SGI SI GLU library

1 files changed, 1479 insertions, 0 deletions

diff --git a/src/glu/sgi/libnurbs/nurbtess/monoTriangulation.cc b/src/glu/sgi/libnurbs/nurbtess/monoTriangulation.cc
new file mode 100644
index 00000000000..6244298fb77
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+++ b/src/glu/sgi/libnurbs/nurbtess/monoTriangulation.cc
@@ -0,0 +1,1479 @@
+/*
+** 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.
+**
+** $Date: 2001/03/17 00:25:41 $ $Revision: 1.1 $
+*/
+/*
+** $Header: /home/krh/git/sync/mesa-cvs-repo/Mesa/src/glu/sgi/libnurbs/nurbtess/monoTriangulation.cc,v 1.1 2001/03/17 00:25:41 brianp Exp $
+*/
+
+#include <stdlib.h>
+#include <stdio.h>
+#include "glimports.h"
+#include "zlassert.h"
+
+#include "monoTriangulation.h"
+#include "polyUtil.h" /*for area*/
+#include "partitionX.h"
+#include "monoPolyPart.h"
+
+
+
+extern  directedLine*  polygonConvert(directedLine* polygon);
+
+/*poly is NOT deleted
+ */
+void monoTriangulationOpt(directedLine* poly, primStream* pStream)
+{
+  Int n_cusps;
+  Int n_edges = poly->numEdges();
+  directedLine** cusps = (directedLine**) malloc(sizeof(directedLine*)*n_edges);
+  assert(cusps);
+  findInteriorCuspsX(poly, n_cusps, cusps);
+  if(n_cusps ==0) //u monotine
+    {
+      monoTriangulationFun(poly, compV2InX, pStream);
+    }
+  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(poly, compV2InX, pStream);
+	}
+      else
+	{
+	  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();	  
+	}
+    }
+  else
+    {
+      //we need a general partitionX funtion (supposed to be in partitionX.C,
+      //not implemented yet. XXX
+      monoTriangulationFun(poly, compV2InY, pStream);    
+    }
+  
+  free(cusps);
+}
+
+void monoTriangulationRecOpt(Real* topVertex, Real* botVertex, 
+			     vertexArray* left_chain, Int left_current,
+			     vertexArray* right_chain, Int right_current,
+			     primStream* pStream)
+{
+  Int i,j;
+  Int n_left = left_chain->getNumElements();
+  Int n_right = right_chain->getNumElements();
+  if(left_current>= n_left-1 ||
+     right_current>= n_right-1)
+    {
+      monoTriangulationRec(topVertex, botVertex, left_chain, left_current,
+			   right_chain, right_current, pStream);
+      return;
+    }
+  //now both left and right  have at least two vertices each.
+  Real left_v = left_chain->getVertex(left_current)[1];
+  Real right_v = right_chain->getVertex(right_current)[1];
+ 
+  if(left_v <= right_v) //first left vertex is below right
+    {
+      //find the last vertex of right which is above or equal to left
+      for(j=right_current; j<=n_right-1; j++)
+	{
+	  if(right_chain->getVertex(j)[1] < left_v)
+	    break;
+	}
+      monoTriangulationRecGen(topVertex, left_chain->getVertex(left_current),
+			      left_chain, left_current, left_current,
+			      right_chain, right_current, j-1,
+			      pStream);
+      monoTriangulationRecOpt(right_chain->getVertex(j-1),
+			      botVertex,
+			      left_chain, left_current,
+			      right_chain, j,
+			      pStream);
+    }
+  else //first right vertex is strictly below left
+    {
+      //find the last vertex of left which is strictly above right
+      for(i=left_current; i<=n_left-1; i++)
+	{
+	  if(left_chain->getVertex(i)[1] <= right_v)
+	    break;
+	}
+      monoTriangulationRecGen(topVertex, right_chain->getVertex(right_current),
+			      left_chain, left_current, i-1,
+			      right_chain, right_current, right_current,
+			      pStream);
+      monoTriangulationRecOpt(left_chain->getVertex(i-1),
+			      botVertex,
+			      left_chain, i,
+			      right_chain, right_current,
+			      pStream);
+    }
+}
+
+
+void monoTriangulationRecGenTBOpt(Real* topVertex, Real* botVertex, 
+			  vertexArray* inc_chain, Int inc_current, Int inc_end,
+			  vertexArray* dec_chain, Int dec_current, Int dec_end,
+			  primStream* pStream)
+{
+  pStream->triangle(topVertex, inc_chain->getVertex(inc_current), dec_chain->getVertex(dec_current));
+  
+/*printf("**(%f,%f)\n", inc_chain->getArray()[0][0],inc_chain->getArray()[0][1]);*/
+  triangulateXYMonoTB(inc_end-inc_current+1, inc_chain->getArray()+inc_current,  dec_end-dec_current+1,  dec_chain->getArray()+dec_current, pStream);
+
+  pStream->triangle(botVertex, dec_chain->getVertex(dec_end), inc_chain->getVertex(inc_end));
+}
+  
+
+/*n_left>=1
+ *n_right>=1
+ *the strip is going top to bottom. compared to the funtion
+ * triangulateXYmono()
+ */
+void triangulateXYMonoTB(Int n_left, Real** leftVerts,
+		       Int n_right, Real** rightVerts,
+		       primStream* pStream)
+{
+
+
+  Int i,j,k,l;
+  Real* topMostV;
+
+  assert(n_left>=1 && n_right>=1);
+  if(leftVerts[0][1] >= rightVerts[0][1])
+    {
+      i=1;
+      j=0;
+      topMostV = leftVerts[0];
+    }
+  else
+    {
+      i=0;
+      j=1;
+      topMostV = rightVerts[0];
+    }
+  
+  while(1)
+    {
+      if(i >= n_left) /*case1: no more in left*/
+	{
+
+	  if(j<n_right-1) /*at least two vertices in right*/
+	    {
+	      pStream->begin();
+	      pStream->insert(topMostV);
