colmap/lib/PoissonRecon/MarchingCubes.cpp

/*
Copyright (c) 2006, Michael Kazhdan and Matthew Bolitho
All rights reserved.

Redistribution and use in source and binary forms, with or without modification,
are permitted provided that the following conditions are met:

Redistributions of source code must retain the above copyright notice, this list of
conditions and the following disclaimer. Redistributions in binary form must reproduce
the above copyright notice, this list of conditions and the following disclaimer
in the documentation and/or other materials provided with the distribution.

Neither the name of the Johns Hopkins University nor the names of its contributors
may be used to endorse or promote products derived from this software without specific
prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY
EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO THE IMPLIED WARRANTIES
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SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
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TO, PROCUREMENT OF SUBSTITUTE  GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
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*/
#include <math.h>
#include "MarchingCubes.h"

////////////
// Square //
////////////
int Square::AntipodalCornerIndex(int idx){
	int x,y;
	FactorCornerIndex(idx,x,y);
	return CornerIndex( (x+1)%2 , (y+1)%2 );
}
int Square::CornerIndex( int x , int y ){ return (y<<1)|x; }
void Square::FactorCornerIndex( int idx , int& x , int& y ){ x=(idx>>0)&1 , y=(idx>>1)&1; }
int Square::EdgeIndex( int orientation , int i )
{
	switch( orientation )
	{
		case 0: // x
			if( !i ) return  0; // (0,0) -> (1,0)
			else     return  2; // (0,1) -> (1,1)
		case 1: // y
			if( !i ) return  3; // (0,0) -> (0,1)
			else     return  1; // (1,0) -> (1,1)
	};
	return -1;
}
void Square::FactorEdgeIndex(int idx,int& orientation,int& i){
	switch(idx){
		case 0: case 2:
			orientation=0;
			i=idx/2;
			return;
		case 1: case 3:
			orientation=1;
			i=((idx/2)+1)%2;
			return;
	};
}
void Square::EdgeCorners(int idx,int& c1,int& c2){
	int orientation,i;
	FactorEdgeIndex(idx,orientation,i);
	switch(orientation){
		case 0:
			c1 = CornerIndex(0,i);
			c2 = CornerIndex(1,i);
			break;
		case 1:
			c1 = CornerIndex(i,0);
			c2 = CornerIndex(i,1);
			break;
	};
}
int Square::ReflectEdgeIndex(int idx,int edgeIndex){
	int orientation=edgeIndex%2;
	int o,i;
	FactorEdgeIndex(idx,o,i);
	if(o!=orientation){return idx;}
	else{return EdgeIndex(o,(i+1)%2);}
}
int Square::ReflectCornerIndex(int idx,int edgeIndex){
	int orientation=edgeIndex%2;
	int x,y;
	FactorCornerIndex(idx,x,y);
	switch(orientation){
		case 0:	return CornerIndex((x+1)%2,y);
		case 1:	return CornerIndex(x,(y+1)%2);
	};
	return -1;
}



//////////
// Cube //
//////////
int Cube::CornerIndex( int x , int y , int z ){ return (z<<2)|(y<<1)|x; }
void Cube::FactorCornerIndex( int idx , int& x , int& y , int& z ){ x = (idx>>0)&1 , y = (idx>>1)&1 , z = (idx>>2)&1; }
int Cube::EdgeIndex(int orientation,int i,int j){return (i | (j<<1))|(orientation<<2);}
void Cube::FactorEdgeIndex( int idx , int& orientation , int& i , int &j )
{
	orientation=idx>>2;
	i = (idx&1);
	j = (idx&2)>>1;
}
int Cube::FaceIndex( int x , int y , int z )
{
	if     ( x<0 ) return  0;
	else if( x>0 ) return  1;
	else if( y<0 ) return  2;
	else if( y>0 ) return  3;
	else if( z<0 ) return  4;
	else if( z>0 ) return  5;
	else           return -1;
}
int Cube::FaceIndex( int dir , int offSet ){ return (dir<<1)|offSet; }

void Cube::FactorFaceIndex( int idx , int& x , int& y , int& z )
{
	x=y=z=0;
	switch( idx )
	{
		case 0: x=-1; break;
		case 1: x= 1; break;
		case 2: y=-1; break;
		case 3: y= 1; break;
		case 4: z=-1; break;
		case 5: z= 1; break;
	};
}
void Cube::FactorFaceIndex( int idx , int& dir , int& offSet )
{
	dir  = idx>>1;
	offSet=idx &1;
}
bool Cube::IsEdgeCorner( int cIndex , int e )
{
	int o , i , j;
	FactorEdgeIndex( e , o , i , j );
	switch( o )
	{
	case 0: return (cIndex && 2)==(i<<1) && (cIndex && 4)==(j<<2);
	case 1: return (cIndex && 1)==(i<<0) && (cIndex && 4)==(j<<2);
	case 2: return (cIndex && 4)==(i<<2) && (cIndex && 2)==(j<<1);
	default: return false;
	}
}
bool Cube::IsFaceCorner( int cIndex , int f )
{
	int dir , off;
	FactorFaceIndex( f , dir , off );
	return ( cIndex & (1<<dir) )==(off<<dir);
}

int Cube::FaceAdjacentToEdges( int eIndex1 , int eIndex2 )
{
	int f1,f2,g1,g2;
	FacesAdjacentToEdge(eIndex1,f1,f2);
	FacesAdjacentToEdge(eIndex2,g1,g2);
	if(f1==g1 || f1==g2){return f1;}
	if(f2==g1 || f2==g2){return f2;}
	return -1;
}

