
+** 
+** http://oss.sgi.com/projects/FreeB
+** 
+** Note that, as provided in the License, the Software is distributed on an
+** "AS IS" basis, with ALL EXPRESS AND IMPLIED WARRANTIES AND CONDITIONS
+** DISCLAIMED, INCLUDING, WITHOUT LIMITATION, ANY IMPLIED WARRANTIES AND
+** CONDITIONS OF MERCHANTABILITY, SATISFACTORY QUALITY, FITNESS FOR A
+** PARTICULAR PURPOSE, AND NON-INFRINGEMENT.
+** 
+** Original Code. The Original Code is: OpenGL Sample Implementation,
+** Version 1.2.1, released January 26, 2000, developed by Silicon Graphics,
+** Inc. The Original Code is Copyright (c) 1991-2000 Silicon Graphics, Inc.
+** Copyright in any portions created by third parties is as indicated
+** elsewhere herein. All Rights Reserved.
+** 
+** Additional Notice Provisions: The application programming interfaces
+** established by SGI in conjunction with the Original Code are The
+** OpenGL(R) Graphics System: A Specification (Version 1.2.1), released
+** April 1, 1999; The OpenGL(R) Graphics System Utility Library (Version
+** 1.3), released November 4, 1998; and OpenGL(R) Graphics with the X
+** Window System(R) (Version 1.3), released October 19, 1998. This software
+** was created using the OpenGL(R) version 1.2.1 Sample Implementation
+** published by SGI, but has not been independently verified as being
+** compliant with the OpenGL(R) version 1.2.1 Specification.
+**
+*/
+/*
+*/
+
+#include "gluos.h"
+#include <stdlib.h>
+#include <stdio.h>
+#include <math.h>
+
+#ifndef max
+#define max(a,b) ((a>b)? a:b)
+#endif
+#ifndef min
+#define min(a,b) ((a>b)? b:a)
+#endif
+
+#include <GL/gl.h>
+
+#include "glimports.h"
+#include "zlassert.h"
+#include "sampleMonoPoly.h"
+#include "sampleComp.h"
+#include "polyDBG.h"
+#include "partitionX.h"
+
+
+#define ZERO 0.00001
+
+//#define  MYDEBUG
+
+//#define SHORTEN_GRID_LINE
+//see work/newtess/internal/test/problems
+
+
+/*split a polygon so that each vertex correcpond to one edge
+ *the head of the first edge of the returned plygon must be the head of the first
+ *edge of the origianl polygon. This is crucial for the code in sampleMonoPoly function
+ */
+ directedLine*  polygonConvert(directedLine* polygon)
+{
+  int i;
+  directedLine* ret;
+  sampledLine* sline;
+  sline = new sampledLine(2);
+  sline->setPoint(0, polygon->getVertex(0));
+  sline->setPoint(1, polygon->getVertex(1));
+  ret=new directedLine(INCREASING, sline);
+  for(i=1; i<= polygon->get_npoints()-2; i++)
+    {
+      sline = new sampledLine(2);
+      sline->setPoint(0, polygon->getVertex(i));
+      sline->setPoint(1, polygon->getVertex(i+1));
+      ret->insert(new directedLine(INCREASING, sline));
+    }
+
+  for(directedLine *temp = polygon->getNext(); temp != polygon; temp = temp->getNext())
+    {
+      for(i=0; i<= temp->get_npoints()-2; i++)
+	{
+	  sline = new sampledLine(2);
+	  sline->setPoint(0, temp->getVertex(i));
+	  sline->setPoint(1, temp->getVertex(i+1));
+	  ret->insert(new directedLine(INCREASING, sline));
+	}
+    }
+  return ret;
+}
+
+void triangulateConvexPolyVertical(directedLine* topV, directedLine* botV, primStream *pStream)
+{
+  Int i,j;
+  Int n_leftVerts;
+  Int n_rightVerts;
+  Real** leftVerts;
+  Real** rightVerts;
+  directedLine* tempV;
+  n_leftVerts = 0;
+  for(tempV = topV; tempV != botV; tempV = tempV->getNext())
+    {
+      n_leftVerts += tempV->get_npoints();
+    }
+  n_rightVerts=0;
+  for(tempV = botV; tempV != topV; tempV = tempV->getNext())
+    {
+      n_rightVerts += tempV->get_npoints();
+    }
+
+  Real2* temp_leftVerts = (Real2 *) malloc(sizeof(Real2) * n_leftVerts);
+  assert(temp_leftVerts);
+  Real2* temp_rightVerts = (Real2 *) malloc(sizeof(Real2) * n_rightVerts);
+  assert(temp_rightVerts);
+
+  leftVerts = (Real**) malloc(sizeof(Real2*) * n_leftVerts);
+  assert(leftVerts);
+  rightVerts = (Real**) malloc(sizeof(Real2*) * n_rightVerts);
+  assert(rightVerts);
+  for(i=0; i<n_leftVerts; i++)
+    leftVerts[i] = temp_leftVerts[i];
+  for(i=0; i<n_rightVerts; i++)
+    rightVerts[i] = temp_rightVerts[i];
+
+  i=0;
+  for(tempV = topV; tempV != botV; tempV = tempV->getNext())
+    {
+      for(j=1; j<tempV->get_npoints(); j++)
+	{
+	  leftVerts[i][0] = tempV->getVertex(j)[0];
+	  leftVerts[i][1] = tempV->getVertex(j)[1];
+	  i++;
+	}
+    }
+  n_leftVerts = i;
+  i=0;
+  for(tempV = topV->getPrev(); tempV != botV->getPrev(); tempV = tempV->getPrev())
+    {
+      for(j=tempV->get_npoints()-1; j>=1; j--)
+	{
+	  rightVerts[i][0] = tempV->getVertex(j)[0];
+	  rightVerts[i][1] = tempV->getVertex(j)[1];
+	  i++;
+	}
+    }
+  n_rightVerts = i;
+  triangulateXYMonoTB(n_leftVerts, leftVerts, n_rightVerts, rightVerts, pStream);
+  free(leftVerts);
+  free(rightVerts);
+  free(temp_leftVerts);
+  free(temp_rightVerts);
+}  
+
+void triangulateConvexPolyHoriz(directedLine* leftV, directedLine* rightV, primStream *pStream)
+{
+  Int i,j;
+  Int n_lowerVerts;
+  Int n_upperVerts;
+  Real2 *lowerVerts;
+  Real2 *upperVerts;
+  directedLine* tempV;
+  n_lowerVerts=0;
+  for(tempV = leftV; tempV != rightV; tempV = tempV->getNext())
+    {
+      n_lowerVerts += tempV->get_npoints();
+    }
+  n_upperVerts=0;
+  for(tempV = rightV; tempV != leftV; tempV = tempV->getNext())
+    {
+      n_upperVerts += tempV->get_npoints();
+    }
+  lowerVerts = (Real2 *) malloc(sizeof(Real2) * n_lowerVerts);
+  assert(n_lowerVerts);
+  upperVerts = (Real2 *) malloc(sizeof(Real2) * n_upperVerts);
+  assert(n_upperVerts);
+  i=0;
+  for(tempV = leftV; tempV != rightV; tempV = tempV->getNext())
+    {
+      for(j=0; j<tempV->get_npoints(); j++)
+	{
+	  lowerVerts[i][0] = tempV->getVertex(j)[0];
+	  lowerVerts[i][1] = tempV->getVertex(j)[1];
+	  i++;
+	}
+    }
+  i=0;
+  for(tempV = leftV->getPrev(); tempV != rightV->getPrev(); tempV = tempV->getPrev())
+    {
+      for(j=tempV->get_npoints()-1; j>=0; j--)
+	{
+	  upperVerts[i][0] = tempV->getVertex(j)[0];
+	  upperVerts[i][1] = tempV->getVertex(j)[1];
+	  i++;
+	}
+    }
+  triangulateXYMono(n_upperVerts, upperVerts, n_lowerVerts, lowerVerts, pStream);
+  free(lowerVerts);
+  free(upperVerts);
+}  
+void triangulateConvexPoly(directedLine* polygon, Int ulinear, Int vlinear, primStream* pStream)
+{
+  /*find left, right, top , bot
+    */
+  directedLine* tempV;
+  directedLine* topV;
+  directedLine* botV;
+  directedLine* leftV;
+  directedLine* rightV;
+  topV = botV = polygon;
+
+  for(tempV = polygon->getNext(); tempV != polygon; tempV = tempV->getNext())
+    {
+      if(compV2InY(topV->head(), tempV->head())<0) {
+
+	topV = tempV;
+      }
+      if(compV2InY(botV->head(), tempV->head())>0) {
+
+	botV = tempV;
+      }
+    }
+  //find leftV
+  for(tempV = topV; tempV != botV; tempV = tempV->getNext())
+    {
+      if(tempV->tail()[0] >= tempV->head()[0])
+	break;
+    }
+  leftV = tempV;
+  //find rightV
+  for(tempV = botV; tempV != topV; tempV = tempV->getNext())
+    {
+      if(tempV->tail()[0] <= tempV->head()[0])
+	break;
+    }
+  rightV = tempV;
+  if(vlinear)
+    {
+      triangulateConvexPolyHoriz( leftV, rightV, pStream);
+    }
+  else if(ulinear)
+    {
+      triangulateConvexPolyVertical(topV, botV, pStream);
+    }
+  else
+    {
+      if(DBG_is_U_direction(polygon))
+	{
+	  triangulateConvexPolyHoriz( leftV, rightV, pStream);
+	}
+      else
+	triangulateConvexPolyVertical(topV, botV, pStream);
+    }
+}	      
+
+/*for debug purpose*/
+void drawCorners(
+		 Real* topV, Real* botV,		 
+		 vertexArray* leftChain,
+		 vertexArray* rightChain,
+		 gridBoundaryChain* leftGridChain,
+		 gridBoundaryChain* rightGridChain,
+		 Int gridIndex1,
+		 Int gridIndex2,
+		 Int leftCornerWhere,
+		 Int leftCornerIndex,
+		 Int rightCornerWhere,
+		 Int rightCornerIndex,
+		 Int bot_leftCornerWhere,
+		 Int bot_leftCornerIndex,
+		 Int bot_rightCornerWhere,
+		 Int bot_rightCornerIndex)
+{
+  Real* leftCornerV;
+  Real* rightCornerV;
+  Real* bot_leftCornerV;
+  Real* bot_rightCornerV;
+
+  if(leftCornerWhere == 1)
+    leftCornerV = topV;
+  else if(leftCornerWhere == 0)
+    leftCornerV = leftChain->getVertex(leftCornerIndex);
+  else
+    leftCornerV = rightChain->getVertex(leftCornerIndex);
+
+  if(rightCornerWhere == 1)
+    rightCornerV = topV;
+  else if(rightCornerWhere == 0)
+    rightCornerV = leftChain->getVertex(rightCornerIndex);
+  else
+    rightCornerV = rightChain->getVertex(rightCornerIndex);
+
+  if(bot_leftCornerWhere == 1)
+    bot_leftCornerV = botV;
+  else if(bot_leftCornerWhere == 0)
+    bot_leftCornerV = leftChain->getVertex(bot_leftCornerIndex);
+  else
+    bot_leftCornerV = rightChain->getVertex(bot_leftCornerIndex);
+
+  if(bot_rightCornerWhere == 1)
+    bot_rightCornerV = botV;
+  else if(bot_rightCornerWhere == 0)
+    bot_rightCornerV = leftChain->getVertex(bot_rightCornerIndex);
+  else
+    bot_rightCornerV = rightChain->getVertex(bot_rightCornerIndex);
+
+  Real topGridV = leftGridChain->get_v_value(gridIndex1);
+  Real topGridU1 = leftGridChain->get_u_value(gridIndex1);