+	      for(k=n_right-1; k>=j; k--)		
+		pStream->insert(rightVerts[j]);
+
+	      pStream->end(PRIMITIVE_STREAM_FAN);
+	      
+	    }
+
+	  break;	
+	}
+      else if(j>= n_right) /*case2: no more in right*/
+	{
+
+	  if(i<n_left-1) /*at least two vertices in left*/
+	    {
+	      pStream->begin();
+	      pStream->insert(topMostV);
+
+	      for(k=i; k<n_left; k++)
+		pStream->insert(leftVerts[k]);
+
+	      pStream->end(PRIMITIVE_STREAM_FAN);	      
+	    }
+
+	  break;
+	}
+      else /* case3: neither is empty, plus the topMostV, there is at least one triangle to output*/
+	{
+
+	  if(leftVerts[i][1] >=  rightVerts[j][1])
+	    {
+	      pStream->begin();
+	      pStream->insert(rightVerts[j]); /*the origin of this fan*/
+
+	      pStream->insert(topMostV);
+
+	      /*find the last k>=i such that 
+	       *leftverts[k][1] >= rightverts[j][1]
+	       */
+	      k=i;
+	      while(k<n_left)
+		{
+		  if(leftVerts[k][1] < rightVerts[j][1])
+		    break;
+		  k++;
+		}
+	      k--;
+	      for(l=i; l<=k; l++)
+		{
+		  pStream->insert(leftVerts[l]);
+		}
+
+	      pStream->end(PRIMITIVE_STREAM_FAN);
+	      //update i for next loop
+	      i = k+1;
+	      topMostV = leftVerts[k];
+
+	    }
+	  else /*leftVerts[i][1] < rightVerts[j][1]*/
+	    {
+	      pStream->begin();
+	      pStream->insert(leftVerts[i]);/*the origion of this fan*/
+
+	      /*find the last k>=j such that
+	       *rightverts[k][1] > leftverts[i][1]*/
+	      k=j;
+	      while(k< n_right)
+		{
+		  if(rightVerts[k][1] <= leftVerts[i][1])
+		    break;
+		  k++;
+		}
+	      k--;
+
+	      for(l=k; l>= j; l--)
+		pStream->insert(rightVerts[l]);
+
+	      pStream->insert(topMostV);
+	      pStream->end(PRIMITIVE_STREAM_FAN);
+	      j=k+1;
+	      topMostV = rightVerts[j-1];
+	    }	  
+	}
+    }
+}
+
+static int chainConvex(vertexArray* inc_chain, Int inc_current, Int inc_end)
+{
+  Int i;
+  //if there are no more than 2 vertices, return 1
+  if(inc_current >= inc_end-1) return 1;
+  for(i=inc_current; i<= inc_end-2; i++)
+    {
+      if(area(inc_chain->getVertex(i), inc_chain->getVertex(i+1), inc_chain->getVertex(i+2)) <0)
+	return 0;
+    }
+  return 1;
+}
+
+static int chainConcave(vertexArray* dec_chain, Int dec_current, Int dec_end)
+{
+  Int i;
+  //if there are no more than 2 vertices, return 1
+  if(dec_current >= dec_end -1) return 1;
+  for(i=dec_current; i<=dec_end-2; i++)
+    {
+      if(area(dec_chain->getVertex(i), dec_chain->getVertex(i+1), dec_chain->getVertex(i+2)) >0)
+	return 0;
+    }
+  return 1;
+}
+ 
+void monoTriangulationRecGenInU(Real* topVertex, Real* botVertex,
+				vertexArray* inc_chain, Int inc_current, Int inc_end,
+				vertexArray* dec_chain, Int dec_current, Int dec_end,
+				primStream* pStream)
+{
+  
+}
+
+void  monoTriangulationRecGenOpt(Real* topVertex, Real* botVertex,
+				 vertexArray* inc_chain, Int inc_current, Int inc_end,
+				 vertexArray* dec_chain, Int dec_current, Int dec_end,
+				 primStream* pStream)
+{
+  Int i;
+  //copy this to a polygon: directedLine Lioop
+  sampledLine* sline;
+  directedLine* dline;
+  directedLine* poly;
+
+  if(inc_current <= inc_end) //at least one vertex in inc_chain
+    {      
+      sline = new sampledLine(topVertex, inc_chain->getVertex(inc_current));
+      poly = new directedLine(INCREASING, sline);
+      for(i=inc_current; i<=inc_end-1; i++)
+	{
+	  sline = new sampledLine(inc_chain->getVertex(i), inc_chain->getVertex(i+1));
+	  dline = new directedLine(INCREASING, sline);
+	  poly->insert(dline);
+	}
+      sline = new sampledLine(inc_chain->getVertex(inc_end), botVertex);
+      dline = new directedLine(INCREASING, sline);
+      poly->insert(dline);
+    }
+  else //inc_chian is empty
+    {
+      sline = new sampledLine(topVertex, botVertex);
+      dline = new directedLine(INCREASING, sline);
+      poly = dline;
+    }
+  
+  assert(poly != NULL);
+
+  if(dec_current <= dec_end) //at least on vertex in dec_Chain
+    {
+      sline = new sampledLine(botVertex, dec_chain->getVertex(dec_end));
+      dline = new directedLine(INCREASING, sline);
+      poly->insert(dline);
+      for(i=dec_end; i>dec_current; i--)
+	{
+	  sline = new sampledLine(dec_chain->getVertex(i), dec_chain->getVertex(i-1));
+	  dline = new directedLine(INCREASING, sline);
+	  poly->insert(dline);
+	}
+      sline = new sampledLine(dec_chain->getVertex(dec_current), topVertex);
+      dline = new directedLine(INCREASING, sline);
+      poly->insert(dline);      
+    }
+  else //dec_chain  is empty
+    {
+      sline = new sampledLine(botVertex, topVertex);
+      dline = new directedLine(INCREASING, sline);
+      poly->insert(dline);
+    }
+
+  {
+    Int n_cusps;
+    Int n_edges = poly->numEdges();
+    directedLine** cusps = (directedLine**) malloc(sizeof(directedLine*)*n_edges);
+    assert(cusps);
+    findInteriorCuspsX(poly, n_cusps, cusps);
+
+    if(n_cusps ==0) //u monotine
+      {
+	monoTriangulationFun(poly, compV2InX, pStream);
+      }