void Cube::FacesAdjacentToEdge( int eIndex , int& f1Index , int& f2Index )
{
	int orientation,i1,i2;
	FactorEdgeIndex(eIndex,orientation,i1,i2);
	i1<<=1;
	i2<<=1;
	i1--;
	i2--;
	switch(orientation){
		case 0:
			f1Index=FaceIndex( 0,i1, 0);
			f2Index=FaceIndex( 0, 0,i2);
			break;
		case 1:
			f1Index=FaceIndex(i1, 0, 0);
			f2Index=FaceIndex( 0, 0,i2);
			break;
		case 2:
			f1Index=FaceIndex(i1, 0, 0);
			f2Index=FaceIndex( 0,i2, 0);
			break;
	};
}
void Cube::EdgeCorners( int idx , int& c1 , int& c2 )
{
	int orientation,i1,i2;
	FactorEdgeIndex(idx,orientation,i1,i2);
	switch(orientation){
		case 0:
			c1=CornerIndex(0,i1,i2);
			c2=CornerIndex(1,i1,i2);
			break;
		case 1:
			c1=CornerIndex(i1,0,i2);
			c2=CornerIndex(i1,1,i2);
			break;
		case 2:
			c1=CornerIndex(i1,i2,0);
			c2=CornerIndex(i1,i2,1);
			break;
	};
}
void Cube::FaceCorners( int idx , int& c1 , int& c2 , int& c3 , int& c4 )
{
	int i=idx%2;
	switch(idx/2){
	case 0:
		c1=CornerIndex(i,0,0);
		c2=CornerIndex(i,1,0);
		c3=CornerIndex(i,0,1);
		c4=CornerIndex(i,1,1);
		return;
	case 1:
		c1=CornerIndex(0,i,0);
		c2=CornerIndex(1,i,0);
		c3=CornerIndex(0,i,1);
		c4=CornerIndex(1,i,1);
		return;
	case 2:
		c1=CornerIndex(0,0,i);
		c2=CornerIndex(1,0,i);
		c3=CornerIndex(0,1,i);
		c4=CornerIndex(1,1,i);
		return;
	}
}
int Cube::AntipodalCornerIndex( int idx )
{
	int x,y,z;
	FactorCornerIndex(idx,x,y,z);
	return CornerIndex((x+1)%2,(y+1)%2,(z+1)%2);
}
int Cube::FaceReflectFaceIndex( int idx , int faceIndex )
{
	if(idx/2!=faceIndex/2) return idx;
	else
	{
		if( idx%2 ) return idx-1;
		else        return idx+1;
	}
}
int Cube::FaceReflectEdgeIndex( int idx , int faceIndex )
{
	int orientation=faceIndex/2;
	int o,i,j;
	FactorEdgeIndex(idx,o,i,j);
	if(o==orientation){return idx;}
	switch(orientation){
		case 0:	return EdgeIndex(o,(i+1)%2,j);
		case 1:
			switch(o){
				case 0:	return EdgeIndex(o,(i+1)%2,j);
				case 2:	return EdgeIndex(o,i,(j+1)%2);
			};
		case 2:	return EdgeIndex(o,i,(j+1)%2);
	};
	return -1;
}
int Cube::FaceReflectCornerIndex( int idx , int faceIndex )
{
	int orientation=faceIndex/2;
	int x,y,z;
	FactorCornerIndex(idx,x,y,z);
	switch(orientation){
		case 0:	return CornerIndex((x+1)%2,y,z);
		case 1:	return CornerIndex(x,(y+1)%2,z);
		case 2: return CornerIndex(x,y,(z+1)%2);
	};
	return -1;
}
int Cube::EdgeReflectCornerIndex( int idx , int edgeIndex )
{
	int orientation , x , y , z;
	FactorEdgeIndex( edgeIndex , orientation , x , y );
	FactorCornerIndex( idx , x , y , z );
	switch(orientation)
	{
		case 0:	return CornerIndex(  x      , (y+1)%2 , (z+1)%2 );
		case 1:	return CornerIndex( (x+1)%2 ,  y      , (z+1)%2 );
		case 2:	return CornerIndex( (x+1)%2 , (y+1)%2 ,  z      );
	};
	return -1;
}
int	Cube::EdgeReflectEdgeIndex( int edgeIndex )
{
	int o , i1 , i2;
	FactorEdgeIndex( edgeIndex , o , i1 , i2 );
	return Cube::EdgeIndex( o , (i1+1)%2 , (i2+1)%2 );
}


/////////////////////
// MarchingSquares //
/////////////////////
#if NEW_ORDERING
// #pragma message ( "[WARNING] Not clear if MarchingSquares::edgeMask and MarchingSquares::edges are set correctly" )
const int MarchingSquares::cornerMap[] = { 0 , 1 , 3 , 2 };
bool MarchingSquares::HasEdgeRoots( unsigned char mcIndex , int edgeIndex )
{
	int c1 , c2;
	Square::EdgeCorners( edgeIndex , c1 , c2 );
	return !(
		( ( mcIndex&(1<<MarchingSquares::cornerMap[c1]) ) &&  ( mcIndex&(1<<MarchingSquares::cornerMap[c2])) )
		||
		(!( mcIndex&(1<<MarchingSquares::cornerMap[c1]) ) && !( mcIndex&(1<<MarchingSquares::cornerMap[c2])) )
		);
}
#endif // NEW_ORDERING
/*
0} // (0,0) -> (1,0)
1} // (1,0) -> (1,1)
2} // (0,1) -> (1,1)
3} // (0,0) -> (0,1)
*/
const int MarchingSquares::edgeMask[1<<Square::CORNERS]=
{
	    0, //  0 ->         ->                         ->
	    9, //  1 -> 0       -> (0,0)                   -> 0,3     ->  9
	    3, //  2 -> 1       -> (1,0)                   -> 0,1     ->  3
	   10, //  3 -> 0,1     -> (0,0) (1,0)             -> 1,3     -> 10
	   12, //  4 -> 2       -> (0,1)                   -> 2,3     -> 12
	    5, //  5 -> 0,2     -> (0,0) (0,1)             -> 0,2     ->  5
	   15, //  6 -> 1,2     -> (1,0) (0,1)             -> 0,1,2,3 -> 15
	    6, //  7 -> 0,1,2   -> (0,0) (1,0) (0,1)       -> 1,2     ->  6
	    6, //  8 -> 3       -> (1,1)                   -> 1,2     ->  6
	   15, //  9 -> 0,3     -> (0,0) (1,1)             -> 0,1,2,3 -> 15
	    5, // 10 -> 1,3     -> (1,0) (1,1)             -> 0,2     ->  5
	   12, // 11 -> 0,1,3   -> (0,0) (1,0) (1,1)       -> 2,3     -> 12
	   10, // 12 -> 2,3     -> (0,1) (1,1)             -> 1,3     -> 10
	    3, // 13 -> 0,2,3   -> (0,0) (0,1) (1,1)       -> 0,1     ->  3
	    9, // 14 -> 1,2,3   -> (1,0) (0,1) (1,1)       -> 0,3     ->  9
	    0, // 15 -> 0,1,2,3 -> (0,0) (1,0) (0,1) (1,1) ->
};
#if NEW_ORDERING
/*
0} // (0,0) -> (1,0)
1} // (1,0) -> (1,1)
2} // (0,1) -> (1,1)
3} // (0,0) -> (0,1)
*/
const int MarchingSquares::edges[1<<Square::CORNERS][MAX_EDGES*2+1] =
{
	// Positive to the right
	/////////////////////////////////// (0,0) (1,0) (1,1) (0,1)
	{ -1 ,  -1 ,  -1 ,  -1 ,  -1 } , //   -     -     -     -
	{  3 ,   0 ,  -1 ,  -1 ,  -1 } , //   +     -     -     -
	{  0 ,   1 ,  -1 ,  -1 ,  -1 } , //   -     +     -     -
	{  3 ,   1 ,  -1 ,  -1 ,  -1 } , //   +     +     -     -
	{  1 ,   2 ,  -1 ,  -1 ,  -1 } , //   -     -     +     -
	{  3 ,   2 ,   1 ,   0 ,  -1 } , //   +     -     +     -
	{  0 ,   2 ,  -1 ,  -1 ,  -1 } , //   -     +     +     -
	{  3 ,   2 ,  -1 ,  -1 ,  -1 } , //   +     +     +     -
	// Now flip the edges
	{  2 ,   3 ,  -1 ,  -1 ,  -1 } , //   -     -     -     +
	{  2 ,   0 ,  -1 ,  -1 ,  -1 } , //   +     -     -     +
	{  2 ,   3 ,   0 ,   1 ,  -1 } , //   -     +     -     +
	{  2 ,   1 ,  -1 ,  -1 ,  -1 } , //   +     +     -     +
	{  1 ,   3 ,  -1 ,  -1 ,  -1 } , //   -     -     +     +
	{  1 ,   0 ,  -1 ,  -1 ,  -1 } , //   +     -     +     +
	{  0 ,   3 ,  -1 ,  -1 ,  -1 } , //   -     +     +     +
	{ -1 ,  -1 ,  -1 ,  -1 ,  -1 } , //   +     +     +     +
};
#else // !NEW_ORDERING
const int MarchingSquares::edges[1<<Square::CORNERS][MAX_EDGES*2+1] = {
	{ -1,  -1,  -1,  -1,  -1}, //
	{  3,   0,  -1,  -1,  -1}, // (0,0)
	{  0,   1,  -1,  -1,  -1}, // (1,0)
	{  3,   1,  -1,  -1,  -1}, // (0,0) (1,0)
	{  2,   3,  -1,  -1,  -1}, // (0,1)
	{  2,   0,  -1,  -1,  -1}, // (0,0) (0,1)
	{  0,   1,   2,   3,  -1}, // (1,0) (0,1)
	{  1,   2,  -1,  -1,  -1}, // (0,0) (1,0) (0,1)
	{  2,   1,  -1,  -1,  -1}, // (1,1)
	{  3,   0,   1,   2,  -1}, // (0,0) (1,1)
	{  0,   2,  -1,  -1,  -1}, // (1,0) (1,1)
	{  3,   2,  -1,  -1,  -1}, // (0,0) (1,0) (1,1)
	{  1,   3,  -1,  -1,  -1}, // (0,1) (1,1)
	{  1,   0,  -1,  -1,  -1}, // (0,0) (0,1) (1,1)
	{  0,   3,  -1,  -1,  -1}, // (1,0) (0,1) (1,1)
	{ -1,  -1,  -1,  -1,  -1}, // (0,0) (1,0) (0,1) (1,1)
};
#endif // NEW_ORDERING