+  Real topGridU2 = rightGridChain->get_u_value(gridIndex1);
+  Real botGridV = leftGridChain->get_v_value(gridIndex2);
+  Real botGridU1 = leftGridChain->get_u_value(gridIndex2);
+  Real botGridU2 = rightGridChain->get_u_value(gridIndex2);
+  
+  glBegin(GL_LINE_STRIP);
+  glVertex2fv(leftCornerV);
+  glVertex2f(topGridU1, topGridV);
+  glEnd();
+
+  glBegin(GL_LINE_STRIP);
+  glVertex2fv(rightCornerV);
+  glVertex2f(topGridU2, topGridV);
+  glEnd();
+
+  glBegin(GL_LINE_STRIP);
+  glVertex2fv(bot_leftCornerV);
+  glVertex2f(botGridU1, botGridV);
+  glEnd();
+
+  glBegin(GL_LINE_STRIP);
+  glVertex2fv(bot_rightCornerV);
+  glVertex2f(botGridU2, botGridV);
+  glEnd();
+
+
+}
+		 
+void toVertexArrays(directedLine* topV, directedLine* botV, vertexArray& leftChain, vertexArray& rightChain)
+{  
+  Int i;
+  directedLine* tempV;
+  for(i=1; i<=topV->get_npoints()-2; i++) { /*the first vertex is the top vertex which doesn't belong to inc_chain*/
+    leftChain.appendVertex(topV->getVertex(i));
+  }
+  for(tempV = topV->getNext(); tempV != botV; tempV = tempV->getNext())
+    {
+      for(i=0; i<=tempV->get_npoints()-2; i++){
+	leftChain.appendVertex(tempV->getVertex(i));
+      }
+    }  
+
+  for(tempV = topV->getPrev(); tempV != botV; tempV = tempV->getPrev())
+    {
+      for(i=tempV->get_npoints()-2; i>=0; i--){
+	rightChain.appendVertex(tempV->getVertex(i));
+      }
+    }
+  for(i=botV->get_npoints()-2; i>=1; i--){ 
+    rightChain.appendVertex(tempV->getVertex(i));
+  }
+}
+
+
+void findTopAndBot(directedLine* polygon, directedLine*& topV, directedLine*& botV)
+{
+  assert(polygon);
+  directedLine* tempV;
+  topV = botV = polygon;
+   for(tempV = polygon->getNext(); tempV != polygon; tempV = tempV->getNext())
+    {
+      if(compV2InY(topV->head(), tempV->head())<0) {
+	topV = tempV;
+      }
+      if(compV2InY(botV->head(), tempV->head())>0) {
+	botV = tempV;
+      }
+    }
+}
+   
+void findGridChains(directedLine* topV, directedLine* botV, 
+		    gridWrap* grid,
+		    gridBoundaryChain*& leftGridChain,
+		    gridBoundaryChain*& rightGridChain)
+{
+  /*find the first(top) and the last (bottom) grid line which intersect the
+   *this polygon
+   */
+  Int firstGridIndex; /*the index in the grid*/
+  Int lastGridIndex;
+
+  firstGridIndex = (Int) ((topV->head()[1] - grid->get_v_min()) / (grid->get_v_max() - grid->get_v_min()) * (grid->get_n_vlines()-1));
+
+  if(botV->head()[1] < grid->get_v_min())
+    lastGridIndex = 0;
+  else
+    lastGridIndex  = (Int) ((botV->head()[1] - grid->get_v_min()) / (grid->get_v_max() - grid->get_v_min()) * (grid->get_n_vlines()-1)) + 1;
+
+  /*find the interval inside  the polygon for each gridline*/
+  Int *leftGridIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(leftGridIndices);
+  Int *rightGridIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(rightGridIndices);
+  Int *leftGridInnerIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(leftGridInnerIndices);
+  Int *rightGridInnerIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(rightGridInnerIndices);
+
+  findLeftGridIndices(topV, firstGridIndex, lastGridIndex, grid,  leftGridIndices, leftGridInnerIndices);
+
+  findRightGridIndices(topV, firstGridIndex, lastGridIndex, grid,  rightGridIndices, rightGridInnerIndices);
+
+  leftGridChain =  new gridBoundaryChain(grid, firstGridIndex, firstGridIndex-lastGridIndex+1, leftGridIndices, leftGridInnerIndices);
+
+  rightGridChain = new gridBoundaryChain(grid, firstGridIndex, firstGridIndex-lastGridIndex+1, rightGridIndices, rightGridInnerIndices);
+
+  free(leftGridIndices);
+  free(rightGridIndices);
+  free(leftGridInnerIndices);
+  free(rightGridInnerIndices);
+}
+
+void findDownCorners(Real *botVertex, 
+		   vertexArray *leftChain, Int leftChainStartIndex, Int leftChainEndIndex,
+		   vertexArray *rightChain, Int rightChainStartIndex, Int rightChainEndIndex,
+		   Real v,
+		   Real uleft,
+		   Real uright,
+		   Int& ret_leftCornerWhere, /*0: left chain, 1: topvertex, 2: rightchain*/
+		   Int& ret_leftCornerIndex, /*useful when ret_leftCornerWhere == 0 or 2*/
+		   Int& ret_rightCornerWhere, /*0: left chain, 1: topvertex, 2: rightchain*/
+		   Int& ret_rightCornerIndex /*useful when ret_leftCornerWhere == 0 or 2*/
+		   )
+{
+#ifdef MYDEBUG
+printf("*************enter find donw corner\n");
+printf("finddownCorner: v=%f, uleft=%f, uright=%f\n", v, uleft, uright);
+printf("(%i,%i,%i,%i)\n", leftChainStartIndex, leftChainEndIndex,rightChainStartIndex, rightChainEndIndex); 
+printf("left chain is\n");
+leftChain->print();
+printf("right chain is\n");
+rightChain->print();
+#endif
+
+  assert(v > botVertex[1]);
+  Real leftGridPoint[2];
+  leftGridPoint[0] = uleft;
+  leftGridPoint[1] = v;
+  Real rightGridPoint[2];
+  rightGridPoint[0] = uright;
+  rightGridPoint[1] = v;
+
+  Int i;
+  Int index1, index2;
+
+  index1 = leftChain->findIndexBelowGen(v, leftChainStartIndex, leftChainEndIndex);
+  index2 = rightChain->findIndexBelowGen(v, rightChainStartIndex, rightChainEndIndex);
+
+  if(index2 <= rightChainEndIndex) //index2 was found above
+    index2 = rightChain->skipEqualityFromStart(v, index2, rightChainEndIndex);
+
+  if(index1>leftChainEndIndex && index2 > rightChainEndIndex) /*no point below v on left chain or right chain*/
+    {
+
+      /*the botVertex is the only vertex below v*/
+      ret_leftCornerWhere = 1;
+      ret_rightCornerWhere = 1;
+    }
+  else if(index1>leftChainEndIndex ) /*index2 <= rightChainEndIndex*/
+    {
+
+      ret_rightCornerWhere = 2; /*on right chain*/
+      ret_rightCornerIndex = index2;
+
+
+      Real tempMin = rightChain->getVertex(index2)[0];
+      Int tempI = index2;
+      for(i=index2+1; i<= rightChainEndIndex; i++)
+	if(rightChain->getVertex(i)[0] < tempMin)
+	  {
+	    tempI = i;
+	    tempMin = rightChain->getVertex(i)[0];
+	  }
+
+
+      //we consider whether we can use botVertex as left corner. First check 
+      //if (leftGirdPoint, botVertex) interesects right chian or not.
+     if(DBG_intersectChain(rightChain, rightChainStartIndex,rightChainEndIndex,
+				    leftGridPoint, botVertex))
+       {
+	 ret_leftCornerWhere = 2;//right
+	 ret_leftCornerIndex = index2; //should use tempI???
+       }
+     else if(botVertex[0] < tempMin)
+       ret_leftCornerWhere = 1; //bot
+     else
+       {
+	 ret_leftCornerWhere = 2; //right
+	 ret_leftCornerIndex = tempI;
+       }
+    }
+  else if(index2> rightChainEndIndex) /*index1<=leftChainEndIndex*/
+    {
+      ret_leftCornerWhere = 0; /*left chain*/
+      ret_leftCornerIndex = index1;
+      
+      /*find the vertex on the left chain with the maximum u,
+       *either this vertex or the botvertex can be used as the right corner
+       */
+
+      Int tempI;
+      //skip those points which are equal to v. (avoid degeneratcy)
+      for(tempI = index1; tempI <= leftChainEndIndex; tempI++)
+	if(leftChain->getVertex(tempI)[1] < v) 
+	  break;
+      if(tempI > leftChainEndIndex)
+	ret_rightCornerWhere = 1;
+      else
+	{
+	  Real tempMax = leftChain->getVertex(tempI)[0];
+	  for(i=tempI; i<= leftChainEndIndex; i++)
+	    if(leftChain->getVertex(i)[0] > tempMax)
+	      {
+		tempI = i;
+		tempMax = leftChain->getVertex(i)[0];
+	      }
+
+
+
+	  //we consider whether we can use botVertex as a corner. So first we check 
+	  //whether (rightGridPoint, botVertex) interescts the left chain or not.
+	  if(DBG_intersectChain(leftChain, leftChainStartIndex,leftChainEndIndex,
+				    rightGridPoint, botVertex))
+	    {
+	      ret_rightCornerWhere = 0;
+	      ret_rightCornerIndex = index1; //should use tempI???
+	    }
+	  else if(botVertex[0] > tempMax)
+	    {
+		      
+              ret_rightCornerWhere = 1;
+	    }
+	  else
+	    {
+	      ret_rightCornerWhere = 0;
+	      ret_rightCornerIndex = tempI;
+	    }
+	}
+      
+    }
+  else /*index1<=leftChainEndIndex and index2 <=rightChainEndIndex*/
+    {
+      if(leftChain->getVertex(index1)[1] >= rightChain->getVertex(index2)[1]) /*left point above right point*/
+	{
+	  ret_leftCornerWhere = 0; /*on left chain*/
+	  ret_leftCornerIndex = index1;
+
+	  Real tempMax;
+	  Int tempI;
+
+	  tempI = index1;
+	  tempMax = leftChain->getVertex(index1)[0];
+
+	  /*find the maximum u for all the points on the left above the right point index2*/
+	  for(i=index1+1; i<= leftChainEndIndex; i++)
+	    {
+	      if(leftChain->getVertex(i)[1] < rightChain->getVertex(index2)[1])
+		break;
+
+	      if(leftChain->getVertex(i)[0]>tempMax)
+		{
+		  tempI = i;
+		  tempMax = leftChain->getVertex(i)[0];
+		}
+	    }
+	  //we consider if we can use rightChain(index2) as right corner
+	  //we check if (rightChain(index2), rightGidPoint) intersecs left chain or not.
+	  if(DBG_intersectChain(leftChain, leftChainStartIndex,leftChainEndIndex, rightGridPoint, rightChain->getVertex(index2)))
+	    {
+	      ret_rightCornerWhere = 0;
+	      ret_rightCornerIndex = index1; //should use tempI???