+    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(poly, compV2InX, pStream);
+	    }
+	  else
+	    {
+	      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();	  
+	    }
+      }
+    else
+      {
+	//we need a general partitionX funtion (supposed to be in partitionX.C,
+	//not implemented yet. XXX
+	//monoTriangulationFun(poly, compV2InY, pStream);    
+	
+	directedLine* new_polygon = polygonConvert(poly);
+	directedLine* list = monoPolyPart(new_polygon);
+	for(directedLine* temp = list; temp != NULL; temp = temp->getNextPolygon())
+	  { 
+	    monoTriangulationFun(temp, compV2InX, pStream);
+	  }
+	//clean up
+	list->deletePolygonListWithSline();
+        
+      }
+
+    free(cusps);
+    /*
+      if(numInteriorCuspsX(poly) == 0) //is u monotone
+	monoTriangulationFun(poly, compV2InX, pStream);
+      else //it is not u motone
+	monoTriangulationFun(poly, compV2InY, pStream);    
+	*/
+    //clean up space
+    poly->deleteSinglePolygonWithSline();
+    return;
+  }
+      
+  //apparently the following code is not reachable, 
+  //it is for test purpose
+  if(inc_current > inc_end || dec_current>dec_end)
+    {
+    monoTriangulationRecGen(topVertex, botVertex, inc_chain, inc_current, inc_end,
+			    dec_chain, dec_current, dec_end,
+			    pStream);    
+    return;
+  }
+  
+
+  if(
+     area(dec_chain->getVertex(dec_current),
+	  topVertex,
+	  inc_chain->getVertex(inc_current)) >=0
+     && chainConvex(inc_chain, inc_current, inc_end)
+     && chainConcave(dec_chain, dec_current, dec_end)
+     && area(inc_chain->getVertex(inc_end), botVertex, dec_chain->getVertex(dec_end)) >=0
+     )
+    {
+      monoTriangulationRecFunGen(topVertex, botVertex, 
+				 inc_chain, inc_current, inc_end,
+				 dec_chain, dec_current, dec_end, 
+				 compV2InX, pStream);
+    }
+  else
+    {
+      monoTriangulationRecGen(topVertex, botVertex, inc_chain, inc_current, inc_end,
+			      dec_chain, dec_current, dec_end,
+			      pStream);
+    }
+}
+
+/*if inc_current>inc_end, then inc_chain has no points to be considered
+ *same for dec_chain
+ */
+void monoTriangulationRecGen(Real* topVertex, Real* botVertex, 
+			  vertexArray* inc_chain, Int inc_current, Int inc_end,
+			  vertexArray* dec_chain, Int dec_current, Int dec_end,
+			  primStream* pStream)
+{
+  Real** inc_array ;
+  Real** dec_array ;
+  Int i;
+
+  if(inc_current > inc_end && dec_current>dec_end)
+    return;
+  else if(inc_current>inc_end) /*no more vertices on inc_chain*/
+    {
+      dec_array = dec_chain->getArray();
+      reflexChain rChain(100,0);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the dec_chain*/
+      for(i=dec_current; i<=dec_end; i++){
+	rChain.processNewVertex(dec_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+    }
+  else if(dec_current> dec_end) /*no more vertices on dec_chain*/
+    {
+      inc_array = inc_chain->getArray();
+
+      reflexChain rChain(100,1);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the inc_chain*/
+      for(i=inc_current; i<=inc_end; i++){
+	rChain.processNewVertex(inc_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+    }
+  else /*neither chain is empty*/
+    {
+      inc_array = inc_chain -> getArray();
+      dec_array = dec_chain -> getArray();
+
+      /*if top of inc_chain is 'lower' than top of dec_chain, process all the 
+       *vertices on the dec_chain which are higher than top of inc_chain
+       */
+      if(compV2InY(inc_array[inc_current], dec_array[dec_current]) <= 0)
+	{
+
+	  reflexChain rChain(100, 0);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=dec_current; i<=dec_end; i++)
+	    {
+	      if(compV2InY(inc_array[inc_current], dec_array[i]) <= 0)
+		rChain.processNewVertex(dec_array[i], pStream);
+	      else 
+		break;
+	    }
+	  rChain.outputFan(inc_array[inc_current], pStream);
+	  monoTriangulationRecGen(dec_array[i-1], botVertex, 
+			       inc_chain, inc_current, inc_end,
+			       dec_chain, i, dec_end,
+			       pStream);
+	}
+      else /*compV2InY(inc_array[inc_current], dec_array[dec_current]) > 0*/
+	{
+
+	  reflexChain rChain(100, 1);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=inc_current; i<=inc_end; i++)
+	    {
+	      if(compV2InY(inc_array[i], dec_array[dec_current]) >0)		
+		rChain.processNewVertex(inc_array[i], pStream);	      
+	      else
+		break;
+	    }
+	  rChain.outputFan(dec_array[dec_current], pStream);
+	  monoTriangulationRecGen(inc_array[i-1], botVertex, 
+			       inc_chain, i, inc_end,
+			       dec_chain, dec_current,dec_end,
+			       pStream);
+	}
+    }/*end case neither is empty*/
+}
+
+void monoTriangulationFun(directedLine* monoPolygon, Int (*compFun)(Real*, Real*), primStream* pStream)
+{
+  Int i;
+  /*find the top vertex, bottom vertex, inccreasing chain, and decreasing chain,
+   *then call monoTriangulationRec
+   */
+  directedLine* tempV;
+  directedLine* topV;
+  directedLine* botV;
+  topV = botV = monoPolygon;
+  for(tempV = monoPolygon->getNext(); tempV != monoPolygon; tempV = tempV->getNext())
