f1->edge[2]->adjface[1] = f0;	   /* Find a fourth, non-coplanar point to form tetrahedron. */   v3 = v2->next;   vol = VolumeSign( f0, v3 );   while ( !vol )   {      if ( ( v3 = v3->next ) == v0 )          printf("DoubleTriangle:  All points are coplanar!\n"), exit(0);      vol = VolumeSign( f0, v3 );   }	   /* Insure that v3 will be the first added. */   vertices = v3;   if ( debug ) {      fprintf(stderr, "DoubleTriangle: finished. Head repositioned at v3.\n");      PrintOut( vertices );   }	}  /*---------------------------------------------------------------------ConstructHull adds the vertices to the hull one at a time.  The hullvertices are those in the list marked as onhull.---------------------------------------------------------------------*/void	ConstructHull( void ){   tVertex  v, vnext;   int 	    vol;   bool	    changed;	/* T if addition changes hull; not used. */   v = vertices;   do {      vnext = v->next;      if ( !v->mark ) {         v->mark = PROCESSED;	 changed = AddOne( v );	 CleanUp();	 if ( check ) {	    fprintf(stderr,"ConstructHull: After Add of %d & Cleanup:\n",                v->vnum);	    Checks();	 }	 if ( debug )            PrintOut( v );      }      v = vnext;   } while ( v != vertices );}/*---------------------------------------------------------------------AddOne is passed a vertex.  It first determines all faces visible from that point.  If none are visible then the point is marked as not onhull.  Next is a loop over edges.  If both faces adjacent to an edgeare visible, then the edge is marked for deletion.  If just one of theadjacent faces is visible then a new face is constructed.---------------------------------------------------------------------*/bool 	AddOne( tVertex p ){   tFace  f;    tEdge  e;   int 	  vol;   bool	  vis = FALSE;   if ( debug ) {      fprintf(stderr, "AddOne: starting to add v%d.\n", p->vnum);      PrintOut( vertices );   }   /* Mark faces visible from p. */   f = faces;   do {      vol = VolumeSign( f, p );      if (debug) fprintf(stderr,          "faddr: %6x   paddr: %6x   Vol = %d\n", f,p,vol);      if ( vol < 0 ) {	 f->visible = VISIBLE;  	 vis = TRUE;                            }      f = f->next;   } while ( f != faces );   /* If no faces are visible from p, then p is inside the hull. */   if ( !vis ) {      p->onhull = !ONHULL;        return FALSE;    }   /* Mark edges in interior of visible region for deletion.      Erect a newface based on each border edge. */   e = edges;   do {      tEdge temp;      temp = e->next;      if ( e->adjface[0]->visible && e->adjface[1]->visible )	 /* e interior: mark for deletion. */	 e->delete = REMOVED;      else if ( e->adjface[0]->visible || e->adjface[1]->visible ) 	 /* e border: make a new face. */	 e->newface = MakeConeFace( e, p );      e = temp;   } while ( e != edges );   return TRUE;}/*---------------------------------------------------------------------VolumeSign returns the sign of the volume of the tetrahedron determined by fand p.  VolumeSign is +1 iff p is on the negative side of f,where the positive side is determined by the rh-rule.  So the volume is positive if the ccw normal to f points outside the tetrahedron.The final fewer-multiplications form is due to Robert Fraczkiewicz.