/*
Copyright (c) 2015, Michael Kazhdan and Fabian Prada
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 
OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT
SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
TO, PROCUREMENT OF SUBSTITUTE  GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
DAMAGE.
*/
#include <math.h>
#include <vector>
#include <unordered_map>
#include <algorithm>
#include <atomic>

/////////
// FEM //
/////////

template< class Real >
inline Real FEM::Area( const SquareMatrix< Real , 2 >& tensor , const Point2D< Real > t[3] )
{
	return Dot( tensor , TangentVector< Real >( t[2] - t[0] ) , Rotate90( tensor , TangentVector< Real >( t[1] - t[0] ) ) ) / (Real)2;
}
template< class Real >
inline Real FEM::Area( const SquareMatrix< Real , 2 >& tensor , Point2D< Real > v1 , Point2D< Real > v2 , Point2D< Real > v3 )
{
	return Dot( tensor , TangentVector< Real >( v3 - v1 ) , Rotate90( tensor , TangentVector< Real >( v2 - v1 ) ) ) / (Real)2;
}
template< class Real >
inline Real FEM::Angle( const SquareMatrix< Real , 2 >& tensor , FEM::TangentVector< Real > v1 , FEM::TangentVector< Real > v2 )
{
	return (Real)acos( Dot( tensor , v1 , v2 ) / sqrt( SquareLength( tensor , v1 ) * SquareLength( tensor , v2 ) ) );
}
template< class Real >
inline Real FEM::Angle( const SquareMatrix< Real , 2 >& tensor , FEM::CotangentVector< Real > v1 , FEM::CotangentVector< Real > v2 )
{
	return (Real)acos( Dot( tensor , v1 , v2 ) / sqrt( SquareLength( tensor , v1 ) * SquareLength( tensor , v2 ) ) );
}
template< class Real >
inline SquareMatrix< Real , 2 > FEM::CoJ( const SquareMatrix< Real , 2 >& tensor )
{
	SquareMatrix< Real , 2 > J;
	J(0,1) = (Real)1 , J(1,0) = (Real)-1;
	return tensor * J / (Real)sqrt( tensor.determinant() );
}
template< class Real >
inline SquareMatrix< Real , 2 > FEM::J( const SquareMatrix< Real , 2 >& tensor )
{
	SquareMatrix< Real , 2 > J;
	J(0,1) = (Real)1 , J(1,0) = (Real)-1;
	return tensor.inverse() * J * (Real)sqrt( tensor.determinant() );
}
template< class Real > inline FEM::CotangentVector< Real > FEM::Rotate90( const SquareMatrix< Real , 2 >& tensor , FEM::CotangentVector< Real > v ){ return CoJ( tensor ) * v; }
template< class Real > inline FEM::  TangentVector< Real > FEM::Rotate90( const SquareMatrix< Real , 2 >& tensor , FEM::  TangentVector< Real > v ){ return   J( tensor ) * v; }

template< class Real > SquareMatrix< Real , 2 > FEM::MakeConformal( const SquareMatrix< Real , 2 >& sourceTensor , const SquareMatrix< Real , 2 >& targetTensor ){ return targetTensor * (Real)sqrt( sourceTensor.determinant() / targetTensor.determinant() ); }
template< class Real > SquareMatrix< Real , 2 > FEM::MakeAuthalic ( const SquareMatrix< Real , 2 >& sourceTensor , const SquareMatrix< Real , 2 >& targetTensor ){ return sourceTensor * (Real)sqrt( targetTensor.determinant() / sourceTensor.determinant() ); }
template< class Real >
SquareMatrix< Real , 2 > FEM::TensorRoot( const SquareMatrix< Real , 2 >& tensor )
{
	// Code borrowed from: https://en.wikipedia.org/wiki/Square_root_of_a_2_by_2_matrix
	SquareMatrix< Real , 2 > root = tensor;
	Real det = tensor.determinant();
	if( det<0 ) fprintf( stderr , "[ERROR] FEM::TensorRoot: Negative determinant: %g\n" , det ) , exit( 0 );
	Real s = (Real)sqrt( det );
	Real disc = (Real)( tensor.trace() + 2. * s );
	if( disc<0 ) fprintf( stderr , "[ERROR] FEM::TensorRoot: Negative discriminant: %g\n" , disc ) , exit( 0 );
	root(0,0) += s , root(1,1) += s;
	return root / (Real)sqrt( disc );
}
////////////////////////
// FEM::RightTriangle //
////////////////////////