double MarchingSquares::vertexList[Square::EDGES][2];

unsigned char MarchingSquares::GetIndex( const double v[Square::CORNERS] , double iso )
{
	unsigned char idx=0;
#if NEW_ORDERING
	if( v[ Square::CornerIndex(0,0) ]<iso ) idx |=   1;
	if( v[ Square::CornerIndex(1,0) ]<iso ) idx |=   2;
	if( v[ Square::CornerIndex(1,1) ]<iso ) idx |=   4;
	if( v[ Square::CornerIndex(0,1) ]<iso ) idx |=   8;
#else // !NEW_ORDERING
	for( int i=0 ; i<Square::CORNERS ; i++ ) if( v[i]<iso ) idx|=(1<<i);
#endif // NEW_ORDERING
	return idx;
}
unsigned char MarchingSquares::GetIndex( const float v[Square::CORNERS] , float iso )
{
	unsigned char idx=0;
#if NEW_ORDERING
	if( v[ Square::CornerIndex(0,0) ]<iso ) idx |=   1;
	if( v[ Square::CornerIndex(1,0) ]<iso ) idx |=   2;
	if( v[ Square::CornerIndex(1,1) ]<iso ) idx |=   4;
	if( v[ Square::CornerIndex(0,1) ]<iso ) idx |=   8;
#else // !NEW_ORDERING
	for( int i=0 ; i<Square::CORNERS ; i++ ) if( v[i]<iso ) idx|=(1<<i);
#endif // NEW_ORDERING
	return idx;
}

bool MarchingSquares::IsAmbiguous( const double v[Square::CORNERS] , double isoValue ){ return IsAmbiguous( GetIndex( v , isoValue ) ); }
bool MarchingSquares::HasRoots( unsigned char idx ){ return !(idx==0 || idx==15); }
#if NEW_ORDERING
bool MarchingSquares::IsAmbiguous( unsigned char idx ){ return (idx==5) || (idx==10); }
#else // !NEW_ORDERING
bool MarchingSquares::IsAmbiguous( unsigned char idx ){ return (idx==6) || (idx==9); }
#endif // NEW_ORDERING
int MarchingSquares::AddEdges(const double v[Square::CORNERS],double iso,Edge* isoEdges)
{
	unsigned char idx;
	int nEdges=0;
	Edge e;

	idx=GetIndex(v,iso);

	/* Cube is entirely in/out of the surface */
	if (!edgeMask[idx]) return 0;

	/* Find the vertices where the surface intersects the cube */
	int i,j,ii=1;
	for(i=0;i<12;i++){
		if(edgeMask[idx] & ii){SetVertex(i,v,iso);}
		ii<<=1;
	}
	/* Create the triangle */
	for (i=0;edges[idx][i]!=-1;i+=2) {
		for(j=0;j<2;j++){
			e.p[0][j]=vertexList[edges[idx][i+0]][j];
			e.p[1][j]=vertexList[edges[idx][i+1]][j];
		}
		isoEdges[nEdges++]=e;
	}
	return nEdges;
}

int MarchingSquares::AddEdgeIndices( unsigned char mcIndex , int* isoIndices )
{
	int nEdges = 0;
	/* Square is entirely in/out of the surface */
	if( !edgeMask[mcIndex] ) return 0;

	/* Create the edges */
	for( int i=0 ; edges[mcIndex][i]!=-1 ; i+=2 )
	{
		for( int j=0 ; j<2 ; j++ ) isoIndices[i+j] = edges[mcIndex][i+j];
		nEdges++;
	}
	return nEdges;
}
int MarchingSquares::AddEdgeIndices( const double v[Square::CORNERS] , double iso , int* isoIndices ){ return AddEdgeIndices( GetIndex( v , iso ) , isoIndices ); }
void MarchingSquares::SetVertex(int e,const double values[Square::CORNERS],double iso){
	int o,i,c1,c2;
	Square::FactorEdgeIndex(e,o,i);
	Square::EdgeCorners(e,c1,c2);
	switch(o){
		case 0:
			vertexList[e][0]=Interpolate(values[c1]-iso,values[c2]-iso);
			vertexList[e][1]=i;
			break;
		case 1:
			vertexList[e][1]=Interpolate(values[c1]-iso,values[c2]-iso);
			vertexList[e][0]=i;
			break;
	}
}
double MarchingSquares::Interpolate(double v1,double v2){return v1/(v1-v2);}