---------------------------------------------------------------------*/int  VolumeSign( tFace f, tVertex p ){   double  vol;   int     voli;   double  ax, ay, az, bx, by, bz, cx, cy, cz, dx, dy, dz;   double  bxdx, bydy, bzdz, cxdx, cydy, czdz;   ax = f->vertex[0]->v[X];   ay = f->vertex[0]->v[Y];   az = f->vertex[0]->v[Z];   bx = f->vertex[1]->v[X];   by = f->vertex[1]->v[Y];   bz = f->vertex[1]->v[Z];   cx = f->vertex[2]->v[X];   cy = f->vertex[2]->v[Y];   cz = f->vertex[2]->v[Z];   dx = p->v[X];   dy = p->v[Y];   dz = p->v[Z];      bxdx=bx-dx;   bydy=by-dy;   bzdz=bz-dz;   cxdx=cx-dx;   cydy=cy-dy;   czdz=cz-dz;   vol =   (az-dz) * (bxdx*cydy - bydy*cxdx)         + (ay-dy) * (bzdz*cxdx - bxdx*czdz)	 + (ax-dx) * (bydy*czdz - bzdz*cydy);   if ( debug )      fprintf(stderr,"Face=%6x; Vertex=%d: vol(int) = %d, vol(double) = %lf\n",	      f,p->vnum,voli,vol);   /* The volume should be an integer. */   if      ( vol > 0.5 )   return  1;   else if ( vol < -0.5 )  return -1;   else                    return  0;}/*---------------------------------------------------------------------*/int  Volumei( tFace f, tVertex p ){   int 	   vol;   int 	   ax, ay, az, bx, by, bz, cx, cy, cz, dx, dy, dz;   int	   bxdx, bydy, bzdz, cxdx, cydy, czdz;   double  vold;   int	   i;   ax = f->vertex[0]->v[X];   ay = f->vertex[0]->v[Y];   az = f->vertex[0]->v[Z];   bx = f->vertex[1]->v[X];   by = f->vertex[1]->v[Y];   bz = f->vertex[1]->v[Z];   cx = f->vertex[2]->v[X];   cy = f->vertex[2]->v[Y];   cz = f->vertex[2]->v[Z];   dx = p->v[X];   dy = p->v[Y];   dz = p->v[Z];      bxdx=bx-dx;   bydy=by-dy;   bzdz=bz-dz;   cxdx=cx-dx;   cydy=cy-dy;   czdz=cz-dz;   vol =   (az-dz)*(bxdx*cydy-bydy*cxdx)         + (ay-dy)*(bzdz*cxdx-bxdx*czdz)	 + (ax-dx)*(bydy*czdz-bzdz*cydy);   return vol;}		/*---------------------------------------------------------------------Volumed is the same as VolumeSign but computed with doubles.  For protection against overflow.---------------------------------------------------------------------*/double 	Volumed( tFace f, tVertex p ){   double  vol;   double  ax, ay, az, bx, by, bz, cx, cy, cz, dx, dy, dz;   double  bxdx, bydy, bzdz, cxdx, cydy, czdz;   ax = f->vertex[0]->v[X];   ay = f->vertex[0]->v[Y];   az = f->vertex[0]->v[Z];   bx = f->vertex[1]->v[X];   by = f->vertex[1]->v[Y];   bz = f->vertex[1]->v[Z];   cx = f->vertex[2]->v[X];   cy = f->vertex[2]->v[Y];   cz = f->vertex[2]->v[Z];   dx = p->v[X];   dy = p->v[Y];   dz = p->v[Z];      bxdx=bx-dx;   bydy=by-dy;   bzdz=bz-dz;   cxdx=cx-dx;   cydy=cy-dy;   czdz=cz-dz;   vol = (az-dz)*(bxdx*cydy-bydy*cxdx)         + (ay-dy)*(bzdz*cxdx-bxdx*czdz)	 + (ax-dx)*(bydy*czdz-bzdz*cydy);   return vol;}/*-------------------------------------------------------------------*/void	PrintPoint( tVertex p ){   int	i;   for ( i = 0; i < 3; i++ )      printf("\t%d", p->v[i]);   putchar('\n');}/*---------------------------------------------------------------------MakeConeFace makes a new face and two new edges between the edge and the point that are passed to it. It returns a pointer tothe new face.---------------------------------------------------------------------*/tFace	MakeConeFace( tEdge e, tVertex p ){   tEdge  new_edge[2];   tFace  new_face;   int 	  i, j;   /* Make two new edges (if don't already exist). */   for ( i=0; i < 2; ++i )       /* If the edge exists, copy it into new_edge. */      if ( !( new_edge[i] = e->endpts[i]->duplicate) ) {	 /* Otherwise (duplicate is NULL), MakeNullEdge. */	 new_edge[i] = MakeNullEdge();	 new_edge[i]->endpts[0] = e->endpts[i];	 new_edge[i]->endpts[1] = p;	 e->endpts[i]->duplicate = new_edge[i];      }   /* Make the new face. */   new_face = MakeNullFace();      new_face->edge[0] = e;   new_face->edge[1] = new_edge[0];   new_face->edge[2] = new_edge[1];   MakeCcw( new_face, e, p );            /* Set the adjacent face pointers. */   for ( i=0; i < 2; ++i )      for ( j=0; j < 2; ++j )  	 /* Only one NULL link should be set to new_face. */	 if ( !new_edge[i]->adjface[j] ) {	    new_edge[i]->adjface[j] = new_face;	    break;	 }           return new_face;}/*---------------------------------------------------------------------MakeCcw puts the vertices in the face structure in counterclock wise order.  