template< class Real >
inline Point2D< Real > FEM::RightTriangle< Real >::Center( const SquareMatrix< Real , 2 >& tensor , int centerType )
{
	static const double SQRT_THREE_QUARTERS = sqrt( 3./4 );
	switch( centerType )
	{
	case CENTER_BARYCENTRIC:
		return Point2D< Real >( (Real)1./3 , (Real)1./3 );
	case CENTER_INCENTRIC:
		{
			Real lengths[] =
			{
				(Real)sqrt( Point2D< Real >::Dot( RightTriangle< Real >::EdgeDirections[0] , tensor * RightTriangle< Real >::EdgeDirections[0] ) ) ,
				(Real)sqrt( Point2D< Real >::Dot( RightTriangle< Real >::EdgeDirections[1] , tensor * RightTriangle< Real >::EdgeDirections[1] ) ) ,
				(Real)sqrt( Point2D< Real >::Dot( RightTriangle< Real >::EdgeDirections[2] , tensor * RightTriangle< Real >::EdgeDirections[2] ) )
			};
			Real lSum = lengths[0] + lengths[1] + lengths[2];
			return Point2D< Real >( lengths[1] / lSum , lengths[2] / lSum );
		}
	case CENTER_CIRCUMCENTRIC:
		{
			Real maxDet = 0;
			Point2D< Real > c;
			for( int j=0 ; j<3 ; j++ )
			{
				Point2D< Real > c1 = RightTriangle< Real >::EdgeMidpoints[(j+1)%3] , c2 = RightTriangle< Real >::EdgeMidpoints[(j+2)%3];
				Point2D< Real > v1 = Rotate90( tensor , RightTriangle< Real >::EdgeDirections[(j+1)%3] ) , v2 = Rotate90( tensor , RightTriangle< Real >::EdgeDirections[(j+2)%3] );
				// Solve for s and t such that:
				//		c1 + s * v1 = c2 + t * v2
				// =>	c1 - c2 = - s * v1 + t * v2
				// =>	c1 - c2 = ( -v1 | v2 ) * ( s , t )^t
				SquareMatrix< Real , 2 > M;
				M( 0 , 0 ) = -v1[0] , M( 0 , 1 ) = -v1[1];
				M( 1 , 0 ) =  v2[0] , M( 1 , 1 ) =  v2[1];
				Real det = (Real)fabs( M.determinant() );
				if( det>maxDet )
				{
					Point2D< Real > x = M.inverse() * ( c1 - c2 );
					c = ( c1 + v1 * x[0] + c2 + v2 * x[1] ) / 2;
					maxDet = det;
				}
			}
			return c;
		}
	case CENTER_ISOGONIC:
		{
			Point2D< Real > eVertices[3];
			// Solve for the positive value of t such that:
			//		|| Corners[j+1] - ( midPoint + t * perpDir ) ||^2 = || Corners[j+1] - Corners[j+2] ||^2
			// <=>	t^2 || perpDir ||^2 = || Corners[j+1] - Corners[j+2] ||^2 - || Corners[j+1] - midPoint ||^2
			// <=>	t^2 || Corners[j+1] - Corners[j+2] ||^2 = || Corners[j+1] - Corners[j+2] ||^2 - || ( Corners[j+1] - Corners[j+2] )/2 ||^2
			// <=>	t^2 || Corners[j+1] - Corners[j+2] ||^2 = 3/4 || Corners[j+1] - Corners[j+2] ||^2
			// <=>	t = sqrt( 3/4 )
			for( int j=0 ; j<3 ; j++ ) eVertices[j] = RightTriangle< Real >::EdgeMidpoints[j] - Rotate90( tensor ,  RightTriangle< Real >::EdgeDirections[j] ) * (Real)SQRT_THREE_QUARTERS;
			Real maxDet = 0;
			Point2D< Real > c;
			for( int j=0 ; j<3 ; j++ )
			{
				Point2D< Real > c1 = eVertices[(j+1)%3] , c2 = eVertices[(j+2)%3];
				Point2D< Real > v1 = RightTriangle< Real >::Corners[(j+1)%3] - c1 , v2 = RightTriangle< Real >::Corners[(j+2)%3] - c2;