+	    }
+	  else if(tempMax >= rightChain->getVertex(index2)[0] ||
+	     tempMax >= uright
+	     )
+	    {
+
+	      ret_rightCornerWhere = 0; /*on left Chain*/
+	      ret_rightCornerIndex = tempI;
+	    }
+	  else
+	    {
+	      ret_rightCornerWhere = 2; /*on right chain*/
+	      ret_rightCornerIndex = index2;
+	    }
+	}
+      else /*left below right*/
+	{
+	  ret_rightCornerWhere = 2; /*on the right*/
+	  ret_rightCornerIndex = index2;
+	  
+	  Real tempMin;
+	  Int tempI;
+	  
+	  tempI = index2;
+	  tempMin = rightChain->getVertex(index2)[0];
+	  
+	  /*find the minimum u for all the points on the right above the left poitn index1*/
+	  for(i=index2+1; i<= rightChainEndIndex; i++)
+	    {
+	      if( rightChain->getVertex(i)[1] < leftChain->getVertex(index1)[1])
+		break;
+	      if(rightChain->getVertex(i)[0] < tempMin)
+		{
+		  tempI = i;
+		  tempMin = rightChain->getVertex(i)[0];
+		}
+	    }
+
+	  //we consider if we can use leftchain(index1) as left corner. 
+	  //we check if (leftChain(index1) intersects right chian or not
+	  if(DBG_intersectChain(rightChain, rightChainStartIndex, rightChainEndIndex, leftGridPoint, leftChain->getVertex(index1)))
+	    {
+	      ret_leftCornerWhere = 2;
+	      ret_leftCornerIndex = index2; //should use tempI???
+	      }
+	  else if(tempMin <= leftChain->getVertex(index1)[0] ||
+	     tempMin <= uleft)				
+	    {
+	      ret_leftCornerWhere = 2; /* on right chain*/
+	      ret_leftCornerIndex = tempI;
+	    }
+	  else
+	    {
+	      ret_leftCornerWhere = 0; /*on left chain*/
+	      ret_leftCornerIndex = index1;
+	    }
+	}
+    }
+
+}
+
+
+void findUpCorners(Real *topVertex, 
+		   vertexArray *leftChain, Int leftChainStartIndex, Int leftChainEndIndex,
+		   vertexArray *rightChain, Int rightChainStartIndex, Int rightChainEndIndex,
+		   Real v,
+		   Real uleft,
+		   Real uright,
+		   Int& ret_leftCornerWhere, /*0: left chain, 1: topvertex, 2: rightchain*/
+		   Int& ret_leftCornerIndex, /*useful when ret_leftCornerWhere == 0 or 2*/
+		   Int& ret_rightCornerWhere, /*0: left chain, 1: topvertex, 2: rightchain*/
+		   Int& ret_rightCornerIndex /*useful when ret_leftCornerWhere == 0 or 2*/
+		   )
+{
+#ifdef MYDEBUG
+printf("***********enter findUpCorners\n");
+#endif
+
+  assert(v < topVertex[1]);
+  Real leftGridPoint[2];
+  leftGridPoint[0] = uleft;
+  leftGridPoint[1] = v;
+  Real rightGridPoint[2];
+  rightGridPoint[0] = uright;
+  rightGridPoint[1] = v;
+
+  Int i;
+  Int index1, index2;
+
+  index1 = leftChain->findIndexFirstAboveEqualGen(v, leftChainStartIndex, leftChainEndIndex);
+
+
+  index2 = rightChain->findIndexFirstAboveEqualGen(v, rightChainStartIndex, rightChainEndIndex);
+
+  if(index2>= leftChainStartIndex)  //index2 was found above  
+    index2 = rightChain->skipEqualityFromStart(v, index2, rightChainEndIndex);
+
+  if(index1<leftChainStartIndex && index2 <rightChainStartIndex) /*no point above v on left chain or right chain*/
+    {
+      /*the topVertex is the only vertex above v*/
+      ret_leftCornerWhere = 1;
+      ret_rightCornerWhere = 1;
+    }
+  else if(index1<leftChainStartIndex ) /*index2 >= rightChainStartIndex*/
+    {
+      ret_rightCornerWhere = 2; /*on right chain*/
+      ret_rightCornerIndex = index2;
+
+      //find the minimum u on right top, either that, or top, or right[index2] is the left corner
+      Real tempMin = rightChain->getVertex(index2)[0];
+      Int tempI = index2;
+      for(i=index2-1; i>=rightChainStartIndex; i--)
+	if(rightChain->getVertex(i)[0] < tempMin)
+	  {
+	    tempMin = rightChain->getVertex(i)[0];
+	    tempI = i;
+	  }
+      //chech whether (leftGridPoint, top) intersects rightchai,
+      //if yes, use right corner as left corner
+      //if not, use top or right[tempI] as left corner
+      if(DBG_intersectChain(rightChain, rightChainStartIndex, rightChainEndIndex,
+			leftGridPoint, topVertex))
+	{
+	  ret_leftCornerWhere = 2; //rightChain
+	  ret_leftCornerIndex = index2; 
+	}
+      else if(topVertex[0] < tempMin)
+	ret_leftCornerWhere = 1; /*topvertex*/
+      else
+	{
+	  ret_leftCornerWhere = 2; //right chain
+	  ret_leftCornerIndex = tempI;
+	}
+	      
+    }
+  else if(index2< rightChainStartIndex) /*index1>=leftChainStartIndex*/
+    {
+      ret_leftCornerWhere = 0; /*left chain*/
+      ret_leftCornerIndex = index1;
+       
+      //find the maximum u on the left top section. either that or topvertex, or left[index1]  is the right corner
+      Real tempMax = leftChain->getVertex(index1)[0];
+      Int tempI = index1;
+
+      for(i=index1-1; i>=leftChainStartIndex; i--){
+
+	if(leftChain->getVertex(i)[0] > tempMax)
+	  {
+
+	    tempMax = leftChain->getVertex(i)[0];
+	    tempI = i;
+	  }
+      }
+      //check whether (rightGridPoint, top) intersects leftChain or not
+      //if yes, we use leftCorner as the right corner
+      //if not, we use either top or left[tempI] as the right corner
+      if(DBG_intersectChain(leftChain, leftChainStartIndex,leftChainEndIndex,
+			    rightGridPoint, topVertex))
+	 {
+	   ret_rightCornerWhere = 0; //left chan
+	   ret_rightCornerIndex = index1;
+	 }
+      else if(topVertex[0] > tempMax)
+	ret_rightCornerWhere = 1;//topVertex
+      else
+	{
+	  ret_rightCornerWhere = 0;//left chain
+	  ret_rightCornerIndex = tempI;
+	}
+    }
+  else /*index1>=leftChainStartIndex and index2 >=rightChainStartIndex*/
+    {
+      if(leftChain->getVertex(index1)[1] <= rightChain->getVertex(index2)[1]) /*left point below right point*/
+	{
+	  ret_leftCornerWhere = 0; /*on left chain*/
+	  ret_leftCornerIndex = index1;
+
+	  Real tempMax;
+	  Int tempI;
+
+	  tempI = index1;
+	  tempMax = leftChain->getVertex(index1)[0];
+
+	  /*find the maximum u for all the points on the left below the right point index2*/
+	  for(i=index1-1; i>= leftChainStartIndex; i--)
+	    {
+	      if(leftChain->getVertex(i)[1] > rightChain->getVertex(index2)[1])
+		break;
+
+	      if(leftChain->getVertex(i)[0]>tempMax)
+		{
+		  tempI = i;
+		  tempMax = leftChain->getVertex(i)[0];
+		}
+	    }
+	  //chek whether (rightChain(index2), rightGridPoint) intersects leftchian or not
+	  if(DBG_intersectChain(leftChain, leftChainStartIndex, leftChainEndIndex, rightGridPoint, rightChain->getVertex(index2)))
+	     {
+	       ret_rightCornerWhere = 0;
+	       ret_rightCornerIndex = index1;
+	     }
+	  else if(tempMax >= rightChain->getVertex(index2)[0] ||
+	     tempMax >= uright)
+	    {
+	      ret_rightCornerWhere = 0; /*on left Chain*/
+	      ret_rightCornerIndex = tempI;
+	    }
+	  else
+	    {
+	      ret_rightCornerWhere = 2; /*on right chain*/
+	      ret_rightCornerIndex = index2;
+	    }
+	}
+      else /*left above right*/
+	{
+	  ret_rightCornerWhere = 2; /*on the right*/
+	  ret_rightCornerIndex = index2;
+	  
+	  Real tempMin;
+	  Int tempI;
+	  
+	  tempI = index2;
+	  tempMin = rightChain->getVertex(index2)[0];
+	  
+	  /*find the minimum u for all the points on the right below the left poitn index1*/
+	  for(i=index2-1; i>= rightChainStartIndex; i--)
+	    {
+	      if( rightChain->getVertex(i)[1] > leftChain->getVertex(index1)[1])
+		break;
+	      if(rightChain->getVertex(i)[0] < tempMin)
+		{
+		  tempI = i;
+		  tempMin = rightChain->getVertex(i)[0];
+		}
+	    }
+          //check whether (leftGRidPoint,left(index1)) interesect right chain 
+	  if(DBG_intersectChain(rightChain, rightChainStartIndex, rightChainEndIndex,
+				leftGridPoint, leftChain->getVertex(index1)))
+	    {
+	      ret_leftCornerWhere = 2; //right
+	      ret_leftCornerIndex = index2;
+	    }
+	  else if(tempMin <= leftChain->getVertex(index1)[0] ||
+	     tempMin <= uleft)
+	    {
+	      ret_leftCornerWhere = 2; /* on right chain*/
+	      ret_leftCornerIndex = tempI;
+	    }
+	  else
+	    {
+	      ret_leftCornerWhere = 0; /*on left chain*/
+	      ret_leftCornerIndex = index1;
+	    }
+	}
+    }
+#ifdef MYDEBUG
+printf("***********leave findUpCorners\n");
+#endif
+}
+
+//return 1 if neck exists, 0 othewise
+Int findNeckF(vertexArray *leftChain, Int botLeftIndex,
+	      vertexArray *rightChain, Int botRightIndex,
+	      gridBoundaryChain* leftGridChain,
+	      gridBoundaryChain* rightGridChain,
+	      Int gridStartIndex,
+	      Int& neckLeft, 
+	      Int& neckRight)
+{
+/*
+printf("enter findNeckF, botleft, botright=%i,%i,gstartindex=%i\n",botLeftIndex,botRightIndex,gridStartIndex);
+printf("leftChain is\n");
+leftChain->print();
+printf("rightChain is\n");
+rightChain->print();
+*/
+
+  Int lowerGridIndex; //the grid below leftChain and rightChian vertices
+  Int i;
+  Int n_vlines = leftGridChain->get_nVlines();
+  Real v;
+  if(botLeftIndex >= leftChain->getNumElements() ||
+     botRightIndex >= rightChain->getNumElements())
+    return 0; //no neck exists
+     
+  v=min(leftChain->getVertex(botLeftIndex)[1], rightChain->getVertex(botRightIndex)[1]);  
+
+ 
+
+
+  for(i=gridStartIndex; i<n_vlines; i++)
+    if(leftGridChain->get_v_value(i) <= v && 
+       leftGridChain->getUlineIndex(i)<= rightGridChain->getUlineIndex(i))
+      break;
+  
+  lowerGridIndex = i;
+
+  if(lowerGridIndex == n_vlines) //the two trm vertex are higher than all gridlines
+    return 0;
+  else 
+    {
+      Int botLeft2, botRight2;
+/*
+printf("leftGridChain->get_v_)value=%f\n",leftGridChain->get_v_value(lowerGridIndex), botLeftIndex); 
+printf("leftChain->get_vertex(0)=(%f,%f)\n", leftChain->getVertex(0)[0],leftChain->getVertex(0)[1]);
+printf("leftChain->get_vertex(1)=(%f,%f)\n", leftChain->getVertex(1)[0],leftChain->getVertex(1)[1]);
+printf("leftChain->get_vertex(2)=(%f,%f)\n", leftChain->getVertex(2)[0],leftChain->getVertex(2)[1]);
+*/
+      botLeft2 = leftChain->findIndexFirstAboveEqualGen(leftGridChain->get_v_value(lowerGridIndex), botLeftIndex, leftChain->getNumElements()-1) -1 ;
+
+/*
+printf("botLeft2=%i\n", botLeft2);
+printf("leftChain->getNumElements=%i\n", leftChain->getNumElements());
+*/
+
+      botRight2 = rightChain->findIndexFirstAboveEqualGen(leftGridChain->get_v_value(lowerGridIndex), botRightIndex, rightChain->getNumElements()-1) -1;
+      if(botRight2 < botRightIndex) botRight2=botRightIndex;
+
+      if(botLeft2 < botLeftIndex) botLeft2 = botLeftIndex;
+
+      assert(botLeft2 >= botLeftIndex);
+      assert(botRight2 >= botRightIndex);
+      //find nectLeft so that it is th erightmost vertex on letChain
+
+      Int tempI = botLeftIndex;
+      Real temp = leftChain->getVertex(tempI)[0];
+      for(i=botLeftIndex+1; i<= botLeft2; i++)
+	if(leftChain->getVertex(i)[0] > temp)
+	  {
+	    temp = leftChain->getVertex(i)[0];
+	    tempI = i;
+	  }
+      neckLeft = tempI;
+
+      tempI = botRightIndex;
+      temp = rightChain->getVertex(tempI)[0];
+      for(i=botRightIndex+1; i<= botRight2; i++)
+	if(rightChain->getVertex(i)[0] < temp)
+	  {
+	    temp = rightChain->getVertex(i)[0];
+	    tempI = i;
+	  }
+      neckRight = tempI;
+      return 1;
+    }
+}
+							
+  
+	          
+/*find i>=botLeftIndex,j>=botRightIndex so that
+ *(leftChain[i], rightChain[j]) is a neck.