+    {
+      if(compFun(topV->head(), tempV->head())<0) {
+	topV = tempV;
+      }
+      if(compFun(botV->head(), tempV->head())>0) {
+	botV = tempV;
+      }
+    }
+
+  /*creat increase and decrease chains*/
+  vertexArray inc_chain(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*/
+    inc_chain.appendVertex(topV->getVertex(i));
+  }
+  for(tempV = topV->getNext(); tempV != botV; tempV = tempV->getNext())
+    {
+      for(i=0; i<=tempV->get_npoints()-2; i++){
+	inc_chain.appendVertex(tempV->getVertex(i));
+      }
+    }
+  
+  vertexArray dec_chain(20);
+  for(tempV = topV->getPrev(); tempV != botV; tempV = tempV->getPrev())
+    {
+      for(i=tempV->get_npoints()-2; i>=0; i--){
+	dec_chain.appendVertex(tempV->getVertex(i));
+      }
+    }
+  for(i=botV->get_npoints()-2; i>=1; i--){ 
+    dec_chain.appendVertex(tempV->getVertex(i));
+  }
+  
+  monoTriangulationRecFun(topV->head(), botV->head(), &inc_chain, 0, &dec_chain, 0, compFun, pStream);
+
+}  
+
+void monoTriangulation(directedLine* monoPolygon, primStream* pStream)
+{
+  Int i;
+  /*find the top vertex, bottom vertex, inccreasing chain, and decreasing chain,
+   *then call monoTriangulationRec
+   */
+  directedLine* tempV;
+  directedLine* topV;
+  directedLine* botV;
+  topV = botV = monoPolygon;
+  for(tempV = monoPolygon->getNext(); tempV != monoPolygon; tempV = tempV->getNext())
+    {
+      if(compV2InY(topV->head(), tempV->head())<0) {
+	topV = tempV;
+      }
+      if(compV2InY(botV->head(), tempV->head())>0) {
+	botV = tempV;
+      }
+    }
+  /*creat increase and decrease chains*/
+  vertexArray inc_chain(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*/
+    inc_chain.appendVertex(topV->getVertex(i));
+  }
+  for(tempV = topV->getNext(); tempV != botV; tempV = tempV->getNext())
+    {
+      for(i=0; i<=tempV->get_npoints()-2; i++){
+	inc_chain.appendVertex(tempV->getVertex(i));
+      }
+    }
+  
+  vertexArray dec_chain(20);
+  for(tempV = topV->getPrev(); tempV != botV; tempV = tempV->getPrev())
+    {
+      for(i=tempV->get_npoints()-2; i>=0; i--){
+	dec_chain.appendVertex(tempV->getVertex(i));
+      }
+    }
+  for(i=botV->get_npoints()-2; i>=1; i--){ 
+    dec_chain.appendVertex(tempV->getVertex(i));
+  }
+  
+  monoTriangulationRec(topV->head(), botV->head(), &inc_chain, 0, &dec_chain, 0, pStream);
+
+}
+
+/*the chain could be increasing or decreasing, although we use the
+ * name inc_chain.
+ *the argument  is_increase_chain indicates whether this chain
+ *is increasing (left chain in V-monotone case) or decreaing (right chain
+ *in V-monotone case).
+ */
+void monoTriangulation2(Real* topVertex, Real* botVertex, 
+			vertexArray* inc_chain, Int inc_smallIndex,
+			Int inc_largeIndex,
+			Int is_increase_chain,
+			primStream* pStream)
+{
+  assert( inc_chain != NULL);
+  Real** inc_array ;
+
+  if(inc_smallIndex > inc_largeIndex)
+    return; //no triangles 
+  if(inc_smallIndex == inc_largeIndex)
+    {
+      if(is_increase_chain)
+	pStream->triangle(inc_chain->getVertex(inc_smallIndex), botVertex, topVertex);
+      else
+	pStream->triangle(inc_chain->getVertex(inc_smallIndex), topVertex, botVertex);	
+      return;
+    }
+  Int i;
+
+  if(is_increase_chain && botVertex[1] == inc_chain->getVertex(inc_largeIndex)[1])
+    {
+      pStream->triangle(botVertex, inc_chain->getVertex(inc_largeIndex-1),
+			inc_chain->getVertex(inc_largeIndex));
+      monoTriangulation2(topVertex, botVertex, inc_chain, inc_smallIndex,
+			 inc_largeIndex-1,
+			 is_increase_chain,
+			 pStream);
+      return;
+    }
+  else if( (!is_increase_chain) && topVertex[1] == inc_chain->getVertex(inc_smallIndex)[1])
+    {
+      pStream->triangle(topVertex, inc_chain->getVertex(inc_smallIndex+1),
+			inc_chain->getVertex(inc_smallIndex));
+      monoTriangulation2(topVertex, botVertex, inc_chain, inc_smallIndex+1,
+			 inc_largeIndex, is_increase_chain, pStream);
+      return ;
+    }		           
+
+  inc_array = inc_chain->getArray();
+
+  reflexChain rChain(20,is_increase_chain); /*1 means the chain is increasing*/
+
+  rChain.processNewVertex(topVertex, pStream);
+
+  for(i=inc_smallIndex; i<=inc_largeIndex; i++){
+    rChain.processNewVertex(inc_array[i], pStream);
+  }
+  rChain.processNewVertex(botVertex, pStream);
+
+}
+ 
+/*if compFun == compV2InY, top to bottom: V-monotone
+ *if compFun == compV2InX, right to left: U-monotone
+ */
+void monoTriangulationRecFunGen(Real* topVertex, Real* botVertex, 
+			  vertexArray* inc_chain, Int inc_current, Int inc_end,
+			  vertexArray* dec_chain, Int dec_current, Int dec_end,
+			  Int  (*compFun)(Real*, Real*),
+			  primStream* pStream)
+{
+  assert( inc_chain != NULL && dec_chain != NULL);
+  assert( ! (inc_current> inc_end &&
+	     dec_current> dec_end));
+  Int inc_nVertices;
+  Int dec_nVertices;
+  Real** inc_array ;
+  Real** dec_array ;
+  Int i;
+  assert( ! ( (inc_chain==NULL) && (dec_chain==NULL)));
+
+  if(inc_current> inc_end) /*no more vertices on inc_chain*/
+    {
+