///////////////////
// MarchingCubes //
///////////////////
const int MarchingCubes::edgeMask[1<<Cube::CORNERS]={
	    0,  273,  545,  816, 2082, 2355, 2563, 2834,
	 1042, 1283, 1587, 1826, 3120, 3361, 3601, 3840,
	  324,   85,  869,  628, 2406, 2167, 2887, 2646,
	 1366, 1095, 1911, 1638, 3444, 3173, 3925, 3652,
	  644,  917,  165,  436, 2726, 2999, 2183, 2454,
	 1686, 1927, 1207, 1446, 3764, 4005, 3221, 3460,
	  960,  721,  481,  240, 3042, 2803, 2499, 2258,
	 2002, 1731, 1523, 1250, 4080, 3809, 3537, 3264,
	 2184, 2457, 2729, 3000,  170,  443,  651,  922,
	 3226, 3467, 3771, 4010, 1208, 1449, 1689, 1928,
	 2508, 2269, 3053, 2812,  494,  255,  975,  734,
	 3550, 3279, 4095, 3822, 1532, 1261, 2013, 1740,
	 2572, 2845, 2093, 2364,  558,  831,   15,  286,
	 3614, 3855, 3135, 3374, 1596, 1837, 1053, 1292,
	 2888, 2649, 2409, 2168,  874,  635,  331,   90,
	 3930, 3659, 3451, 3178, 1912, 1641, 1369, 1096,
	 1096, 1369, 1641, 1912, 3178, 3451, 3659, 3930,
	   90,  331,  635,  874, 2168, 2409, 2649, 2888,
	 1292, 1053, 1837, 1596, 3374, 3135, 3855, 3614,
	  286,   15,  831,  558, 2364, 2093, 2845, 2572,
	 1740, 2013, 1261, 1532, 3822, 4095, 3279, 3550,
	  734,  975,  255,  494, 2812, 3053, 2269, 2508,
	 1928, 1689, 1449, 1208, 4010, 3771, 3467, 3226,
	  922,  651,  443,  170, 3000, 2729, 2457, 2184,
	 3264, 3537, 3809, 4080, 1250, 1523, 1731, 2002,
	 2258, 2499, 2803, 3042,  240,  481,  721,  960,
	 3460, 3221, 4005, 3764, 1446, 1207, 1927, 1686,
	 2454, 2183, 2999, 2726,  436,  165,  917,  644,
	 3652, 3925, 3173, 3444, 1638, 1911, 1095, 1366,
	 2646, 2887, 2167, 2406,  628,  869,   85,  324,
	 3840, 3601, 3361, 3120, 1826, 1587, 1283, 1042,
	 2834, 2563, 2355, 2082,  816,  545,  273,    0
};
const int MarchingCubes::triangles[1<<Cube::CORNERS][MAX_TRIANGLES*3+1] = {
	{  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   4,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   0,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   9,   5,   8,   5,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   5,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   4,   8,   1,   5,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,  11,   1,   9,   1,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   9,  11,   8,  11,   1,   8,   1,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   1,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   8,   0,  10,   0,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   0,   9,   4,   1,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   8,   9,  10,   9,   5,  10,   5,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,  10,   4,  11,   4,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,  10,   8,  11,   8,   0,  11,   0,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,  11,  10,   9,  10,   4,   9,   4,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   9,  11,   8,  11,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   6,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   6,   2,   0,   4,   6,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   6,   2,   8,   5,   0,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   4,   6,   9,   5,   6,   2,   9,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   5,  11,   8,   6,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   5,  11,   6,   2,   0,   4,   6,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   6,   2,   8,   9,  11,   1,   9,   1,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,  11,   2,   2,  11,   1,   2,   1,   6,   6,   1,   4,  -1,  -1,  -1,  -1},
	{   1,  10,   4,   2,   8,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   0,   1,   6,   2,   1,  10,   6,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   0,   9,   4,   1,  10,   8,   6,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   2,   9,   5,   6,   2,   5,   1,   6,   1,  10,   6,  -1,  -1,  -1,  -1},
	{   2,   8,   6,   4,   5,  11,   4,  11,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   2,   0,   6,   2,   5,  11,   6,   5,  10,   6,  11,  -1,  -1,  -1,  -1},
	{   9,  11,  10,   9,  10,   4,   9,   4,   0,   8,   6,   2,  -1,  -1,  -1,  -1},
	{   9,  11,   2,   2,  11,   6,  10,   6,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   2,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   9,   2,   4,   8,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   2,   7,   0,   7,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   5,   4,   2,   7,   4,   8,   2,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   9,   2,   5,  11,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   5,  11,   0,   4,   8,   9,   2,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   0,   2,   1,   2,   7,   1,   7,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   7,  11,   1,   2,   7,   1,   4,   2,   4,   8,   2,  -1,  -1,  -1,  -1},
	{   4,   1,  10,   9,   2,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   9,   2,   0,   1,  10,   0,  10,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   1,  10,   2,   7,   5,   0,   2,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,  10,   8,   1,  10,   2,   7,   1,   2,   5,   1,   7,  -1,  -1,  -1,  -1},
	{   7,   9,   2,  10,   4,   5,  11,  10,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,  10,   8,  11,   8,   0,  11,   0,   5,   9,   2,   7,  -1,  -1,  -1,  -1},
	{  11,  10,   7,   7,  10,   4,   7,   4,   2,   2,   4,   0,  -1,  -1,  -1,  -1},
	{  11,  10,   7,   7,  10,   2,   8,   2,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   9,   8,   6,   7,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   6,   7,   0,   4,   7,   9,   0,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   6,   7,   5,   8,   6,   5,   0,   8,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   6,   7,   5,   4,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,  11,   1,   8,   6,   7,   9,   8,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   6,   7,   0,   4,   7,   9,   0,   7,  11,   1,   5,  -1,  -1,  -1,  -1},
	{   8,   1,   0,  11,   1,   8,   6,  11,   8,   7,  11,   6,  -1,  -1,  -1,  -1},
	{  11,   6,   7,   1,   6,  11,   6,   1,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,  10,   4,   6,   7,   9,   6,   9,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   1,   9,   9,   1,  10,   9,  10,   7,   7,  10,   6,  -1,  -1,  -1,  -1},
	{   6,   7,   5,   8,   6,   5,   0,   8,   5,   1,  10,   4,  -1,  -1,  -1,  -1},
	{   1,   7,   5,  10,   7,   1,   7,  10,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,  10,   4,  11,   4,   5,   7,   9,   8,   6,   7,   8,  -1,  -1,  -1,  -1},
	{   0,   6,   9,   9,   6,   7,   6,   0,   5,   5,  11,  10,   5,  10,   6,  -1},
	{   8,   7,   0,   6,   7,   8,   4,   0,   7,  11,  10,   4,   7,  11,   4,  -1},
	{  11,  10,   6,  11,   6,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   7,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   4,   8,  11,   7,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   5,   0,  11,   7,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   7,   3,   4,   8,   9,   5,   4,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   1,   5,   3,   5,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   4,   8,   7,   3,   1,   5,   7,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   1,   0,   3,   0,   9,   3,   9,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   8,   9,   4,   8,   7,   3,   4,   7,   1,   4,   3,  -1,  -1,  -1,  -1},
	{   1,  10,   4,   3,  11,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,  11,   7,   8,   0,   1,  10,   8,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   1,  10,   5,   0,   9,  11,   7,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   8,   9,  10,   9,   5,  10,   5,   1,  11,   7,   3,  -1,  -1,  -1,  -1},
	{   4,   5,   7,   4,   7,   3,   4,   3,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   8,   3,   3,   8,   0,   3,   0,   7,   7,   0,   5,  -1,  -1,  -1,  -1},
	{   4,   3,  10,   4,   7,   3,   4,   0,   7,   0,   9,   7,  -1,  -1,  -1,  -1},
	{  10,   8,   3,   3,   8,   7,   9,   7,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   7,   3,   8,   6,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   7,   3,   2,   0,   4,   2,   4,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   7,   3,   8,   6,   2,   5,   0,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   4,   6,   9,   5,   6,   2,   9,   6,   3,  11,   7,  -1,  -1,  -1,  -1},
	{   8,   6,   2,   3,   1,   5,   3,   5,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   1,   5,   3,   5,   7,   6,   2,   0,   4,   6,   0,  -1,  -1,  -1,  -1},
	{   3,   1,   0,   3,   0,   9,   3,   9,   7,   2,   8,   6,  -1,  -1,  -1,  -1},
	{   9,   4,   2,   2,   4,   6,   4,   9,   7,   7,   3,   1,   7,   1,   4,  -1},
	{   8,   6,   2,  11,   7,   3,   4,   1,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   0,   1,   6,   2,   1,  10,   6,   1,  11,   7,   3,  -1,  -1,  -1,  -1},
	{   5,   0,   9,   4,   1,  10,   8,   6,   2,  11,   7,   3,  -1,  -1,  -1,  -1},
	{  11,   7,   3,   5,   2,   9,   5,   6,   2,   5,   1,   6,   1,  10,   6,  -1},
	{   4,   5,   7,   4,   7,   3,   4,   3,  10,   6,   2,   8,  -1,  -1,  -1,  -1},
	{  10,   5,   3,   3,   5,   7,   5,  10,   6,   6,   2,   0,   6,   0,   5,  -1},
	{   8,   6,   2,   4,   3,  10,   4,   7,   3,   4,   0,   7,   0,   9,   7,  -1},
	{   9,   7,  10,  10,   7,   3,  10,   6,   9,   6,   2,   9,  -1,  -1,  -1,  -1},
	{   3,  11,   9,   2,   3,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   8,   0,   2,   3,  11,   2,  11,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   2,   3,   0,   3,  11,   0,  11,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   3,   8,   8,   3,  11,   8,  11,   4,   4,  11,   5,  -1,  -1,  -1,  -1},
	{   2,   3,   1,   2,   1,   5,   2,   5,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   3,   1,   2,   1,   5,   2,   5,   9,   0,   4,   8,  -1,  -1,  -1,  -1},
	{   0,   2,   3,   0,   3,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   3,   8,   8,   3,   4,   1,   4,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,  10,   4,   9,   2,   3,  11,   9,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   8,   0,  10,   0,   1,   3,  11,   9,   2,   3,   9,  -1,  -1,  -1,  -1},
	{   0,   2,   3,   0,   3,  11,   0,  11,   5,   1,  10,   4,  -1,  -1,  -1,  -1},
	{   5,   2,  11,  11,   2,   3,   2,   5,   1,   1,  10,   8,   1,   8,   2,  -1},
	{  10,   2,   3,   9,   2,  10,   4,   9,  10,   5,   9,   4,  -1,  -1,  -1,  -1},
	{   5,  10,   0,   0,  10,   8,  10,   5,   9,   9,   2,   3,   9,   3,  10,  -1},
	{   0,   2,   4,   4,   2,  10,   3,  10,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   8,   2,  10,   2,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   9,   8,   3,  11,   8,   6,   3,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,  11,   9,   3,  11,   0,   4,   3,   0,   6,   3,   4,  -1,  -1,  -1,  -1},
	{  11,   5,   3,   5,   0,   3,   0,   6,   3,   0,   8,   6,  -1,  -1,  -1,  -1},
	{   3,   4,   6,  11,   4,   3,   4,  11,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   1,   6,   6,   1,   5,   6,   5,   8,   8,   5,   9,  -1,  -1,  -1,  -1},
	{   0,   6,   9,   4,   6,   0,   5,   9,   6,   3,   1,   5,   6,   3,   5,  -1},