+ */
+void findNeck(vertexArray *leftChain, Int botLeftIndex, 
+	      vertexArray *rightChain, Int botRightIndex, 
+	      Int& leftLastIndex, /*left point of the neck*/
+	      Int& rightLastIndex /*right point of the neck*/
+	      )
+{
+  assert(botLeftIndex < leftChain->getNumElements() &&
+     botRightIndex < rightChain->getNumElements());
+     
+  /*now the neck exists for sure*/
+
+  if(leftChain->getVertex(botLeftIndex)[1] <= rightChain->getVertex(botRightIndex)[1]) //left below right
+    {
+
+      leftLastIndex = botLeftIndex;
+      
+      /*find i so that rightChain[i][1] >= leftchainbotverte[1], and i+1<
+       */
+      rightLastIndex=rightChain->findIndexAboveGen(leftChain->getVertex(botLeftIndex)[1], botRightIndex+1, rightChain->getNumElements()-1);    
+    }
+  else  //left above right
+    {
+
+      rightLastIndex = botRightIndex;
+     
+      leftLastIndex = leftChain->findIndexAboveGen(rightChain->getVertex(botRightIndex)[1], 
+						  botLeftIndex+1,
+						  leftChain->getNumElements()-1);
+    }
+}
+      
+      
+
+void findLeftGridIndices(directedLine* topEdge, Int firstGridIndex, Int lastGridIndex, gridWrap* grid,  Int* ret_indices, Int* ret_innerIndices)
+{
+
+  Int i,k,isHoriz = 0;
+  Int n_ulines = grid->get_n_ulines();
+  Real uMin = grid->get_u_min();
+  Real uMax = grid->get_u_max();
+  /*
+  Real vMin = grid->get_v_min();
+  Real vMax = grid->get_v_max();
+  */
+  Real slop = 0.0, uinterc;
+
+#ifdef SHORTEN_GRID_LINE
+  //uintercBuf stores all the interction u value for each grid line
+  //notice that lastGridIndex<= firstGridIndex
+  Real *uintercBuf = (Real *) malloc (sizeof(Real) * (firstGridIndex-lastGridIndex+1));
+  assert(uintercBuf);
+#endif
+
+  /*initialization to make vtail bigger than grid->...*/
+  directedLine* dLine = topEdge;
+  Real vtail = grid->get_v_value(firstGridIndex) + 1.0; 
+  Real tempMaxU = grid->get_u_min();
+
+
+  /*for each grid line*/
+  for(k=0, i=firstGridIndex; i>=lastGridIndex; i--, k++)
+    {
+
+      Real grid_v_value = grid->get_v_value(i);
+
+      /*check whether this grid line is below the current trim edge.*/
+      if(vtail > grid_v_value)
+	{
+	  /*since the grid line is below the trim edge, we 
+	   *find the trim edge which will contain the trim line
+	   */
+	  while( (vtail=dLine->tail()[1]) > grid_v_value){
+
+	    tempMaxU = max(tempMaxU, dLine->tail()[0]);
+	    dLine = dLine -> getNext();
+	  }
+
+	  if( fabs(dLine->head()[1] - vtail) < ZERO)
+	    isHoriz = 1;
+	  else
+	    {
+	      isHoriz = 0;
+	      slop = (dLine->head()[0] - dLine->tail()[0]) / (dLine->head()[1]-vtail);
+	    }
+	}
+
+      if(isHoriz)
+	{
+	  uinterc = max(dLine->head()[0], dLine->tail()[0]);
+	}
+      else
+	{
+	  uinterc = slop * (grid_v_value - vtail) + dLine->tail()[0];
+	}
+      
+      tempMaxU = max(tempMaxU, uinterc);
+
+      if(uinterc < uMin && uinterc >= uMin - ZERO)
+	uinterc = uMin;
+      if(uinterc > uMax && uinterc <= uMax + ZERO)
+	uinterc = uMax;
+
+#ifdef SHORTEN_GRID_LINE
+      uintercBuf[k] = uinterc;
+#endif
+
+      assert(uinterc >= uMin && uinterc <= uMax);
+       if(uinterc == uMax)
+         ret_indices[k] = n_ulines-1;
+       else
+	 ret_indices[k] = (Int)(((uinterc-uMin)/(uMax - uMin)) * (n_ulines-1)) + 1;
+      if(ret_indices[k] >= n_ulines)
+	ret_indices[k] = n_ulines-1;
+
+
+      ret_innerIndices[k] = (Int)(((tempMaxU-uMin)/(uMax - uMin)) * (n_ulines-1)) + 1;
+
+      /*reinitialize tempMaxU for next grdiLine*/
+      tempMaxU = uinterc;
+    }
+#ifdef SHORTEN_GRID_LINE
+  //for each grid line, compare the left grid point with the 
+  //intersection point. If the two points are too close, then
+  //we should move the grid point one grid to the right
+  //and accordingly we should update the inner index.
+  for(k=0, i=firstGridIndex; i>=lastGridIndex; i--, k++)
+    {
+      //check gridLine i
+      //check ret_indices[k]
+      Real a = grid->get_u_value(ret_indices[k]-1);
+      Real b = grid->get_u_value(ret_indices[k]);
+      assert(uintercBuf[k] >= a && uintercBuf < b);
+      if( (b-uintercBuf[k]) <= 0.2 * (b-a)) //interc is very close to b
+	{
+	  ret_indices[k]++;
+	}
+
+      //check ret_innerIndices[k]
+      if(k>0)
+	{
+	  if(ret_innerIndices[k] < ret_indices[k-1])
+	    ret_innerIndices[k] = ret_indices[k-1];
+	  if(ret_innerIndices[k] < ret_indices[k])
+	    ret_innerIndices[k] = ret_indices[k];
+	}
+    }
+  //clean up
+  free(uintercBuf);
+#endif
+}
+
+void findRightGridIndices(directedLine* topEdge, Int firstGridIndex, Int lastGridIndex, gridWrap* grid,  Int* ret_indices, Int* ret_innerIndices)
+{
+
+  Int i,k;
+  Int n_ulines = grid->get_n_ulines();
+  Real uMin = grid->get_u_min();
+  Real uMax = grid->get_u_max();
+  /*
+  Real vMin = grid->get_v_min();
+  Real vMax = grid->get_v_max();
+  */
+  Real slop = 0.0, uinterc;
+
+#ifdef SHORTEN_GRID_LINE
+  //uintercBuf stores all the interction u value for each grid line
+  //notice that firstGridIndex >= lastGridIndex
+  Real *uintercBuf = (Real *) malloc (sizeof(Real) * (firstGridIndex-lastGridIndex+1));
+  assert(uintercBuf);
+#endif
+
+  /*initialization to make vhead bigger than grid->v_value...*/
+  directedLine* dLine = topEdge->getPrev();
+  Real vhead = dLine->tail()[1];
+  Real tempMinU = grid->get_u_max();
+
+  /*for each grid line*/
+  for(k=0, i=firstGridIndex; i>=lastGridIndex; i--, k++)
+    {
+
+      Real grid_v_value = grid->get_v_value(i);
+
+
+      /*check whether this grid line is below the current trim edge.*/
+      if(vhead >= grid_v_value)
+	{
+	  /*since the grid line is below the tail of the trim edge, we 
+	   *find the trim edge which will contain the trim line
+	   */
+	  while( (vhead=dLine->head()[1]) > grid_v_value){
+	    tempMinU = min(tempMinU, dLine->head()[0]);
+	    dLine = dLine -> getPrev();
+	  }
+
+	  /*skip the equality in the case of degenerat case: horizontal */
+	  while(dLine->head()[1] == grid_v_value)
+	    dLine = dLine->getPrev();
+	    
+	  assert( dLine->tail()[1] != dLine->head()[1]);
+	  slop = (dLine->tail()[0] - dLine->head()[0]) / (dLine->tail()[1]-dLine->head()[1]);
+	  /*
+	   if(dLine->tail()[1] == vhead)
+	     isHoriz = 1;
+	     else
+	    {
+	      isHoriz = 0;
+	      slop = (dLine->tail()[0] - dLine->head()[0]) / (dLine->tail()[1]-vhead);
+	    }
+	    */
+	}
+	uinterc = slop * (grid_v_value - dLine->head()[1]) + dLine->head()[0];
+
+      //in case unterc is outside of the grid due to floating point
+      if(uinterc < uMin)
+	uinterc = uMin;
+      else if(uinterc > uMax)
+	uinterc = uMax;
+
+#ifdef SHORTEN_GRID_LINE
+      uintercBuf[k] = uinterc;
+#endif      
+
+      tempMinU = min(tempMinU, uinterc);
+
+      assert(uinterc >= uMin && uinterc <= uMax);      
+
+      if(uinterc == uMin)
+	ret_indices[k] = 0;
+      else
+	ret_indices[k] = (int)ceil((((uinterc-uMin)/(uMax - uMin)) * (n_ulines-1))) -1;
+/*
+if(ret_indices[k] >= grid->get_n_ulines())
+  {
+  printf("ERROR3\n");
+  exit(0);
+}
+if(ret_indices[k] < 0)    
+  {
+  printf("ERROR4\n");
+  exit(0);
+}
+*/
+      ret_innerIndices[k] = (int)ceil ((((tempMinU-uMin)/(uMax - uMin)) * (n_ulines-1))) -1;
+
+      tempMinU = uinterc;
+    }
+#ifdef SHORTEN_GRID_LINE
+  //for each grid line, compare the left grid point with the 
+  //intersection point. If the two points are too close, then
+  //we should move the grid point one grid to the right
+  //and accordingly we should update the inner index.