+      dec_array = dec_chain->getArray();
+      reflexChain rChain(20,0);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the dec_chain*/
+      for(i=dec_current; i<=dec_end; i++){
+	rChain.processNewVertex(dec_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+
+    }
+  else if(dec_current> dec_end) /*no more vertices on dec_chain*/
+    {
+      inc_array = inc_chain->getArray();
+      reflexChain rChain(20,1);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the inc_chain*/
+      for(i=inc_current; i<=inc_end; i++){
+	rChain.processNewVertex(inc_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+    }
+  else /*neither chain is empty*/
+    {
+      inc_array = inc_chain -> getArray();
+      dec_array = dec_chain -> getArray();
+
+      /*if top of inc_chain is 'lower' than top of dec_chain, process all the 
+       *vertices on the dec_chain which are higher than top of inc_chain
+       */
+      if(compFun(inc_array[inc_current], dec_array[dec_current]) <= 0)
+	{
+
+	  reflexChain rChain(20, 0);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=dec_current; i<=dec_end; i++)
+	    {
+	      if(compFun(inc_array[inc_current], dec_array[i]) <= 0)
+		rChain.processNewVertex(dec_array[i], pStream);
+	      else 
+		break;
+	    }
+	  rChain.outputFan(inc_array[inc_current], pStream);
+	  monoTriangulationRecFunGen(dec_array[i-1], botVertex, 
+			       inc_chain, inc_current, inc_end,
+			       dec_chain, i, dec_end,
+			       compFun,
+			       pStream);
+	}
+      else /*compFun(inc_array[inc_current], dec_array[dec_current]) > 0*/
+	{
+
+	  reflexChain rChain(20, 1);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=inc_current; i<=inc_end; i++)
+	    {
+	      if(compFun(inc_array[i], dec_array[dec_current]) >0)		
+		rChain.processNewVertex(inc_array[i], pStream);	      
+	      else
+		break;
+	    }
+	  rChain.outputFan(dec_array[dec_current], pStream);
+	  monoTriangulationRecFunGen(inc_array[i-1], botVertex, 
+			       inc_chain, i,inc_end,
+			       dec_chain, dec_current,dec_end,
+			       compFun,
+			       pStream);
+	}
+    }/*end case neither is empty*/
+}
+   
+/*if compFun == compV2InY, top to bottom: V-monotone
+ *if compFun == compV2InX, right to left: U-monotone
+ */
+void monoTriangulationRecFun(Real* topVertex, Real* botVertex, 
+			  vertexArray* inc_chain, Int inc_current,
+			  vertexArray* dec_chain, Int dec_current,
+			  Int  (*compFun)(Real*, Real*),
+			  primStream* pStream)
+{
+  assert( inc_chain != NULL && dec_chain != NULL);
+  assert( ! (inc_current>=inc_chain->getNumElements() &&
+	     dec_current>=dec_chain->getNumElements()));
+  Int inc_nVertices;
+  Int dec_nVertices;
+  Real** inc_array ;
+  Real** dec_array ;
+  Int i;
+  assert( ! ( (inc_chain==NULL) && (dec_chain==NULL)));
+
+  if(inc_current>=inc_chain->getNumElements()) /*no more vertices on inc_chain*/
+    {
+
+      dec_array = dec_chain->getArray();
+      dec_nVertices = dec_chain->getNumElements();      
+      reflexChain rChain(20,0);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the dec_chain*/
+      for(i=dec_current; i<dec_nVertices; i++){
+	rChain.processNewVertex(dec_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+
+    }
+  else if(dec_current>= dec_chain->getNumElements()) /*no more vertices on dec_chain*/
+    {
+      inc_array = inc_chain->getArray();
+      inc_nVertices= inc_chain->getNumElements();
+      reflexChain rChain(20,1);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the inc_chain*/
+      for(i=inc_current; i<inc_nVertices; i++){
+	rChain.processNewVertex(inc_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+    }
+  else /*neither chain is empty*/
+    {
+      inc_array = inc_chain -> getArray();
+      dec_array = dec_chain -> getArray();
+      inc_nVertices= inc_chain->getNumElements();
+      dec_nVertices= dec_chain->getNumElements();
+      /*if top of inc_chain is 'lower' than top of dec_chain, process all the 
+       *vertices on the dec_chain which are higher than top of inc_chain
+       */
+      if(compFun(inc_array[inc_current], dec_array[dec_current]) <= 0)
+	{
+
+	  reflexChain rChain(20, 0);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=dec_current; i<dec_nVertices; i++)
+	    {
+	      if(compFun(inc_array[inc_current], dec_array[i]) <= 0)
+		rChain.processNewVertex(dec_array[i], pStream);
+	      else 
+		break;
+	    }
+	  rChain.outputFan(inc_array[inc_current], pStream);
+	  monoTriangulationRecFun(dec_array[i-1], botVertex, 
+			       inc_chain, inc_current,
+			       dec_chain, i,
+			       compFun,
+			       pStream);
+	}
+      else /*compFun(inc_array[inc_current], dec_array[dec_current]) > 0*/
+	{
+
+	  reflexChain rChain(20, 1);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=inc_current; i<inc_nVertices; i++)
+	    {
+	      if(compFun(inc_array[i], dec_array[dec_current]) >0)		
+		rChain.processNewVertex(inc_array[i], pStream);	      
+	      else
+		break;
+	    }
+	  rChain.outputFan(dec_array[dec_current], pStream);