We want to store the vertices in the same order as in the visible face.  The third vertex is always p.---------------------------------------------------------------------*/void	MakeCcw( tFace f, tEdge e, tVertex p ){   tFace  fv;   /* The visible face adjacent to e */   int    i;    /* Index of e->endpoint[0] in fv. */   tEdge  s;	/* Temporary, for swapping */         if  ( e->adjface[0]->visible )              fv = e->adjface[0];   else fv = e->adjface[1];          /* Set vertex[0] & [1] of f to have the same orientation      as do the corresponding vertices of fv. */    for ( i=0; fv->vertex[i] != e->endpts[0]; ++i )      ;   /* Orient f the same as fv. */   if ( fv->vertex[ (i+1) % 3 ] != e->endpts[1] ) {      f->vertex[0] = e->endpts[1];        f->vertex[1] = e->endpts[0];       }   else {                                     f->vertex[0] = e->endpts[0];         f->vertex[1] = e->endpts[1];            SWAP( s, f->edge[1], f->edge[2] );   }   /* This swap is tricky. e is edge[0]. edge[1] is based on endpt[0],      edge[2] on endpt[1].  So if e is oriented "forwards," we      need to move edge[1] to follow [0], because it precedes. */      f->vertex[2] = p;} /*---------------------------------------------------------------------MakeNullEdge creates a new cell and initializes all pointers to NULLand sets all flags to off.  It returns a pointer to the empty cell.---------------------------------------------------------------------*/tEdge 	MakeNullEdge( void ){   tEdge  e;   NEW( e, tsEdge );   e->adjface[0] = e->adjface[1] = e->newface = NULL;   e->endpts[0] = e->endpts[1] = NULL;   e->delete = !REMOVED;   ADD( edges, e );   return e;}/*--------------------------------------------------------------------MakeNullFace creates a new face structure and initializes all of itsflags to NULL and sets all the flags to off.  It returns a pointerto the empty cell.---------------------------------------------------------------------*/tFace 	MakeNullFace( void ){   tFace  f;   int    i;   NEW( f, tsFace);   for ( i=0; i < 3; ++i ) {      f->edge[i] = NULL;      f->vertex[i] = NULL;   }   f->visible = !VISIBLE;   ADD( faces, f );   return f;}/*---------------------------------------------------------------------MakeFace creates a new face structure from three vertices (in ccworder).  It returns a pointer to the face.---------------------------------------------------------------------*/tFace   MakeFace( tVertex v0, tVertex v1, tVertex v2, tFace fold ){   tFace  f;   tEdge  e0, e1, e2;   /* Create edges of the initial triangle. */   if( !fold ) {     e0 = MakeNullEdge();     e1 = MakeNullEdge();     e2 = MakeNullEdge();   }   else { /* Copy from fold, in reverse order. */     e0 = fold->edge[2];     e1 = fold->edge[1];     e2 = fold->edge[0];   }   e0->endpts[0] = v0;              e0->endpts[1] = v1;   e1->endpts[0] = v1;              e1->endpts[1] = v2;   e2->endpts[0] = v2;              e2->endpts[1] = v0;	   /* Create face for triangle. */   f = MakeNullFace();   f->edge[0]   = e0;  f->edge[1]   = e1; f->edge[2]   = e2;   f->vertex[0] = v0;  f->vertex[1] = v1; f->vertex[2] = v2;	   /* Link edges to face. */   e0->adjface[0] = e1->adjface[0] = e2->adjface[0] = f;