+  for(k=0, i=firstGridIndex; i>=lastGridIndex; i--, k++)
+    {
+      //check gridLine i
+      //check ret_indices[k]
+      Real a = grid->get_u_value(ret_indices[k]);
+      Real b = grid->get_u_value(ret_indices[k]+1);
+      assert(uintercBuf[k] > a && uintercBuf <= b);
+      if( (uintercBuf[k]-a) <= 0.2 * (b-a)) //interc is very close to a
+	{
+	  ret_indices[k]--;
+	}
+
+      //check ret_innerIndices[k]
+      if(k>0)
+	{
+	  if(ret_innerIndices[k] > ret_indices[k-1])
+	    ret_innerIndices[k] = ret_indices[k-1];
+	  if(ret_innerIndices[k] > ret_indices[k])
+	    ret_innerIndices[k] = ret_indices[k];
+	}
+    }
+  //clean up
+  free(uintercBuf);
+#endif
+}
+
+
+void sampleMonoPoly(directedLine* polygon, gridWrap* grid, Int ulinear, Int vlinear, primStream* pStream, rectBlockArray* rbArray)
+{
+/*
+{
+grid->print();
+polygon->writeAllPolygons("zloutputFile");
+exit(0);
+}
+*/
+
+if(grid->get_n_ulines() == 2 ||
+   grid->get_n_vlines() == 2)
+{ 
+  if(ulinear && grid->get_n_ulines() == 2)
+    {
+      monoTriangulationFun(polygon, compV2InY, pStream);   
+      return;
+    }
+  else if(DBG_isConvex(polygon) && polygon->numEdges() >=4)
+     {
+       triangulateConvexPoly(polygon, ulinear, vlinear, pStream);
+       return;
+     }
+  else if(vlinear || DBG_is_U_direction(polygon))
+    {
+      Int n_cusps;//num interior cusps
+      Int n_edges = polygon->numEdges();
+      directedLine** cusps = (directedLine**) malloc(sizeof(directedLine*) * n_edges);
+      assert(cusps);
+      findInteriorCuspsX(polygon, n_cusps, cusps);
+
+      if(n_cusps == 0) //u_monotone
+	{
+
+	  monoTriangulationFun(polygon, compV2InX, pStream);
+
+          free(cusps);
+          return;          
+	}
+      else if(n_cusps == 1) //one interior cusp
+	{
+
+	  directedLine* new_polygon = polygonConvert(cusps[0]);
+
+	  directedLine* other = findDiagonal_singleCuspX( new_polygon);
+
+
+
+	  //<other> should NOT be null unless there are self-intersecting
+          //trim curves. In that case, we don't want to core dump, instead,
+          //we triangulate anyway, and print out error message.
+	  if(other == NULL)
+	    {
+	      monoTriangulationFun(polygon, compV2InX, pStream);
+	      free(cusps);
+	      return;
+	    }
+
+	  directedLine* ret_p1;
+	  directedLine* ret_p2;
+
+	  new_polygon->connectDiagonal_2slines(new_polygon, other, 
+					   &ret_p1,
+					   &ret_p2,
+					   new_polygon);
+
+	  monoTriangulationFun(ret_p1, compV2InX, pStream);
+	  monoTriangulationFun(ret_p2, compV2InX, pStream);
+
+	  ret_p1->deleteSinglePolygonWithSline();	      
+	  ret_p2->deleteSinglePolygonWithSline();	  
+
+          free(cusps);
+          return;
+         }
+     free(cusps);
+     }
+}
+
+  /*find the top and bottom of the polygon. It is supposed to be
+   *a V-monotone polygon
+   */
+
+  directedLine* tempV;
+  directedLine* topV;
+  directedLine* botV;
+  topV = botV = polygon;
+
+  for(tempV = polygon->getNext(); tempV != polygon; tempV = tempV->getNext())
+    {
+      if(compV2InY(topV->head(), tempV->head())<0) {
+
+	topV = tempV;
+      }
+      if(compV2InY(botV->head(), tempV->head())>0) {
+
+	botV = tempV;
+      }
+    }
+  
+  /*find the first(top) and the last (bottom) grid line which intersect the
+   *this polygon
+   */
+  Int firstGridIndex; /*the index in the grid*/
+  Int lastGridIndex;
+  firstGridIndex = (Int) ((topV->head()[1] - grid->get_v_min()) / (grid->get_v_max() - grid->get_v_min()) * (grid->get_n_vlines()-1));
+  lastGridIndex  = (Int) ((botV->head()[1] - grid->get_v_min()) / (grid->get_v_max() - grid->get_v_min()) * (grid->get_n_vlines()-1)) + 1;
+
+
+  /*find the interval inside  the polygon for each gridline*/
+  Int *leftGridIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(leftGridIndices);
+  Int *rightGridIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(rightGridIndices);
+  Int *leftGridInnerIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(leftGridInnerIndices);
+  Int *rightGridInnerIndices = (Int*) malloc(sizeof(Int) * (firstGridIndex - lastGridIndex +1));
+  assert(rightGridInnerIndices);
+
+  findLeftGridIndices(topV, firstGridIndex, lastGridIndex, grid,  leftGridIndices, leftGridInnerIndices);
+
+  findRightGridIndices(topV, firstGridIndex, lastGridIndex, grid,  rightGridIndices, rightGridInnerIndices);
+
+  gridBoundaryChain leftGridChain(grid, firstGridIndex, firstGridIndex-lastGridIndex+1, leftGridIndices, leftGridInnerIndices);
+
+  gridBoundaryChain rightGridChain(grid, firstGridIndex, firstGridIndex-lastGridIndex+1, rightGridIndices, rightGridInnerIndices);
+
+
+
+//  leftGridChain.draw();
+//  leftGridChain.drawInner();
+//  rightGridChain.draw();
+//  rightGridChain.drawInner();
+  /*(1) determine the grid boundaries (left and right).
+   *(2) process polygon  into two monotone chaines: use vertexArray
+   *(3) call sampleMonoPolyRec
+   */
+
+  /*copy the two chains into vertexArray datastructure*/
+  Int i;
+  vertexArray leftChain(20); /*this is a dynamic array*/
+  for(i=1; i<=topV->get_npoints()-2; i++) { /*the first vertex is the top vertex which doesn't belong to inc_chain*/
+    leftChain.appendVertex(topV->getVertex(i));
+  }
+  for(tempV = topV->getNext(); tempV != botV; tempV = tempV->getNext())
+    {
+      for(i=0; i<=tempV->get_npoints()-2; i++){
+	leftChain.appendVertex(tempV->getVertex(i));
+      }
+    }
+  
+  vertexArray rightChain(20);
+  for(tempV = topV->getPrev(); tempV != botV; tempV = tempV->getPrev())
+    {
+      for(i=tempV->get_npoints()-2; i>=0; i--){
+	rightChain.appendVertex(tempV->getVertex(i));
+      }
+    }
+  for(i=botV->get_npoints()-2; i>=1; i--){ 
+    rightChain.appendVertex(tempV->getVertex(i));
+  }
+
+  sampleMonoPolyRec(topV->head(),
+		    botV->head(),
+		    &leftChain,
+		    0,
+		    &rightChain,
+		    0,
+		    &leftGridChain,
+		    &rightGridChain,
+		    0,
+		    pStream,
+		    rbArray);
+
+
+  /*cleanup space*/
+  free(leftGridIndices);
+  free(rightGridIndices);
+  free(leftGridInnerIndices);
+  free(rightGridInnerIndices);
+}
+
+void sampleMonoPolyRec(
+		       Real* topVertex,
+		       Real* botVertex,
+		       vertexArray* leftChain, 
+		       Int leftStartIndex,
+		       vertexArray* rightChain,
+		       Int rightStartIndex,
+		       gridBoundaryChain* leftGridChain, 
+		       gridBoundaryChain* rightGridChain, 
+		       Int gridStartIndex,
+		       primStream* pStream,
+		       rectBlockArray* rbArray)
+{
+
+  /*find the first connected component, and the four corners.
+   */
+  Int index1, index2; /*the first and last grid line of the first connected component*/
+
+  if(topVertex[1] <= botVertex[1])
+    return;
+    
+  /*find i so that the grid line is below the top vertex*/
+  Int i=gridStartIndex;
+  while (i < leftGridChain->get_nVlines())
+    {
+      if(leftGridChain->get_v_value(i) < topVertex[1])
+	break;
+      i++;
+    }
+
+  /*find the first connected component starting with i*/
+  /*find index1 so that left_uline_index <= right_uline_index, that is, this
+   *grid line contains at least one inner grid point
+   */
+  index1=i;
+  int num_skipped_grid_lines=0;
+  while(index1 < leftGridChain->get_nVlines())
+    {
+      if(leftGridChain->getUlineIndex(index1) <= rightGridChain->getUlineIndex(index1))
+	break;
+      num_skipped_grid_lines++;
+      index1++;
+    }
+
+
+
+  if(index1 >= leftGridChain->get_nVlines()) /*no grid line exists which has inner point*/
+    {
+      /*stop recursion, ...*/
+      /*monotone triangulate it...*/
+//      printf("no grid line exists\n");      
+/*
+      monoTriangulationRecOpt(topVertex, botVertex, leftChain, leftStartIndex,
+			   rightChain, rightStartIndex, pStream);
+*/
+
+if(num_skipped_grid_lines <2)
+  {
+    monoTriangulationRecGenOpt(topVertex, botVertex, leftChain, leftStartIndex,
+			       leftChain->getNumElements()-1,
+			       rightChain, rightStartIndex, 
+			       rightChain->getNumElements()-1,
+			       pStream);
+  }
+else
+  {
+    //the optimum way to triangulate is top-down since this polygon
+    //is narrow-long.