	{   3,   1,   6,   6,   1,   8,   0,   8,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   1,   4,   3,   4,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   9,   8,   3,  11,   8,   6,   3,   8,   4,   1,  10,  -1,  -1,  -1,  -1},
	{   3,   9,   6,  11,   9,   3,  10,   6,   9,   0,   1,  10,   9,   0,  10,  -1},
	{   4,   1,  10,  11,   5,   3,   5,   0,   3,   0,   6,   3,   0,   8,   6,  -1},
	{   5,  10,   6,   1,  10,   5,   6,  11,   5,   6,   3,  11,  -1,  -1,  -1,  -1},
	{  10,   5,   3,   4,   5,  10,   6,   3,   5,   9,   8,   6,   5,   9,   6,  -1},
	{   6,   3,  10,   9,   0,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,  10,   0,   0,  10,   4,   0,   8,   3,   8,   6,   3,  -1,  -1,  -1,  -1},
	{   6,   3,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   3,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   6,  10,   0,   4,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   0,   9,  10,   3,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   6,  10,   8,   9,   5,   8,   5,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   1,   5,  10,   3,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   4,   8,   1,   5,  11,  10,   3,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   3,   6,   0,   9,  11,   1,   0,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   9,  11,   8,  11,   1,   8,   1,   4,  10,   3,   6,  -1,  -1,  -1,  -1},
	{   4,   1,   3,   6,   4,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   1,   3,   8,   0,   3,   6,   8,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   0,   9,   3,   6,   4,   1,   3,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   9,   6,   6,   9,   5,   6,   5,   3,   3,   5,   1,  -1,  -1,  -1,  -1},
	{   6,   4,   5,   6,   5,  11,   6,  11,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   6,   8,   0,   3,   6,   0,   5,   3,   5,  11,   3,  -1,  -1,  -1,  -1},
	{   3,   9,  11,   0,   9,   3,   6,   0,   3,   4,   0,   6,  -1,  -1,  -1,  -1},
	{   8,   9,   6,   6,   9,   3,  11,   3,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   8,  10,   3,   2,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   2,   0,  10,   3,   0,   4,  10,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   0,   9,   8,  10,   3,   8,   3,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   3,   2,  10,   3,   9,   5,  10,   9,   4,  10,   5,  -1,  -1,  -1,  -1},
	{  11,   1,   5,   2,   8,  10,   3,   2,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   2,   0,  10,   3,   0,   4,  10,   0,   5,  11,   1,  -1,  -1,  -1,  -1},
	{   9,  11,   1,   9,   1,   0,   2,   8,  10,   3,   2,  10,  -1,  -1,  -1,  -1},
	{  10,   2,   4,   3,   2,  10,   1,   4,   2,   9,  11,   1,   2,   9,   1,  -1},
	{   1,   3,   2,   4,   1,   2,   8,   4,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   1,   3,   2,   0,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   3,   2,   4,   1,   2,   8,   4,   2,   9,   5,   0,  -1,  -1,  -1,  -1},
	{   9,   3,   2,   5,   3,   9,   3,   5,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   2,  11,  11,   2,   8,  11,   8,   5,   5,   8,   4,  -1,  -1,  -1,  -1},
	{   5,   2,   0,  11,   2,   5,   2,  11,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   3,   8,   8,   3,   2,   3,   4,   0,   0,   9,  11,   0,  11,   3,  -1},
	{   9,  11,   3,   9,   3,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   3,   6,   9,   2,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   2,   7,  10,   3,   6,   0,   4,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   3,   6,   7,   5,   0,   7,   0,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   5,   4,   2,   7,   4,   8,   2,   4,  10,   3,   6,  -1,  -1,  -1,  -1},
	{  10,   3,   6,   9,   2,   7,   1,   5,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   3,   6,   9,   2,   7,   1,   5,  11,   0,   4,   8,  -1,  -1,  -1,  -1},
	{   1,   0,   2,   1,   2,   7,   1,   7,  11,   3,   6,  10,  -1,  -1,  -1,  -1},
	{  10,   3,   6,   1,   7,  11,   1,   2,   7,   1,   4,   2,   4,   8,   2,  -1},
	{   9,   2,   7,   6,   4,   1,   6,   1,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   1,   3,   8,   0,   3,   6,   8,   3,   7,   9,   2,  -1,  -1,  -1,  -1},
	{   0,   2,   7,   0,   7,   5,   4,   1,   3,   6,   4,   3,  -1,  -1,  -1,  -1},
	{   2,   5,   8,   7,   5,   2,   6,   8,   5,   1,   3,   6,   5,   1,   6,  -1},
	{   6,   4,   5,   6,   5,  11,   6,  11,   3,   7,   9,   2,  -1,  -1,  -1,  -1},
	{   9,   2,   7,   0,   6,   8,   0,   3,   6,   0,   5,   3,   5,  11,   3,  -1},
	{   3,   4,  11,   6,   4,   3,   7,  11,   4,   0,   2,   7,   4,   0,   7,  -1},
	{  11,   3,   8,   8,   3,   6,   8,   2,  11,   2,   7,  11,  -1,  -1,  -1,  -1},
	{   9,   8,  10,   7,   9,  10,   3,   7,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   0,   7,   0,   4,   7,   4,   3,   7,   4,  10,   3,  -1,  -1,  -1,  -1},
	{   8,  10,   0,   0,  10,   3,   0,   3,   5,   5,   3,   7,  -1,  -1,  -1,  -1},
	{  10,   5,   4,   3,   5,  10,   5,   3,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   8,  10,   7,   9,  10,   3,   7,  10,   1,   5,  11,  -1,  -1,  -1,  -1},
	{   1,   5,  11,   9,   0,   7,   0,   4,   7,   4,   3,   7,   4,  10,   3,  -1},
	{  11,   0,   7,   1,   0,  11,   3,   7,   0,   8,  10,   3,   0,   8,   3,  -1},
	{   7,   1,   4,  11,   1,   7,   4,   3,   7,   4,  10,   3,  -1,  -1,  -1,  -1},
	{   4,   9,   8,   7,   9,   4,   1,   7,   4,   3,   7,   1,  -1,  -1,  -1,  -1},
	{   7,   1,   3,   9,   1,   7,   1,   9,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   7,   0,   0,   7,   5,   7,   8,   4,   4,   1,   3,   4,   3,   7,  -1},
	{   5,   1,   3,   7,   5,   3,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   4,  11,  11,   4,   5,   4,   3,   7,   7,   9,   8,   7,   8,   4,  -1},
	{   3,   9,   0,   7,   9,   3,   0,  11,   3,   0,   5,  11,  -1,  -1,  -1,  -1},
	{   3,   7,  11,   8,   4,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   3,   7,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   6,  10,  11,   7,   6,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,   4,   8,  10,  11,   7,  10,   7,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   5,   0,   6,  10,  11,   7,   6,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   9,   5,   8,   5,   4,   6,  10,  11,   7,   6,  11,  -1,  -1,  -1,  -1},
	{   5,   7,   6,   5,   6,  10,   5,  10,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   7,   6,   5,   6,  10,   5,  10,   1,   4,   8,   0,  -1,  -1,  -1,  -1},
	{   1,   0,  10,  10,   0,   9,  10,   9,   6,   6,   9,   7,  -1,  -1,  -1,  -1},
	{   1,   7,  10,  10,   7,   6,   7,   1,   4,   4,   8,   9,   4,   9,   7,  -1},
	{   7,   6,   4,   7,   4,   1,   7,   1,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   0,   1,   8,   0,  11,   7,   8,  11,   6,   8,   7,  -1,  -1,  -1,  -1},
	{   7,   6,   4,   7,   4,   1,   7,   1,  11,   5,   0,   9,  -1,  -1,  -1,  -1},
	{  11,   6,   1,   7,   6,  11,   5,   1,   6,   8,   9,   5,   6,   8,   5,  -1},
	{   4,   5,   7,   4,   7,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   7,   0,   0,   7,   8,   6,   8,   7,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   6,   9,   9,   6,   0,   4,   0,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   9,   7,   8,   7,   6,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,  10,  11,   2,   8,  11,   7,   2,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,  11,   4,   4,  11,   7,   4,   7,   0,   0,   7,   2,  -1,  -1,  -1,  -1},
	{   8,  10,  11,   2,   8,  11,   7,   2,  11,   5,   0,   9,  -1,  -1,  -1,  -1},
	{   9,   4,   2,   5,   4,   9,   7,   2,   4,  10,  11,   7,   4,  10,   7,  -1},
	{   1,   8,  10,   2,   8,   1,   5,   2,   1,   7,   2,   5,  -1,  -1,  -1,  -1},
	{   1,   7,  10,   5,   7,   1,   4,  10,   7,   2,   0,   4,   7,   2,   4,  -1},
	{   7,   1,   9,   9,   1,   0,   1,   7,   2,   2,   8,  10,   2,  10,   1,  -1},
	{   7,   2,   9,  10,   1,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   4,   2,   4,   1,   2,   1,   7,   2,   1,  11,   7,  -1,  -1,  -1,  -1},
	{  11,   0,   1,   7,   0,  11,   0,   7,   2,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   0,   9,   8,   4,   2,   4,   1,   2,   1,   7,   2,   1,  11,   7,  -1},
	{   2,   5,   1,   9,   5,   2,   1,   7,   2,   1,  11,   7,  -1,  -1,  -1,  -1},
	{   4,   5,   8,   8,   5,   2,   7,   2,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   2,   0,   5,   7,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   7,   2,   4,   4,   2,   8,   4,   0,   7,   0,   9,   7,  -1,  -1,  -1,  -1},
	{   7,   2,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,  11,   9,   6,  10,   9,   2,   6,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,  11,   9,   6,  10,   9,   2,   6,   9,   0,   4,   8,  -1,  -1,  -1,  -1},
	{   5,  10,  11,   6,  10,   5,   0,   6,   5,   2,   6,   0,  -1,  -1,  -1,  -1},
	{   2,   5,   8,   8,   5,   4,   5,   2,   6,   6,  10,  11,   6,  11,   5,  -1},
	{  10,   1,   6,   1,   5,   6,   5,   2,   6,   5,   9,   2,  -1,  -1,  -1,  -1},
	{   0,   4,   8,  10,   1,   6,   1,   5,   6,   5,   2,   6,   5,   9,   2,  -1},
	{   1,   0,  10,  10,   0,   6,   2,   6,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   6,   1,   1,   6,  10,   1,   4,   2,   4,   8,   2,  -1,  -1,  -1,  -1},
	{  11,   9,   1,   1,   9,   2,   1,   2,   4,   4,   2,   6,  -1,  -1,  -1,  -1},
	{   8,   1,   6,   0,   1,   8,   2,   6,   1,  11,   9,   2,   1,  11,   2,  -1},
	{  11,   6,   1,   1,   6,   4,   6,  11,   5,   5,   0,   2,   5,   2,   6,  -1},
	{   2,   6,   8,  11,   5,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   6,   4,   2,   2,   4,   9,   5,   9,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   9,   6,   6,   9,   2,   6,   8,   5,   8,   0,   5,  -1,  -1,  -1,  -1},
	{   0,   2,   6,   0,   6,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   2,   6,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,  10,  11,   9,   8,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   0,  11,   9,   4,  11,   0,  11,   4,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,  10,  11,   0,  10,   5,  10,   0,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,  10,  11,   5,   4,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,   8,  10,   5,   8,   1,   8,   5,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   4,  10,   0,   4,   9,  10,   5,   9,  10,   1,   5,  -1,  -1,  -1,  -1},
	{   0,   8,  10,   1,   0,  10,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  10,   1,   4,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   4,   9,   8,   1,   9,   4,   9,   1,  11,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   1,  11,   9,   0,   1,   9,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  11,   0,   8,   5,   0,  11,   8,   1,  11,   8,   4,   1,  -1,  -1,  -1,  -1},
	{  11,   5,   1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   5,   9,   8,   4,   5,   8,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   9,   0,   5,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{   8,   4,   0,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1},
	{  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1,  -1}
};
const int MarchingCubes::cornerMap[Cube::CORNERS]={0,1,3,2,4,5,7,6};
double MarchingCubes::vertexList[Cube::EDGES][3];