+	  monoTriangulationRecFun(inc_array[i-1], botVertex, 
+			       inc_chain, i,
+			       dec_chain, dec_current,
+			       compFun,
+			       pStream);
+	}
+    }/*end case neither is empty*/
+}
+
+
+void monoTriangulationRec(Real* topVertex, Real* botVertex, 
+			  vertexArray* inc_chain, Int inc_current,
+			  vertexArray* dec_chain, Int dec_current,
+			  primStream* pStream)
+{
+  assert( inc_chain != NULL && dec_chain != NULL);
+  assert( ! (inc_current>=inc_chain->getNumElements() &&
+	     dec_current>=dec_chain->getNumElements()));
+  Int inc_nVertices;
+  Int dec_nVertices;
+  Real** inc_array ;
+  Real** dec_array ;
+  Int i;
+  assert( ! ( (inc_chain==NULL) && (dec_chain==NULL)));
+
+  if(inc_current>=inc_chain->getNumElements()) /*no more vertices on inc_chain*/
+    {
+
+      dec_array = dec_chain->getArray();
+      dec_nVertices = dec_chain->getNumElements();      
+      reflexChain rChain(20,0);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the dec_chain*/
+      for(i=dec_current; i<dec_nVertices; i++){
+	rChain.processNewVertex(dec_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+
+    }
+  else if(dec_current>= dec_chain->getNumElements()) /*no more vertices on dec_chain*/
+    {
+      inc_array = inc_chain->getArray();
+      inc_nVertices= inc_chain->getNumElements();
+      reflexChain rChain(20,1);
+      /*put the top vertex into the reflex chain*/
+      rChain.processNewVertex(topVertex, pStream);
+      /*process all the vertices on the inc_chain*/
+      for(i=inc_current; i<inc_nVertices; i++){
+	rChain.processNewVertex(inc_array[i], pStream);
+      }
+      /*process the bottom vertex*/
+      rChain.processNewVertex(botVertex, pStream);
+    }
+  else /*neither chain is empty*/
+    {
+      inc_array = inc_chain -> getArray();
+      dec_array = dec_chain -> getArray();
+      inc_nVertices= inc_chain->getNumElements();
+      dec_nVertices= dec_chain->getNumElements();
+      /*if top of inc_chain is 'lower' than top of dec_chain, process all the 
+       *vertices on the dec_chain which are higher than top of inc_chain
+       */
+      if(compV2InY(inc_array[inc_current], dec_array[dec_current]) <= 0)
+	{
+
+	  reflexChain rChain(20, 0);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=dec_current; i<dec_nVertices; i++)
+	    {
+	      if(compV2InY(inc_array[inc_current], dec_array[i]) <= 0)
+		rChain.processNewVertex(dec_array[i], pStream);
+	      else 
+		break;
+	    }
+	  rChain.outputFan(inc_array[inc_current], pStream);
+	  monoTriangulationRec(dec_array[i-1], botVertex, 
+			       inc_chain, inc_current,
+			       dec_chain, i,
+			       pStream);
+	}
+      else /*compV2InY(inc_array[inc_current], dec_array[dec_current]) > 0*/
+	{
+
+	  reflexChain rChain(20, 1);
+	  rChain.processNewVertex(topVertex, pStream);
+	  for(i=inc_current; i<inc_nVertices; i++)
+	    {
+	      if(compV2InY(inc_array[i], dec_array[dec_current]) >0)		
+		rChain.processNewVertex(inc_array[i], pStream);	      
+	      else
+		break;
+	    }
+	  rChain.outputFan(dec_array[dec_current], pStream);
+	  monoTriangulationRec(inc_array[i-1], botVertex, 
+			       inc_chain, i,
+			       dec_chain, dec_current,
+			       pStream);
+	}
+    }/*end case neither is empty*/
+}
+    
+
+
+/* the name here assumes that the polygon is Y-monotone, but 
+ *this function also works for X-monotone polygons.
+ * a monotne polygon consists of two extrem verteices: topVertex and botVertex, and
+ *two monotone chains: inc_chain, and dec_chain. The edges of the increasing chain (inc_chain)
+ *is ordered by following pointer: next, while  the edges of the decreasing chain (dec_chain)
+ *is ordered by following pointer: prev
+ * inc_index index the vertex which is the toppest of the inc_chain which we are handling currently.
+ * dec_index index the vertex which is the toppest of the dec_chain which we are handling currently.
+ */
+void monoTriangulationRec(directedLine* inc_chain, Int inc_index, 
+			  directedLine* dec_chain, Int dec_index, 
+			  directedLine* topVertex, Int top_index,
+			  directedLine* botVertex,
+			  primStream* pStream)
+{
+  Int i;
+  directedLine *temp, *oldtemp;
+  Int tempIndex, oldtempIndex;
+  
+  assert(inc_chain != NULL && dec_chain != NULL);
+  
+  if(inc_chain == botVertex) {
+    reflexChain rChain(20, 0);
+    rChain.processNewVertex(topVertex->getVertex(top_index), pStream);
+    for(i=dec_index; i< dec_chain->get_npoints(); i++){
+      rChain.processNewVertex(dec_chain->getVertex(i), pStream);
+    }
+    for(temp = dec_chain->getPrev(); temp != botVertex; temp = temp->getPrev())
+      {
+	for(i=0; i<temp->get_npoints(); i++){
+	  rChain.processNewVertex(temp->getVertex(i), pStream);
+	}
+      }
+  }
+  else if(dec_chain==botVertex) {
+    reflexChain rChain(20, 1);
+    rChain.processNewVertex(topVertex->getVertex(top_index), pStream);
+    for(i=inc_index; i< inc_chain->get_npoints(); i++){
+      rChain.processNewVertex(inc_chain->getVertex(i), pStream);