				// Solve for s and t such that:
				//		c1 + s * v1 = c2 + t * v2
				// =>	c1 - c2 = - s * v1 + t * v2
				// =>	c1 - c2 = ( -v1 | v2 ) * ( s , t )^t
				SquareMatrix< Real , 2 > M;
				M( 0 , 0 ) = -v1[0] , M( 0 , 1 ) = -v1[1];
				M( 1 , 0 ) =  v2[0] , M( 1 , 1 ) =  v2[1];
				Real det = (Real)fabs( M.determinant() );
				if( det>maxDet )
				{
					Point2D< Real > x = M.inverse() * ( c1 - c2 );
					c = ( c1 + v1 * x[0] + c2 + v2 * x[1] ) / 2;
					maxDet = det;
				}
			}
			return c;
		}
	default:
		fprintf( stderr , "[ERROR] FEM::RightTriangle::Center: Unrecognized center type: %d\n" , centerType ) , exit( 0 );
	}
}
template< class Real >
inline Point3D< Real > FEM::RightTriangle< Real >::SubTriangleAreas( const SquareMatrix< Real , 2 >& tensor , Point2D< Real > center )
{
	Point2D< Real > triangle[3];
	triangle[2] = center;
	Point3D< Real > areas;
	for( int i=0 ; i<3 ; i++ )
	{
		triangle[0] = RightTriangle< Real >::Corners[(i+1)%3] , triangle[1] = RightTriangle< Real >::Corners[(i+2)%3];
		areas[i] = Area( tensor , triangle );
	}
	return areas;
}
template< class Real >
inline Point3D< Real > FEM::RightTriangle< Real >::CenterAreas( const SquareMatrix< Real , 2 >& tensor , int centerType ){ return SubTriangleAreas( tensor , Center( tensor , centerType ) ); }

template< class Real >
inline Point2D< Real > FEM::RightTriangle< Real >::EdgeReflect( const SquareMatrix< Real , 2 >& tensor , int e , Point2D< Real > p )
{
	Point2D< Real > c = Corners[(e+1)%3] , v = p - c ,  perp = Rotate90( tensor , EdgeDirections[e] );
	return c + v - ( 2 * Point2D< Real >::Dot( perp , tensor * v ) / Point2D< Real >::Dot( perp , tensor * perp ) ) * perp;
}