unsigned char MarchingCubes::GetIndex(const double v[Cube::CORNERS],double iso)
{
	unsigned char idx=0;
	if (v[Cube::CornerIndex(0,0,0)] < iso) idx |=   1;
	if (v[Cube::CornerIndex(1,0,0)] < iso) idx |=   2;
	if (v[Cube::CornerIndex(1,1,0)] < iso) idx |=   4;
	if (v[Cube::CornerIndex(0,1,0)] < iso) idx |=   8;
	if (v[Cube::CornerIndex(0,0,1)] < iso) idx |=  16;
	if (v[Cube::CornerIndex(1,0,1)] < iso) idx |=  32;
	if (v[Cube::CornerIndex(1,1,1)] < iso) idx |=  64;
	if (v[Cube::CornerIndex(0,1,1)] < iso) idx |= 128;
	return idx;
}
unsigned char MarchingCubes::GetFaceIndex( const double values[Cube::CORNERS] , double iso , int faceIndex )
{
	int i,j,x,y,z;
	unsigned char idx=0;
	double v[2][2];
	Cube::FactorFaceIndex(faceIndex,x,y,z);
	if		(x<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(0,i,j)];}}}
	else if	(x>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(1,i,j)];}}}
	else if	(y<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,0,j)];}}}
	else if	(y>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,1,j)];}}}
	else if	(z<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,j,0)];}}}
	else if	(z>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,j,1)];}}}
	if (v[0][0] < iso) idx |=   1;
	if (v[1][0] < iso) idx |=   2;
	if (v[1][1] < iso) idx |=   4;
	if (v[0][1] < iso) idx |=   8;
	return idx;
}
bool MarchingCubes::IsAmbiguous( const double v[Cube::CORNERS] , double isoValue , int faceIndex ){ return MarchingSquares::IsAmbiguous( GetFaceIndex( v , isoValue , faceIndex ) ); }
bool MarchingCubes::HasRoots( const double v[Cube::CORNERS] , double isoValue , int faceIndex ){ return MarchingSquares::HasRoots( GetFaceIndex( v , isoValue , faceIndex ) ); }
bool MarchingCubes::HasRoots( const double v[Cube::CORNERS] , double isoValue ){ return HasRoots( GetIndex( v , isoValue ) ); }
bool MarchingCubes::HasRoots( unsigned char mcIndex ){ return !(mcIndex==0 || mcIndex==255); }
int MarchingCubes::AddTriangles( const double v[Cube::CORNERS] , double iso , Triangle* isoTriangles )
{
	unsigned char idx;
	int ntriang=0;
	Triangle tri;