+    monoTriangulationRec(topVertex, botVertex, leftChain, leftStartIndex,
+			 rightChain, rightStartIndex, pStream);
+  }
+
+/*
+      monoTriangulationRec(topVertex, botVertex, leftChain, leftStartIndex,
+			   rightChain, rightStartIndex, pStream);
+*/
+
+/*      monoTriangulationRecGenTBOpt(topVertex, botVertex, 
+				   leftChain, leftStartIndex, leftChain->getNumElements()-1,
+				   rightChain, rightStartIndex, rightChain->getNumElements()-1, 
+				   pStream);*/
+      
+
+
+    }
+  else
+    {
+
+      /*find index2 so that left_inner_index <= right_inner_index holds until index2*/
+      index2=index1+1;
+      if(index2 < leftGridChain->get_nVlines())
+	while(leftGridChain->getInnerIndex(index2) <= rightGridChain->getInnerIndex(index2))
+	  {
+	    index2++;
+	    if(index2 >= leftGridChain->get_nVlines())
+	      break;
+	  }
+      
+      index2--;
+      
+
+      
+      /*the neck*/
+      Int neckLeftIndex;
+      Int neckRightIndex;
+
+      /*the four corners*/
+      Int up_leftCornerWhere;
+      Int up_leftCornerIndex;
+      Int up_rightCornerWhere;
+      Int up_rightCornerIndex;
+      Int down_leftCornerWhere;
+      Int down_leftCornerIndex;
+      Int down_rightCornerWhere;
+      Int down_rightCornerIndex;
+
+      Real* tempBotVertex; /*the bottom vertex for this component*/
+      Real* nextTopVertex=NULL; /*for the recursion*/
+      Int nextLeftStartIndex=0;
+      Int nextRightStartIndex=0;
+
+      /*find the points below the grid line index2 on both chains*/
+      Int botLeftIndex = leftChain->findIndexStrictBelowGen(
+						      leftGridChain->get_v_value(index2),
+						      leftStartIndex,
+						      leftChain->getNumElements()-1);
+      Int botRightIndex = rightChain->findIndexStrictBelowGen(
+							rightGridChain->get_v_value(index2),
+							rightStartIndex,
+							rightChain->getNumElements()-1);
+      /*if either botLeftIndex>= numelements,
+       *        or botRightIndex >= numelemnet,
+       *then there is no neck exists. the bottom vertex is botVertex, 
+       */
+      if(! findNeckF(leftChain, botLeftIndex, rightChain, botRightIndex,
+		   leftGridChain, rightGridChain, index2, neckLeftIndex, neckRightIndex))
+	 /*
+      if(botLeftIndex == leftChain->getNumElements() ||
+	 botRightIndex == rightChain->getNumElements())
+	 */
+	{
+#ifdef MYDEBUG
+	  printf("neck NOT exists, botRightIndex=%i\n", botRightIndex);
+#endif
+
+	  tempBotVertex =  botVertex;
+	  nextTopVertex = botVertex;
+	  botLeftIndex = leftChain->getNumElements()-1;
+	  botRightIndex = rightChain->getNumElements()-1;
+	}
+      else /*neck exists*/
+	{
+#ifdef MYDEBUG
+	  printf("neck exists\n");
+#endif
+
+          /*
+	  findNeck(leftChain, botLeftIndex,
+		   rightChain, botRightIndex,
+		   neckLeftIndex,
+		   neckRightIndex);
+		   */
+#ifdef MYDEBUG
+printf("neck is found, neckLeftIndex=%i, neckRightIndex=%i\n", neckLeftIndex, neckRightIndex);
+glBegin(GL_LINES);
+glVertex2fv(leftChain->getVertex(neckLeftIndex));
+glVertex2fv(rightChain->getVertex(neckRightIndex));
+glEnd();
+#endif
+
+	  if(leftChain->getVertex(neckLeftIndex)[1] <= rightChain->getVertex(neckRightIndex)[1])
+	    {
+	      tempBotVertex = leftChain->getVertex(neckLeftIndex);
+	      botLeftIndex = neckLeftIndex-1;
+	      botRightIndex = neckRightIndex;
+	      nextTopVertex = rightChain->getVertex(neckRightIndex);
+	      nextLeftStartIndex = neckLeftIndex;
+	      nextRightStartIndex = neckRightIndex+1;
+	    }
+	  else
+	    {
+	      tempBotVertex = rightChain->getVertex(neckRightIndex);
+	      botLeftIndex = neckLeftIndex;
+	      botRightIndex = neckRightIndex-1;		  
+	      nextTopVertex = leftChain->getVertex(neckLeftIndex);
+	      nextLeftStartIndex = neckLeftIndex+1;
+	      nextRightStartIndex = neckRightIndex;
+	    }
+	}
+
+      findUpCorners(topVertex,
+		    leftChain,
+		    leftStartIndex, botLeftIndex,
+		    rightChain,
+		    rightStartIndex, botRightIndex,
+		    leftGridChain->get_v_value(index1),
+		    leftGridChain->get_u_value(index1),
+		    rightGridChain->get_u_value(index1),
+		    up_leftCornerWhere,
+		    up_leftCornerIndex,
+		    up_rightCornerWhere,
+		    up_rightCornerIndex);
+
+      findDownCorners(tempBotVertex,
+		      leftChain,
+		      leftStartIndex, botLeftIndex,
+		      rightChain,
+		      rightStartIndex, botRightIndex,
+		      leftGridChain->get_v_value(index2),
+		      leftGridChain->get_u_value(index2),
+		      rightGridChain->get_u_value(index2),
+		      down_leftCornerWhere,
+		      down_leftCornerIndex,
+		      down_rightCornerWhere,
+		      down_rightCornerIndex);	      
+#ifdef MYDEBUG
+      printf("find corners done, down_leftwhere=%i, down_righwhere=%i,\n",down_leftCornerWhere, down_rightCornerWhere );
+      printf("find corners done, up_leftwhere=%i, up_righwhere=%i,\n",up_leftCornerWhere, up_rightCornerWhere );
+      printf("find corners done, up_leftindex=%i, up_righindex=%i,\n",up_leftCornerIndex, up_rightCornerIndex );
+      printf("find corners done, down_leftindex=%i, down_righindex=%i,\n",down_leftCornerIndex, down_rightCornerIndex );
+#endif
+
+/*
+      drawCorners(topVertex,
+		  tempBotVertex,
+		  leftChain,
+		  rightChain,
+		  leftGridChain,
+		  rightGridChain,
+		  index1,
+		  index2,
+		  up_leftCornerWhere,
+		  up_leftCornerIndex,
+		  up_rightCornerWhere,
+		  up_rightCornerIndex,
+		  down_leftCornerWhere,
+		  down_leftCornerIndex,
+		  down_rightCornerWhere,
+		  down_rightCornerIndex);
+*/
+
+
+      sampleConnectedComp(topVertex, tempBotVertex,
+			  leftChain, 
+			  leftStartIndex, botLeftIndex,
+			  rightChain,
+			  rightStartIndex, botRightIndex,
+			  leftGridChain,
+			  rightGridChain,
+			  index1, index2,
+			  up_leftCornerWhere,
+			  up_leftCornerIndex,
+			  up_rightCornerWhere,
+			  up_rightCornerIndex,
+			  down_leftCornerWhere,
+			  down_leftCornerIndex,
+			  down_rightCornerWhere,
+			  down_rightCornerIndex,
+			  pStream,
+			  rbArray
+			  );
+
+      /*recursion*/
+
+      sampleMonoPolyRec(
+			nextTopVertex,
+			botVertex,
+			leftChain, 
+			nextLeftStartIndex,
+			rightChain,
+			nextRightStartIndex,
+			leftGridChain, 
+			rightGridChain, 
+			index2+1,
+			pStream, rbArray);
+			
+
+    }
+
+}
+
+void sampleLeftStrip(vertexArray* leftChain,
+		     Int topLeftIndex,
+		     Int botLeftIndex,
+		     gridBoundaryChain* leftGridChain,
+		     Int leftGridChainStartIndex,
+		     Int leftGridChainEndIndex,
+		     primStream* pStream
+		     )
+{
+  assert(leftChain->getVertex(topLeftIndex)[1] > leftGridChain->get_v_value(leftGridChainStartIndex));
+  assert(leftChain->getVertex(topLeftIndex+1)[1] <= leftGridChain->get_v_value(leftGridChainStartIndex));
+  assert(leftChain->getVertex(botLeftIndex)[1] <= leftGridChain->get_v_value(leftGridChainEndIndex));
+  assert(leftChain->getVertex(botLeftIndex-1)[1] > leftGridChain->get_v_value(leftGridChainEndIndex));
+
+  /*
+   *(1)find the last grid line which doesn'; pass below
+   * this first edge, sample this region: one trim edge and 
+   * possily multiple grid lines.
+   */
+  Real *upperVert, *lowerVert; /*the end points of the first trim edge*/
+  upperVert = leftChain->getVertex(topLeftIndex);
+  lowerVert = leftChain->getVertex(topLeftIndex+1);
+  
+  Int index = leftGridChainStartIndex;
+  while(leftGridChain->get_v_value(index) >= lowerVert[1]){
+    index++;
+    if(index > leftGridChainEndIndex) 
+      break;
+  }
+  index--;
+  
+  sampleLeftSingleTrimEdgeRegion(upperVert, lowerVert,
+				 leftGridChain,
+				 leftGridChainStartIndex,
+				 index,
+				 pStream);
+  sampleLeftStripRec(leftChain, topLeftIndex+1, botLeftIndex,
+		     leftGridChain, index, leftGridChainEndIndex,
+		     pStream);
+
+}
+
+void sampleLeftStripRec(vertexArray* leftChain,
+		     Int topLeftIndex,
+		     Int botLeftIndex,
+		     gridBoundaryChain* leftGridChain,
+		     Int leftGridChainStartIndex,
+		     Int leftGridChainEndIndex,
+			primStream* pStream
+		     )
+{
+  /*now top left trim vertex is below the top grid line.
+   */
+  /*stop condition: if topLeftIndex >= botLeftIndex, then stop.
+   */
+  if(topLeftIndex >= botLeftIndex) 
+    return;
+
+  /*find the last trim vertex which is above the second top grid line:
+   * index1.
+   *and sampleLeftOneGridStep(leftchain, topLeftIndex, index1, leftGridChain,
+   * leftGridChainStartIndex).
+   * index1 could be equal to topLeftIndex.
+   */
+  Real secondGridChainV = leftGridChain->get_v_value(leftGridChainStartIndex+1);
+  assert(leftGridChainStartIndex < leftGridChainEndIndex);
+  Int index1 = topLeftIndex;
+  while(leftChain->getVertex(index1)[1] > secondGridChainV)
+    index1++;
+  index1--;
+  
+  sampleLeftOneGridStep(leftChain, topLeftIndex, index1, leftGridChain, leftGridChainStartIndex, pStream);
+
+
+  /* 
+   * Let the next trim vertex be nextTrimVertIndex (which should be  
+   *  below the second grid line).
+   * Find the last grid line index2 which is above nextTrimVert.
+   * sampleLeftSingleTrimEdgeRegion(uppervert[2], lowervert[2], 
+   *                      leftGridChain, leftGridChainStartIndex+1, index2).
+   */
+  Real *uppervert, *lowervert;
+  uppervert = leftChain->getVertex(index1);
+  lowervert = leftChain->getVertex(index1+1);
+  Int index2 = leftGridChainStartIndex+1;
+
+  while(leftGridChain->get_v_value(index2) >= lowervert[1])
+    {
+      index2++;
+      if(index2 > leftGridChainEndIndex)
+	break;
+    }
+  index2--;
+  sampleLeftSingleTrimEdgeRegion(uppervert, lowervert, leftGridChain, leftGridChainStartIndex+1, index2,  pStream);
+
+   /* sampleLeftStripRec(leftChain, 
+                        nextTrimVertIndex,
+                        botLeftIndex,
+                        leftGridChain,
+			index2,
+                        leftGridChainEndIndex
+			)
+   *
+   */
+  sampleLeftStripRec(leftChain, index1+1, botLeftIndex, leftGridChain, index2, leftGridChainEndIndex, pStream);
+  
+}
+
+
+/***************begin RecF***********************/
+/* the gridlines from leftGridChainStartIndex to 
+ * leftGridChainEndIndex are assumed to form a
+ * connected component.
+ * the trim vertex of topLeftIndex is assumed to
+ * be below the first gridline, and the tim vertex
+ * of botLeftIndex is assumed to be above the last
+ * grid line.
+ * If botLeftIndex < topLeftIndex, then no connected componeent exists, and this funcion returns without
+ * outputing any triangles.
+ * Otherwise botLeftIndex >= topLeftIndex, there is at least one triangle to output.
+ */
+void sampleLeftStripRecF(vertexArray* leftChain,
+		     Int topLeftIndex,
+		     Int botLeftIndex,
+		     gridBoundaryChain* leftGridChain,
+		     Int leftGridChainStartIndex,
+		     Int leftGridChainEndIndex,
+			primStream* pStream
+		     )
+{
+  /*now top left trim vertex is below the top grid line.