template< class Real >
template< unsigned int BasisType , class V >
inline V FEM::RightTriangle< Real >::EvaluateScalarField( ConstPointer( V ) c , const Point2D< Real >& p )
{
	switch( BasisType )
	{
		case BASIS_0_WHITNEY: return c[0] * (Real)( 1. - p[0] - p[1] ) + c[1] * p[0] + c[2] * p[1];
		default: fprintf( stderr , "[ERROR] FEM::RightTriangle::EvaluateScalarField: unrecognized basis type: %d\n" , BasisType ) , exit( 0 );
	}
}
template< class Real >
template< unsigned int BasisType , class V >
FEM::CotangentVector< V > FEM::RightTriangle< Real >::EvaluateScalarFieldGradient( ConstPointer( V ) c , const Point2D< Real >& p )
{
	switch( BasisType )
	{
		case BASIS_0_WHITNEY: return CotangentVector< V >( c[0]*CornerGradients[0][0] + c[1]*CornerGradients[1][0] + c[2]*CornerGradients[2][0] , c[0]*CornerGradients[0][1] + c[1]*CornerGradients[1][1] + c[2]*CornerGradients[2][1] );
		default: fprintf( stderr , "[ERROR] FEM::RightTriangle::EvaluateScalarFieldGradient: unrecognized basis type: %d\n" , BasisType ) , exit( 0 );
	}
}
template< class Real >
template< unsigned int BasisType , class V >
inline FEM::CotangentVector< V > FEM::RightTriangle< Real >::EvaluateCovectorField( const SquareMatrix< Real , 2 >& tensor , ConstPointer( V ) c , const Point2D< Real >& p )
{
	switch( BasisType )
	{
		case BASIS_1_CONFORMING:
		{
			SquareMatrix< Real , 2 > J = CoJ( tensor );
			CotangentVector< Real > g[] = { CornerGradients[0] , J *CornerGradients[0] , CornerGradients[1] , J * CornerGradients[1] , CornerGradients[2] , J * CornerGradients[2] };
			return CotangentVector< V >( c[0]*g[0][0] + c[1]*g[1][0] + c[2]*g[2][0] + c[3]*g[3][0] + c[4]*g[4][0] + c[5]*g[5][0] , c[0]*g[0][1] + c[1]*g[1][1] + c[2]*g[2][1] + c[3]*g[3][1] + c[4]*g[4][1] + c[5]*g[5][1] );
		}
		case BASIS_1_WHITNEY:
		{
			Point3D< Real > v( (Real)( 1. - p[0] - p[1] ) , p[0] , p[1] );
			V n[] = { c[1]*v[2] - c[2]*v[1] , c[2]*v[0] - c[0]*v[2] , c[0]*v[1] - c[1]*v[0] };
			return CotangentVector< V >( n[0]*CornerGradients[0][0] + n[1]*CornerGradients[1][0] + n[2]*CornerGradients[2][0] , n[0]*CornerGradients[0][0] + n[1]*CornerGradients[1][1] + n[2]*CornerGradients[2][1] );
		}
		case BASIS_1_TRIANGLE_CONSTANT: return CotangentVector< V >( c[0] , c[1] );
		default: fprintf( stderr , "[ERROR] FEM::RightTriangle::EvaluateCovectorField: unrecognized basis type: %d\n" , BasisType ) , exit( 0 );
	}
}
template< class Real >
template< unsigned int BasisType , class V >
inline V FEM::RightTriangle< Real >::EvaluateDensityField( const SquareMatrix< Real , 2 >& tensor , ConstPointer( V ) c , const Point2D< Real >& p )
{
	switch( BasisType )
	{
		case BASIS_2_WHITNEY: return c[0] / Area( tensor );
		default: fprintf( stderr , "[ERROR] FEM::RightTriangle::EvaluateDensityField: unrecognized basis type: %d\n" , BasisType ) , exit( 0 );
	}
}
template< class Real >
template< unsigned int InBasisType , unsigned int OutBasisType >
typename FEM::BasisInfoSystem2< Real , InBasisType , OutBasisType >::MaskMatrix FEM::RightTriangle< Real >::GetDMask( bool& redundant )
{
	auto Fail = [&] ( void ){ fprintf( stderr , "[ERROR] FEM::RightTriangle::GetDMask: %s -> %s\n" , BasisNames[ InBasisType ] , BasisNames[ OutBasisType ] ) , exit( 0 ); };

	TestBasisType(  InBasisType , "FEM::RightTriangle::GetDMask" , false );
	TestBasisType( OutBasisType , "FEM::RightTriangle::GetDMask" , false );