	idx=GetIndex(v,iso);

	/* Cube is entirely in/out of the surface */
	if (!edgeMask[idx]) return 0;

	/* Find the vertices where the surface intersects the cube */
	int i,j,ii=1;
	for(i=0;i<12;i++){
		if(edgeMask[idx] & ii){SetVertex(i,v,iso);}
		ii<<=1;
	}
	/* Create the triangle */
	for( i=0 ; triangles[idx][i]!=-1 ; i+=3 )
	{
		for(j=0;j<3;j++){
			tri.p[0][j]=vertexList[triangles[idx][i+0]][j];
			tri.p[1][j]=vertexList[triangles[idx][i+1]][j];
			tri.p[2][j]=vertexList[triangles[idx][i+2]][j];
		}
		isoTriangles[ntriang++]=tri;
	}
	return ntriang;
}

int MarchingCubes::AddTriangleIndices(const double v[Cube::CORNERS],double iso,int* isoIndices){
	unsigned char idx;
	int ntriang=0;

	idx=GetIndex(v,iso);

	/* Cube is entirely in/out of the surface */
	if (!edgeMask[idx]) return 0;

	/* Create the triangle */
	for(int i=0;triangles[idx][i]!=-1;i+=3){
		for(int j=0;j<3;j++){isoIndices[i+j]=triangles[idx][i+j];}
		ntriang++;
	}
	return ntriang;
}

void MarchingCubes::SetVertex( int e , const double values[Cube::CORNERS] , double iso )
{
	double t;
	int o , i1 , i2;
	Cube::FactorEdgeIndex( e , o , i1 , i2 );
	switch( o )
	{
	case 0:
		t = Interpolate( values[ Cube::CornerIndex( 0 , i1 , i2 ) ] - iso , values[ Cube::CornerIndex( 1 , i1 , i2 ) ] - iso );
		vertexList[e][0] = t , vertexList[e][1] = i1  , vertexList[e][2] = i2;
		break;
	case 1:
		t = Interpolate( values[ Cube::CornerIndex( i1 , 0 , i2 ) ] - iso , values[ Cube::CornerIndex( i1 , 1 , i2 ) ] - iso );
		vertexList[e][0] = i1 , vertexList[e][1] = t  , vertexList[e][2] = i2;
		break;
	case 2:
		t = Interpolate( values[ Cube::CornerIndex( i1 , i2 , 0 ) ] - iso , values[ Cube::CornerIndex( i1 , i2 , 1 ) ] - iso );
		vertexList[e][0] = i1 , vertexList[e][1] = i2  , vertexList[e][2] = t;
		break;
	}
}
double MarchingCubes::Interpolate( double v1 , double v2 ) { return v1/(v1-v2); }