+    }
+    for(temp = inc_chain->getPrev(); temp != botVertex; temp = temp->getNext())
+      {
+	for(i=0; i<temp->get_npoints(); i++){
+	  rChain.processNewVertex(temp->getVertex(i), pStream);
+	}
+      }
+  }
+  else /*neither reached the bottom*/{
+    if(compV2InY(inc_chain->getVertex(inc_index), dec_chain->getVertex(dec_index)) <=0) {
+      reflexChain rChain(20, 0);
+      rChain.processNewVertex(topVertex -> getVertex(top_index), pStream);
+      temp = dec_chain;
+      tempIndex = dec_index;
+      while( compV2InY(inc_chain->getVertex(inc_index), temp->getVertex(tempIndex))<=0) {
+	oldtemp = temp;
+	oldtempIndex = tempIndex;
+	rChain.processNewVertex(temp->getVertex(tempIndex), pStream);
+	
+	if(tempIndex == temp->get_npoints()-1){
+	  tempIndex = 0;
+	  temp = temp->getPrev();
+	}
+	else{
+	  tempIndex++;
+	}
+      }
+      rChain.outputFan(inc_chain->getVertex(inc_index), pStream);
+      monoTriangulationRec(inc_chain, inc_index, temp, tempIndex, oldtemp, oldtempIndex, botVertex, pStream);
+    }
+    else /* >0*/ {
+      reflexChain rChain(20, 1);
+      rChain.processNewVertex(topVertex -> getVertex(top_index), pStream);
+      temp = inc_chain;
+      tempIndex = inc_index;
+      while( compV2InY(temp->getVertex(tempIndex), dec_chain->getVertex(dec_index))>0){
+	oldtemp = temp;
+	oldtempIndex = tempIndex;
+	rChain.processNewVertex(temp->getVertex(tempIndex), pStream);
+	
+	if(tempIndex == temp->get_npoints()-1){
+	  tempIndex = 0;
+	  temp = temp->getNext();
+	}
+	else{
+	  tempIndex++;
+	}
+      }
+      rChain.outputFan(dec_chain->getVertex(dec_index), pStream);
+      monoTriangulationRec(temp, tempIndex, dec_chain, dec_index, oldtemp, oldtempIndex, botVertex, pStream); 
+    }
+  } /*end case neither reached the bottom*/
+}
+
+/***************************vertexArray begin here**********************************/
+vertexArray::vertexArray(Real2* vertices, Int nVertices)
+{
+  Int i;
+  size = index = nVertices;
+  array = (Real**) malloc(sizeof(Real*) * nVertices);
+  assert(array);
+  for(i=0; i<nVertices; i++)
+    {
+      array[i] = vertices[i];
+      array[i] = vertices[i];
+    }
+}
+
+vertexArray::vertexArray(Int s)
+{
+  size = s;
+  array = (Real**) malloc(sizeof(Real*) * s);
+  assert(array);
+  index = 0;
+}
+
+vertexArray::~vertexArray()
+{
+  free(array);
+}
+
+void vertexArray::appendVertex(Real* ptr)
+{
+  Int i;
+  if(index >= size){
+    Real** temp = (Real**) malloc(sizeof(Real*) * (2*size +1));
+    assert(temp);
+    for(i=0; i<index; i++)
+      temp[i] = array[i];
+    free(array);
+    array = temp;
+    size = 2*size+1;
+  }
+  array[index++] = ptr;
+}
+
+void vertexArray::print()
+{
+  printf("vertex Array:index=%i, size=%i\n", index, size);
+  for(Int i=0; i<index; i++)
+    {
+      printf("(%f,%f) ", array[i][0], array[i][1]);
+    }
+  printf("\n");
+}
+
+/*find the first i such that array[i][1] >= v
+ * and array[i+1][1] <v
+ * if index == 0 (the array is empty, return -1.
+ * if v is above all, return -1.
+ * if v is below all, return index-1.
+ */
+Int vertexArray::findIndexAbove(Real v)
+{
+  Int i;
+  if(index == 0) 
+    return -1;
+  else if(array[0][1] < v)
+    return -1;
+  else
+    {
+      for(i=1; i<index; i++)
+	{
+	  if(array[i][1] < v)
+	    break;
+	}
+      return i-1;
+    }
+}
+
+/*find the first i<=endIndex such that array[i][1] <= v
+ * and array[i-1][1] > v
+ *if sartIndex>endIndex, then return endIndex+1.
+ *otherwise, startIndex<=endIndex, it is assumed that
+ * 0<=startIndex<=endIndex<index.
+ * if v is below all, return endIndex+1
+ * if v is above all, return startIndex.
+ */
+Int vertexArray::findIndexBelowGen(Real v, Int startIndex, Int endIndex)
+{
+  Int i;
+  if(startIndex > endIndex) 
+    return endIndex+1;
+  else if(array[endIndex][1] >  v)
+    return endIndex+1;
+  else //now array[endIndex][1] <= v
+    {
+      for(i=endIndex-1; i>=startIndex; i--)
+	{
+	  if(array[i][1] > v)
+	    break;
+	}
+      return i+1;
+    }
+}
+
+/*find the first i<=endIndex such that array[i-1][1] >= v
+ * and array[i][1] < v
+ *if sartIndex>endIndex, then return endIndex+1.
+ *otherwise, startIndex<=endIndex, it is assumed that
+ * 0<=startIndex<=endIndex<index.
+ * if v is below or equal to all, return endIndex+1
+ * if v is strictly above all, return startIndex.
+ */
+Int vertexArray::findIndexStrictBelowGen(Real v, Int startIndex, Int endIndex)
+{
+  Int i;
+  if(startIndex > endIndex) 
+    return endIndex+1;
+  else if(array[endIndex][1] >=  v)
+    return endIndex+1;
+  else //now array[endIndex][1] < v
+    {
+      for(i=endIndex-1; i>=startIndex; i--)
+	{
+	  if(array[i][1] >= v)
+	    break;
+	}
+      return i+1;
+    }
+}
+
+/*find the first i>startIndex such that array[i-1][1] > v
+ * and array[i][1] >=v
+ *if sartIndex>endIndex, then return startIndex-1.
+ *otherwise, startIndex<=endIndex, it is assumed that
+ * 0<=startIndex<=endIndex<index.
+ * if v is strictly above all, return startIndex-1
+ * if v is strictly below all, return endIndex.