+   */
+  /*stop condition: if topLeftIndex > botLeftIndex, then stop.
+   */
+  if(topLeftIndex > botLeftIndex) 
+    return;
+
+  /*if there is only one grid Line, return.*/
+
+  if(leftGridChainStartIndex>=leftGridChainEndIndex)
+    return;
+
+
+  assert(leftChain->getVertex(topLeftIndex)[1] <= leftGridChain->get_v_value(leftGridChainStartIndex) &&
+	 leftChain->getVertex(botLeftIndex)[1] >= leftGridChain->get_v_value(leftGridChainEndIndex));
+
+  /*firs find the first trim vertex which is below or equal to the second top grid line:
+   * index1.
+   */
+  Real secondGridChainV = leftGridChain->get_v_value(leftGridChainStartIndex+1);
+
+
+  Int index1 = topLeftIndex;
+
+  while(leftChain->getVertex(index1)[1] > secondGridChainV){
+    index1++;
+    if(index1>botLeftIndex)
+      break;
+  }
+
+  /*now leftChain->getVertex(index-1)[1] > secondGridChainV and
+   *    leftChain->getVertex(index)[1] <= secondGridChainV
+   *If equality holds, then we should include the vertex index1, otherwise we include only index1-1, to
+   *perform sampleOneGridStep.
+   */
+  if(index1>botLeftIndex) 
+    index1--;
+  else if(leftChain->getVertex(index1)[1] < secondGridChainV)
+    index1--;
+
+  /*now we have leftChain->getVertex(index1)[1] >= secondGridChainV, and 
+   *  leftChain->getVertex(index1+1)[1] <= secondGridChainV
+   */
+
+
+  sampleLeftOneGridStep(leftChain, topLeftIndex, index1, leftGridChain, leftGridChainStartIndex, pStream);
+
+
+  /*if leftChain->getVertex(index1)[1] == secondGridChainV, then we can recursively do the rest.
+   */
+  if(leftChain->getVertex(index1)[1] == secondGridChainV)
+    {
+
+      sampleLeftStripRecF(leftChain, index1, botLeftIndex,leftGridChain, leftGridChainStartIndex+1, leftGridChainEndIndex, pStream);
+    }
+  else if(index1 < botLeftIndex)
+    {
+
+      /* Otherwise, we have leftChain->getVertex(index1)[1] > secondGridChainV,
+       * let the next trim vertex be nextTrimVertIndex (which should be  strictly
+       *  below the second grid line).
+       * Find the last grid line index2 which is above nextTrimVert.
+       * sampleLeftSingleTrimEdgeRegion(uppervert[2], lowervert[2], 
+       *                      leftGridChain, leftGridChainStartIndex+1, index2).
+       */
+      Real *uppervert, *lowervert;
+      uppervert = leftChain->getVertex(index1);
+      lowervert = leftChain->getVertex(index1+1); //okay since index1<botLeftIndex
+      Int index2 = leftGridChainStartIndex+1;
+
+      
+      while(leftGridChain->get_v_value(index2) >= lowervert[1])
+	{
+	  index2++;
+	  if(index2 > leftGridChainEndIndex)
+	    break;
+	}
+      index2--;
+
+      
+      sampleLeftSingleTrimEdgeRegion(uppervert, lowervert, leftGridChain, leftGridChainStartIndex+1, index2,  pStream);
+      
+      /*recursion*/
+
+      sampleLeftStripRecF(leftChain, index1+1, botLeftIndex, leftGridChain, index2, leftGridChainEndIndex, pStream);
+    }
+  
+}
+
+/***************End RecF***********************/
+
+/*sample the left area in between one trim edge and multiple grid lines.
+ * all the grid lines should be in between the two end poins of the
+ *trim edge.
+ */
+void sampleLeftSingleTrimEdgeRegion(Real upperVert[2], Real lowerVert[2],
+				    gridBoundaryChain* gridChain,
+				    Int beginIndex,
+				    Int endIndex,
+				    primStream* pStream)
+{
+  Int i,j,k;
+
+  vertexArray vArray(endIndex-beginIndex+1);  
+  vArray.appendVertex(gridChain->get_vertex(beginIndex));
+
+  for(k=1, i=beginIndex+1; i<=endIndex; i++, k++)
+    {
+      vArray.appendVertex(gridChain->get_vertex(i));
+
+      /*output the fan of the grid points of the (i)th and (i-1)th grid line.
+       */
+      if(gridChain->getUlineIndex(i) < gridChain->getUlineIndex(i-1))
+	{
+	  pStream->begin();	  
+	  pStream->insert(gridChain->get_vertex(i-1));
+	  for(j=gridChain->getUlineIndex(i); j<= gridChain->getUlineIndex(i-1); j++)
+	    pStream->insert(gridChain->getGrid()->get_u_value(j), gridChain->get_v_value(i));
+	  pStream->end(PRIMITIVE_STREAM_FAN);
+	}
+      else if(gridChain->getUlineIndex(i) > gridChain->getUlineIndex(i-1))
+	{
+	  pStream->begin();
+	  pStream->insert(gridChain->get_vertex(i));
+	  for(j=gridChain->getUlineIndex(i); j>= gridChain->getUlineIndex(i-1); j--)
+	    pStream->insert(gridChain->getGrid()->get_u_value(j), gridChain->get_v_value(i-1));
+	  pStream->end(PRIMITIVE_STREAM_FAN);
+	}
+      /*otherwisem, the two are equal, so there is no fan to outout*/	  
+    }
+  
+  monoTriangulation2(upperVert, lowerVert, &vArray, 0, endIndex-beginIndex, 
+		     0, /*decreasing chain*/
+		     pStream);
+}
+  
+/*return i, such that from begin to i-1 the chain is strictly u-monotone. 
+ */				 
+Int findIncreaseChainFromBegin(vertexArray* chain, Int begin ,Int end)
+{
+  Int i=begin;
+  Real prevU = chain->getVertex(i)[0];
+  Real thisU;
+  for(i=begin+1; i<=end; i++){
+    thisU = chain->getVertex(i)[0];
+
+    if(prevU < thisU){
+      prevU = thisU;
+    }
+    else
+      break;
+  }
+  return i;
+}
+  
+/*check whether there is a vertex whose v value is strictly 
+ *inbetween vup vbelow
+ *if no middle exists return -1, else return the idnex.
+ */
+Int checkMiddle(vertexArray* chain, Int begin, Int end, 
+		Real vup, Real vbelow)
+{
+  Int i;
+  for(i=begin; i<=end; i++)
+    {
+      if(chain->getVertex(i)[1] < vup && chain->getVertex(i)[1]>vbelow)
+	return i;
+    }
+  return -1;
+}
+  
+/*the degenerat case of sampleLeftOneGridStep*/
+void sampleLeftOneGridStepNoMiddle(vertexArray* leftChain,
+				   Int beginLeftIndex,
+				   Int endLeftIndex,
+				   gridBoundaryChain* leftGridChain,
+				   Int leftGridChainStartIndex,
+				   primStream* pStream)
+{
+  /*since there is no middle, there is at most one point which is on the 
+   *second grid line, there could be multiple points on the first (top)
+   *grid line.
+   */
+
+  leftGridChain->leftEndFan(leftGridChainStartIndex+1, pStream);
+
+  monoTriangulation2(leftGridChain->get_vertex(leftGridChainStartIndex),
+		     leftGridChain->get_vertex(leftGridChainStartIndex+1),
+		     leftChain,
+		     beginLeftIndex,
+		     endLeftIndex,
+		     1, //is increase chain.
+		     pStream);
+}
+
+
+
+/*sampling the left area in between two grid lines.
+ */
+void sampleLeftOneGridStep(vertexArray* leftChain,
+		  Int beginLeftIndex,
+		  Int endLeftIndex,
+		  gridBoundaryChain* leftGridChain,
+		  Int leftGridChainStartIndex,
+		  primStream* pStream
+		  )
+{
+  if(checkMiddle(leftChain, beginLeftIndex, endLeftIndex, 
+		 leftGridChain->get_v_value(leftGridChainStartIndex),
+		 leftGridChain->get_v_value(leftGridChainStartIndex+1))<0)
+    
+    {
+      
+      sampleLeftOneGridStepNoMiddle(leftChain, beginLeftIndex, endLeftIndex, leftGridChain, leftGridChainStartIndex, pStream);
+      return;
+    }
+  
+  //copy into a polygon
+  {
+    directedLine* poly = NULL;
+    sampledLine* sline;
+    directedLine* dline;
+    gridWrap* grid = leftGridChain->getGrid();
+    Real vert1[2];
+    Real vert2[2];
+    Int i;
+    
+    Int innerInd = leftGridChain->getInnerIndex(leftGridChainStartIndex+1);
+    Int upperInd = leftGridChain->getUlineIndex(leftGridChainStartIndex);
+    Int lowerInd = leftGridChain->getUlineIndex(leftGridChainStartIndex+1);
+    Real upperV = leftGridChain->get_v_value(leftGridChainStartIndex);
+    Real lowerV = leftGridChain->get_v_value(leftGridChainStartIndex+1);
+
+    //the upper gridline
+    vert1[1] = vert2[1] = upperV;
+    for(i=innerInd;	i>upperInd;	i--)
+      {
+	vert1[0]=grid->get_u_value(i);
+	vert2[0]=grid->get_u_value(i-1);
+	sline = new sampledLine(vert1, vert2);
+	dline = new directedLine(INCREASING, sline);
+	if(poly == NULL)
+	  poly = dline;
+	else
+	  poly->insert(dline);
+      }
+
+    //the edge connecting upper grid with left chain
+    vert1[0] = grid->get_u_value(upperInd);
+    vert1[1] = upperV;
+    sline = new sampledLine(vert1, leftChain->getVertex(beginLeftIndex));
+    dline = new directedLine(INCREASING, sline);
+    if(poly == NULL)
+      poly = dline;
+    else
+      poly->insert(dline);    
+ 
+    //the left chain
+    for(i=beginLeftIndex; i<endLeftIndex; i++)
+      {
+	sline = new sampledLine(leftChain->getVertex(i), leftChain->getVertex(i+1));
+	dline = new directedLine(INCREASING, sline);
+	poly->insert(dline);
+      }
+
+    //the edge connecting left chain with lower gridline
+    vert2[0] = grid->get_u_value(lowerInd);
+    vert2[1] = lowerV;
+    sline = new sampledLine(leftChain->getVertex(endLeftIndex), vert2);
+    dline = new directedLine(INCREASING, sline);
+    poly->insert(dline);
+
+    //the lower grid line
+    vert1[1] = vert2[1] = lowerV;
+    for(i=lowerInd; i<innerInd; i++)
+      {
+	vert1[0] = grid->get_u_value(i);
+	vert2[0] = grid->get_u_value(i+1);
+	sline = new sampledLine(vert1, vert2);
+	dline = new directedLine(INCREASING, sline);
+	poly->insert(dline);       
+      }
+
+    //the vertical grid line segement
+    vert1[0]=vert2[0] = grid->get_u_value(innerInd);
+    vert2[1]=upperV;
+    vert1[1]=lowerV;
+    sline=new sampledLine(vert1, vert2);
+    dline=new directedLine(INCREASING, sline);
+    poly->insert(dline);