///////////////////////////////////
unsigned char MarchingCubes::GetIndex(const float v[Cube::CORNERS],float iso){
	unsigned char idx=0;
	if (v[Cube::CornerIndex(0,0,0)] < iso) idx |=   1;
	if (v[Cube::CornerIndex(1,0,0)] < iso) idx |=   2;
	if (v[Cube::CornerIndex(1,1,0)] < iso) idx |=   4;
	if (v[Cube::CornerIndex(0,1,0)] < iso) idx |=   8;
	if (v[Cube::CornerIndex(0,0,1)] < iso) idx |=  16;
	if (v[Cube::CornerIndex(1,0,1)] < iso) idx |=  32;
	if (v[Cube::CornerIndex(1,1,1)] < iso) idx |=  64;
	if (v[Cube::CornerIndex(0,1,1)] < iso) idx |= 128;
	return idx;
}
unsigned char MarchingCubes::GetFaceIndex( const float values[Cube::CORNERS] , float iso , int faceIndex )
{
	int i,j,x,y,z;
	unsigned char idx=0;
	double v[2][2];
	Cube::FactorFaceIndex(faceIndex,x,y,z);
	if		(x<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(0,i,j)];}}}
	else if	(x>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(1,i,j)];}}}
	else if	(y<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,0,j)];}}}
	else if	(y>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,1,j)];}}}
	else if	(z<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,j,0)];}}}
	else if	(z>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=values[Cube::CornerIndex(i,j,1)];}}}
	if (v[0][0] < iso) idx |=   1;
	if (v[1][0] < iso) idx |=   2;
	if (v[1][1] < iso) idx |=   4;
	if (v[0][1] < iso) idx |=   8;
	return idx;
}
unsigned char MarchingCubes::GetFaceIndex( unsigned char mcIndex , int faceIndex )
{
	int i,j,x,y,z;
	unsigned char idx=0;
	int v[2][2];
	Cube::FactorFaceIndex(faceIndex,x,y,z);
	if		(x<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=mcIndex&(1<<MarchingCubes::cornerMap[Cube::CornerIndex(0,i,j)]);}}}
	else if	(x>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=mcIndex&(1<<MarchingCubes::cornerMap[Cube::CornerIndex(1,i,j)]);}}}
	else if	(y<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=mcIndex&(1<<MarchingCubes::cornerMap[Cube::CornerIndex(i,0,j)]);}}}
	else if	(y>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=mcIndex&(1<<MarchingCubes::cornerMap[Cube::CornerIndex(i,1,j)]);}}}
	else if	(z<0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=mcIndex&(1<<MarchingCubes::cornerMap[Cube::CornerIndex(i,j,1)]);}}}
	else if	(z>0){for(i=0;i<2;i++){for(j=0;j<2;j++){v[i][j]=mcIndex&(1<<MarchingCubes::cornerMap[Cube::CornerIndex(i,j,1)]);}}}
	if (v[0][0]) idx |=   1;
	if (v[1][0]) idx |=   2;
	if (v[1][1]) idx |=   4;
	if (v[0][1]) idx |=   8;
	return idx;
}
bool MarchingCubes::IsAmbiguous( const float v[Cube::CORNERS] , float isoValue , int faceIndex ){ return MarchingSquares::IsAmbiguous( GetFaceIndex( v , isoValue , faceIndex ) ); }
bool MarchingCubes::IsAmbiguous( unsigned char mcIndex , int faceIndex ){ return MarchingSquares::IsAmbiguous( GetFaceIndex( mcIndex , faceIndex ) ); }
bool MarchingCubes::HasRoots( const float v[Cube::CORNERS] , float isoValue ){ return HasRoots( GetIndex( v , isoValue ) ); }
bool MarchingCubes::HasRoots( const float v[Cube::CORNERS] , float isoValue , int faceIndex){ return MarchingSquares::HasRoots( GetFaceIndex( v , isoValue , faceIndex ) ); }
bool MarchingCubes::HasFaceRoots( unsigned char mcIndex , int faceIndex ){ return MarchingSquares::HasRoots( GetFaceIndex( mcIndex , faceIndex ) ); }
bool MarchingCubes::HasEdgeRoots( unsigned char mcIndex , int edgeIndex )
{
	int c1 , c2;
	Cube::EdgeCorners( edgeIndex , c1 , c2 );
	return !(
		( ( mcIndex&(1<<MarchingCubes::cornerMap[c1]) ) &&  ( mcIndex&(1<<MarchingCubes::cornerMap[c2])) )
		||
		(!( mcIndex&(1<<MarchingCubes::cornerMap[c1]) ) && !( mcIndex&(1<<MarchingCubes::cornerMap[c2])) )
		);
}
int MarchingCubes::AddTriangles(const float v[Cube::CORNERS],float iso,Triangle* isoTriangles){
	unsigned char idx;
	int ntriang=0;
	Triangle tri;

	idx = GetIndex( v , iso );

	/* Cube is entirely in/out of the surface */
	if( !edgeMask[idx] ) return 0;

	/* Find the vertices where the surface intersects the cube */
	int i,j,ii=1;
	for( i=0 ; i<12 ; i++ )
	{
		if( edgeMask[idx] & ii ) SetVertex( i , v , iso );
		ii<<=1;
	}
	/* Create the triangle */
	for (i=0;triangles[idx][i]!=-1;i+=3) {
		for(j=0;j<3;j++){
			tri.p[0][j]=vertexList[triangles[idx][i+0]][j];
			tri.p[1][j]=vertexList[triangles[idx][i+1]][j];
			tri.p[2][j]=vertexList[triangles[idx][i+2]][j];
		}
		isoTriangles[ntriang++]=tri;
	}
	return ntriang;
}

int MarchingCubes::AddTriangleIndices( const float v[Cube::CORNERS] , float iso , int* isoIndices ){ return AddTriangleIndices( GetIndex( v , iso ) , isoIndices ); }
int MarchingCubes::AddTriangleIndices( int idx , int* isoIndices )
{
	int ntriang=0;

	/* Cube is entirely in/out of the surface */
	if (!edgeMask[idx]) return 0;

	/* Create the triangle */
	for(int i=0;triangles[idx][i]!=-1;i+=3){
		for(int j=0;j<3;j++){isoIndices[i+j]=triangles[idx][i+j];}
		ntriang++;
	}
	return ntriang;
}

void MarchingCubes::SetVertex( int e , const float values[Cube::CORNERS] , float iso )
{
	double t;
	int o , i1 , i2;
	Cube::FactorEdgeIndex( e , o , i1 , i2 );
	switch( o )
	{
	case 0:
		t = Interpolate( values[ Cube::CornerIndex( 0 , i1 , i2 ) ] - iso , values[ Cube::CornerIndex( 1 , i1 , i2 ) ] - iso );
		vertexList[e][0] = t , vertexList[e][1] = i1  , vertexList[e][2] = i2;
		break;
	case 1:
		t = Interpolate( values[ Cube::CornerIndex( i1 , 0 , i2 ) ] - iso , values[ Cube::CornerIndex( i1 , 1 , i2 ) ] - iso );
		vertexList[e][0] = i1 , vertexList[e][1] = t  , vertexList[e][2] = i2;
		break;
	case 2:
		t = Interpolate( values[ Cube::CornerIndex( i1 , i2 , 0 ) ] - iso , values[ Cube::CornerIndex( i1 , i2 , 1 ) ] - iso );
		vertexList[e][0] = i1 , vertexList[e][1] = i2  , vertexList[e][2] = t;
		break;
	}
}
float MarchingCubes::Interpolate( float v1 , float v2 ){ return v1/(v1-v2); }