+ */
+Int vertexArray::findIndexFirstAboveEqualGen(Real v, Int startIndex, Int endIndex)
+{
+
+  Int i;
+  if(startIndex > endIndex) 
+    return startIndex-1;
+  else if(array[startIndex][1] < v)
+    return startIndex-1;
+  else //now array[startIndex][1] >= v
+    {
+
+      for(i=startIndex; i<=endIndex; i++)
+	{
+	  if(array[i][1] <= v)
+	    break;
+	}
+      if(i>endIndex) // v is strictly below all
+	return endIndex;
+      else if(array[i][1] == v)
+	return i;
+      else
+	return i-1;
+    }
+
+}
+
+
+/*find the first i>=startIndex such that array[i][1] >= v
+ * and array[i+1][1] <v
+ *if sartIndex>endIndex, then return startIndex-1.
+ *otherwise, startIndex<=endIndex, it is assumed that
+ * 0<=startIndex<=endIndex<index.
+ * if v is above all, return startIndex-1
+ * if v is below all, return endIndex.
+ */
+Int vertexArray::findIndexAboveGen(Real v, Int startIndex, Int endIndex)
+{
+  Int i;
+  if(startIndex > endIndex) 
+    return startIndex-1;
+  else if(array[startIndex][1] < v)
+    return startIndex-1;
+  else //now array[startIndex][1] >= v
+    {
+      for(i=startIndex+1; i<=endIndex; i++)
+	{
+	  if(array[i][1] < v)
+	    break;
+	}
+      return i-1;
+    }
+}
+
+Int vertexArray::findDecreaseChainFromEnd(Int begin, Int end)
+{
+  Int i = end;
+  Real prevU = array[i][0];
+  Real thisU;
+  for(i=end-1; i>=begin; i--){
+    thisU = array[i][0];
+    if(thisU < prevU)
+      prevU = thisU;
+    else
+      break;
+  }
+  return i;
+}
+
+//if(V(start) == v, return start, other wise return the
+//last i so that V(i)==v
+Int vertexArray::skipEqualityFromStart(Real v, Int start, Int end)
+{
+  Int i;
+  if(array[start][1] != v)
+    return start;
+  //now array[start][1] == v
+  for(i=start+1; i<= end; i++)
+    if(array[i][1] != v) 
+      break;
+  return i-1;
+}
+  
+
+/***************************vertexArray end****************************************/
+
+
+
+/***************************relfex chain stuff begin here*****************************/
+
+reflexChain::reflexChain(Int size, Int is_increasing)
+{
+  queue = (Real2*) malloc(sizeof(Real2) * size);
+  assert(queue);
+  index_queue = 0;
+  size_queue = size;
+  isIncreasing = is_increasing;
+}
+
+reflexChain::~reflexChain()
+{
+  free(queue);
+}
+
+/*put (u,v) at the end of the queue
+ *pay attention to space
+ */
+void reflexChain::insert(Real u, Real v)
+{
+  Int i;
+  if(index_queue >= size_queue) {
+    Real2 *temp = (Real2*) malloc(sizeof(Real2) * (2*size_queue+1));
+    assert(temp);
+
+    /*copy*/
+    for(i=0; i<index_queue; i++){
+      temp[i][0] = queue[i][0];
+      temp[i][1] = queue[i][1];
+    }
+    
+    free(queue);
+    queue = temp;
+    size_queue = 2*size_queue + 1;
+  }
+
+  queue[index_queue][0] = u;
+  queue[index_queue][1] = v;
+  index_queue ++;
+}
+
+void reflexChain::insert(Real v[2])
+{
+  insert(v[0], v[1]);
+}
+
+/*
+static Real area(Real A[2], Real B[2], Real C[2])
+{
+  Real Bx, By, Cx, Cy;
+  Bx = B[0] - A[0];
+  By = B[1] - A[1];
+  Cx = C[0] - A[0];
+  Cy = C[1] - A[1];
+  return Bx*Cy - Cx*By;
+}
+*/
+
+/*the chain is reflex, and the vertex v is
+ *on the other side of the chain, so that
+ *we can outout the fan with v as the
+ *the center
+ */
+void reflexChain::outputFan(Real v[2], primStream* pStream)
+{
+  Int i;
+  pStream->begin();
+  pStream->insert(v);
+  if(isIncreasing) {
+    for(i=0; i<index_queue; i++)
+      pStream->insert(queue[i]);
+  }
+  else {
+    for(i=index_queue-1; i>=0; i--)
+      pStream->insert(queue[i]);    
+  }
+  pStream->end(PRIMITIVE_STREAM_FAN);
+}
+
+void reflexChain::processNewVertex(Real v[2], primStream* pStream)
+{
+  Int i,j,k;
+  Int isReflex;
+  /*if there are at most one vertex in the queue, then simply insert
+   */
+  if(index_queue <=1){
+    insert(v);
+    return;
+  }
+  
+  /*there are at least two vertices in the queue*/
+  j=index_queue-1;
+  
+  for(i=j; i>=1; i--) {
+    if(isIncreasing) {
+      isReflex = (area(queue[i-1], queue[i], v) <= 0.0);
+    }
+    else /*decreasing*/{
+      isReflex = (area(v, queue[i], queue[i-1]) <= 0.0);	  
+    }
+    if(isReflex) {
+      break;
+    }
+  }
+
+  /*
+   *if i<j then vertices: i+1--j are convex
+   * output triangle fan: 
+   *  v, and queue[i], i+1, ..., j
+   */
+  if(i<j) 
+    {
+      pStream->begin();
+      pStream->insert(v);
+      if(isIncreasing) {
+	for(k=i; k<=j; k++)
+	  pStream->insert(queue[k]);
+      }
+      else {
+	for(k=j; k>=i; k--)
+	  pStream->insert(queue[k]);
+      }
+      
+      pStream->end(PRIMITIVE_STREAM_FAN);
+    }
+
+  /*delete vertices i+1--j from the queue*/
+  index_queue = i+1;
+  /*finally insert v at the end of the queue*/
+  insert(v);
+
+}
+
+void reflexChain::print()
+{
+  Int i;
+  printf("reflex chain: isIncreasing=%i\n", isIncreasing);
+  for(i=0; i<index_queue; i++) {
+    printf("(%f,%f) ", queue[i][0], queue[i][1]);
+  }
+  printf("\n");
+}