+    monoTriangulationOpt(poly, pStream);
+    //cleanup
+    poly->deleteSinglePolygonWithSline();
+    return;
+  }
+ 
+
+
+ 
+
+  Int i;
+  if(1/*leftGridChain->getUlineIndex(leftGridChainStartIndex) >= 
+     leftGridChain->getUlineIndex(leftGridChainStartIndex+1)*/
+     ) /*the second grid line is beyond the first one to the left*/
+    {
+      /*find the maximal U-monotone chain 
+       * of endLeftIndex, endLeftIndex-1, ..., 
+       */
+      i=endLeftIndex;
+      Real prevU = leftChain->getVertex(i)[0];
+      for(i=endLeftIndex-1; i>=beginLeftIndex; i--){
+	Real thisU = leftChain->getVertex(i)[0];
+	if( prevU < thisU){
+	  prevU = thisU;
+	}
+	else 
+	  break;
+      }
+      /*from endLeftIndex to i+1 is strictly U- monotone */
+      /*if i+1==endLeftIndex and the vertex and leftchain is on the second gridline, then
+       *we should use 2 vertices on the leftchain. If we only use one (endLeftIndex), then we
+       *we would output degenerate triangles
+       */
+      if(i+1 == endLeftIndex && leftChain->getVertex(endLeftIndex)[1] == leftGridChain->get_v_value(1+leftGridChainStartIndex))
+	i--;
+
+      Int j = beginLeftIndex/*endLeftIndex*/+1;
+
+
+      if(leftGridChain->getInnerIndex(leftGridChainStartIndex+1) > leftGridChain->getUlineIndex(leftGridChainStartIndex))
+	{
+	  j = findIncreaseChainFromBegin(leftChain, beginLeftIndex, i+1/*endLeftIndex*/);
+
+	  Int temp = beginLeftIndex;
+	  /*now from begin to j-1 is strictly u-monotone*/
+	  /*if j-1 is on the first grid line, then we want to skip to the vertex which is strictly
+	   *below the grid line. This vertexmust exist since there is a 'corner turn' inbetween the two grid lines
+	   */
+	  if(j-1 == beginLeftIndex)
+	    {
+	      while(leftChain->getVertex(j-1)[1] == leftGridChain->get_v_value(leftGridChainStartIndex))
+		j++;	        
+
+	      Real vert[2];
+	      vert[0] = leftGridChain->get_u_value(leftGridChainStartIndex);
+	      vert[1] = leftGridChain->get_v_value(leftGridChainStartIndex);
+
+	      monoTriangulation2(
+				 vert/*leftChain->getVertex(beginLeftIndex)*/,
+				 leftChain->getVertex(j-1),
+				 leftChain,
+				 beginLeftIndex,
+				 j-2,
+				 1,
+				 pStream //increase chain
+				 );
+				 
+	      temp = j-1;
+	    }
+				 
+	  stripOfFanLeft(leftChain, j-1, temp/*beginLeftIndex*/, leftGridChain->getGrid(),
+			 leftGridChain->getVlineIndex(leftGridChainStartIndex),
+			 leftGridChain->getUlineIndex(leftGridChainStartIndex),
+			 leftGridChain->getInnerIndex(leftGridChainStartIndex+1),
+			 pStream,
+			 1 /*the grid line is above the trim line*/
+			 );
+	}
+      
+      stripOfFanLeft(leftChain, endLeftIndex, i+1, leftGridChain->getGrid(), 
+		     leftGridChain->getVlineIndex(leftGridChainStartIndex+1),
+		     leftGridChain->getUlineIndex(leftGridChainStartIndex+1),
+		     leftGridChain->getInnerIndex(leftGridChainStartIndex+1),
+		     pStream,
+		     0 /*the grid line is below the trim lines*/
+		     );
+      
+      /*monotone triangulate the remaining left chain togther with the
+       *two vertices on the two grid v-lines.
+       */
+      Real vert[2][2];
+      vert[0][0]=vert[1][0] = leftGridChain->getInner_u_value(leftGridChainStartIndex+1);
+      vert[0][1] = leftGridChain->get_v_value(leftGridChainStartIndex);      
+      vert[1][1] = leftGridChain->get_v_value(leftGridChainStartIndex+1);
+
+//      vertexArray right(vert, 2);
+
+      monoTriangulation2(
+			 &vert[0][0], /*top vertex */
+			 &vert[1][0], /*bottom vertex*/
+			 leftChain, 
+			 /*beginLeftIndex*/j-1,
+			 i+1,
+			 1, /*an increasing chain*/
+			 pStream);
+    }
+  else /*the second one is shorter than the first one to the left*/
+    {
+      /*find the maximal U-monotone chain of beginLeftIndex, beginLeftIndex+1,...,
+       */
+      i=beginLeftIndex;
+      Real prevU = leftChain->getVertex(i)[0];
+      for(i=beginLeftIndex+1; i<=endLeftIndex; i++){
+	Real thisU = leftChain->getVertex(i)[0];
+
+	if(prevU < thisU){
+	  prevU = thisU;
+	}
+	else
+	  break;
+      }
+      /*from beginLeftIndex to i-1 is strictly U-monotone*/
+
+
+      stripOfFanLeft(leftChain, i-1, beginLeftIndex, leftGridChain->getGrid(),
+			 leftGridChain->getVlineIndex(leftGridChainStartIndex),
+			 leftGridChain->getUlineIndex(leftGridChainStartIndex),
+			 leftGridChain->getUlineIndex(leftGridChainStartIndex+1),
+			 pStream,
+		         1 /*the grid line is above the trim lines*/
+			 );
+      /*monotone triangulate the remaining left chain together with the 
+       *two vertices on the two grid v-lines.
+       */
+      Real vert[2][2];
+      vert[0][0]=vert[1][0] = leftGridChain->get_u_value(leftGridChainStartIndex+1);
+      vert[0][1] = leftGridChain->get_v_value(leftGridChainStartIndex);      
+      vert[1][1] = leftGridChain->get_v_value(leftGridChainStartIndex+1);
+
+      vertexArray right(vert, 2);
+
+      monoTriangulation2(
+			 &vert[0][0], //top vertex 
+			 &vert[1][0], //bottom vertex
+			 leftChain, 
+			 i-1,
+			 endLeftIndex,
+			 1, /*an increase chain*/
+			 pStream);
+
+    }
+}
+  
+/*n_upper>=1
+ *n_lower>=1
+ */
+void triangulateXYMono(Int n_upper, Real upperVerts[][2],
+		       Int n_lower, Real lowerVerts[][2],
+		       primStream* pStream)
+{
+  Int i,j,k,l;
+  Real* leftMostV;
+
+  assert(n_upper>=1 && n_lower>=1);
+  if(upperVerts[0][0] <= lowerVerts[0][0])
+    {
+      i=1;
+      j=0;
+      leftMostV = upperVerts[0];
+    }
+  else
+    {
+      i=0;
+      j=1;
+      leftMostV = lowerVerts[0];
+    }
+  
+  while(1)
+    {
+      if(i >= n_upper) /*case1: no more in upper*/
+	{
+
+	  if(j<n_lower-1) /*at least two vertices in lower*/
+	    {
+	      pStream->begin();
+	      pStream->insert(leftMostV);
+	      while(j<n_lower){
+		pStream->insert(lowerVerts[j]);
+		j++;
+	      }
+	      pStream->end(PRIMITIVE_STREAM_FAN);
+	    }
+
+	  break;	
+	}
+      else if(j>= n_lower) /*case2: no more in lower*/
+	{
+
+	  if(i<n_upper-1) /*at least two vertices in upper*/
+	    {
+	      pStream->begin();
+	      pStream->insert(leftMostV);
+
+	      for(k=n_upper-1; k>=i; k--)
+		pStream->insert(upperVerts[k]);
+
+	      pStream->end(PRIMITIVE_STREAM_FAN);
+	    }
+
+	  break;
+	}
+      else /* case3: neither is empty, plus the leftMostV, there is at least one triangle to output*/
+	{
+
+	  if(upperVerts[i][0] <= lowerVerts[j][0])
+	    {
+	      pStream->begin();
+	      pStream->insert(lowerVerts[j]); /*the origin of this fan*/
+
+	      /*find the last k>=i such that 
+	       *upperverts[k][0] <= lowerverts[j][0]
+	       */
+	      k=i;
+	      while(k<n_upper)
+		{
+		  if(upperVerts[k][0] > lowerVerts[j][0])
+		    break;
+		  k++;
+		}
+	      k--;
+	      for(l=k; l>=i; l--)/*the reverse is for two-face lighting*/
+		{
+		  pStream->insert(upperVerts[l]);
+		}
+	      pStream->insert(leftMostV);
+
+	      pStream->end(PRIMITIVE_STREAM_FAN);
+	      //update i for next loop
+	      i = k+1;
+	      leftMostV = upperVerts[k];
+
+	    }
+	  else /*upperVerts[i][0] > lowerVerts[j][0]*/
+	    {
+	      pStream->begin();
+	      pStream->insert(upperVerts[i]);/*the origion of this fan*/
+	      pStream->insert(leftMostV);
+	      /*find the last k>=j such that
+	       *lowerverts[k][0] < upperverts[i][0]*/
+	      k=j;
+	      while(k< n_lower)
+		{
+		  if(lowerVerts[k][0] >= upperVerts[i][0])
+		    break;
+		  pStream->insert(lowerVerts[k]);
+		  k++;
+		}
+	      pStream->end(PRIMITIVE_STREAM_FAN);
+	      j=k;
+	      leftMostV = lowerVerts[j-1];
+	    }	  
+	}
+    }
+}
+		       
+
+void stripOfFanLeft(vertexArray* leftChain, 
+		    Int largeIndex,
+		    Int smallIndex,
+		    gridWrap* grid,
+		    Int vlineIndex,
+		    Int ulineSmallIndex,
+		    Int ulineLargeIndex,
+		    primStream* pStream,
+		    Int gridLineUp /*1 if the grid line is above the trim lines*/
+		     )
+{
+  assert(largeIndex >= smallIndex);
+
+  Real grid_v_value;
+  grid_v_value = grid->get_v_value(vlineIndex);
+
+  Real2* trimVerts=(Real2*) malloc(sizeof(Real2)* (largeIndex-smallIndex+1));
+  assert(trimVerts);
+
+
+  Real2* gridVerts=(Real2*) malloc(sizeof(Real2)* (ulineLargeIndex-ulineSmallIndex+1));
+  assert(gridVerts);
+
+  Int k,i;
+  if(gridLineUp) /*trim line is below grid line, so trim vertices are going right when index increases*/
+    for(k=0, i=smallIndex; i<=largeIndex; i++, k++)
+      {
+      trimVerts[k][0] = leftChain->getVertex(i)[0];
+      trimVerts[k][1] = leftChain->getVertex(i)[1];
+    }
+  else
+    for(k=0, i=largeIndex; i>=smallIndex; i--, k++)
+      {
+	trimVerts[k][0] = leftChain->getVertex(i)[0];
+	trimVerts[k][1] = leftChain->getVertex(i)[1];
+      }
+
+  for(k=0, i=ulineSmallIndex; i<= ulineLargeIndex; i++, k++)
+    {
+      gridVerts[k][0] = grid->get_u_value(i);
+      gridVerts[k][1] = grid_v_value;
+    }
+
+  if(gridLineUp)
+    triangulateXYMono(
+		      ulineLargeIndex-ulineSmallIndex+1, gridVerts,
+		      largeIndex-smallIndex+1, trimVerts,
+		      pStream);
+  else
+    triangulateXYMono(largeIndex-smallIndex+1, trimVerts,
+		      ulineLargeIndex-ulineSmallIndex+1, gridVerts,
+		      pStream);
+  free(trimVerts);
+  free(gridVerts);
+}
+
+  
+
+
+