	Matrix< unsigned char , FEM::BasisInfo< InBasisType >::Coefficients , FEM::BasisInfo< OutBasisType >::Coefficients > D;
	D *= 0;

	if( InBasisType==BASIS_0_WHITNEY )
	{
		if     ( OutBasisType==BASIS_1_WHITNEY           ) for( int e=0 ; e<3 ; e++ ) D( (e+1)%3 , e ) = D( (e+2)%3 , e ) = 1;
		else if( OutBasisType==BASIS_1_CONFORMING        ) for( int v=0 ; v<3 ; v++ ) D( v , v*2 ) = 1;
		else if( OutBasisType==BASIS_1_TRIANGLE_CONSTANT ) for( int i=0 ; i<2 ; i++ ) D( 0 , i ) = D( 1+i , i ) = 1;
		else Fail();
	}
	else if( OutBasisType==BASIS_2_VERTEX_CONSTANT )
	{
		if     ( InBasisType==BASIS_1_CONFORMING                                        ) for( int v=0 ; v<3 ; v++ ) for( int i=0 ; i<3 ; i++ ) D( i*2+1 , v ) = 1;
		else if( InBasisType==BASIS_1_WHITNEY || InBasisType==BASIS_1_TRIANGLE_CONSTANT ) for( int v=0 ; v<3 ; v++ ) for( int i=0 ; i<BasisInfo< InBasisType >::Coefficients ; i++ ) D(i,v) = 1;
		else Fail();
	}
	else if( InBasisType==BASIS_1_WHITNEY )
	{
		if( OutBasisType==BASIS_2_WHITNEY ) D(0,0) = D(1,0) = D(2,0) = 1;
		else Fail();
	}
	else Fail();
	redundant = ( OutBasisType == BASIS_1_WHITNEY || OutBasisType==BASIS_1_CONFORMING );
	return D;
}
template< class Real >
template< unsigned int InBasisType , unsigned int OutBasisType >
typename FEM::BasisInfoSystem2< Real , InBasisType , OutBasisType >::Matrix FEM::RightTriangle< Real >::GetDMatrix( const SquareMatrix< Real , 2 >& tensor )
{
	auto Fail = [&] ( void ){ fprintf( stderr , "[ERROR] FEM::RightTriangle::GetDMatrix: %s -> %s\n" , BasisNames[ InBasisType ] , BasisNames[ OutBasisType ] ) , exit( 0 ); };

	TestBasisType(  InBasisType , "FEM::RightTriangle::GetDMatrix" , false );
	TestBasisType( OutBasisType , "FEM::RightTriangle::GetDMatrix" , false );

	Matrix< Real , FEM::BasisInfo< InBasisType >::Coefficients , FEM::BasisInfo< OutBasisType >::Coefficients > D;

	if( InBasisType==BASIS_0_WHITNEY )
	{
		if     ( OutBasisType==BASIS_1_WHITNEY           ) for( int e=0 ; e<3 ; e++ ) D( (e+1)%3 , e ) = -1 , D( (e+2)%3 , e ) = 1;
		else if( OutBasisType==BASIS_1_CONFORMING        ) for( int v=0 ; v<3 ; v++ ) D( v , v*2 ) = 1;
		else if( OutBasisType==BASIS_1_TRIANGLE_CONSTANT ) for( int i=0 ; i<2 ; i++ ) D( 0 , i ) = -1 , D( 1+i , i ) = 1;
		else Fail();
	}
	else if( OutBasisType==BASIS_2_VERTEX_CONSTANT )
	{
		if( InBasisType==BASIS_1_WHITNEY || InBasisType==BASIS_1_CONFORMING || InBasisType==BASIS_1_TRIANGLE_CONSTANT )
		{
			Point2D< Real > center = Center( tensor , CENTER_CIRCUMCENTRIC );
			for( int v=0 ; v<3 ; v++ )
			{
				Point< Real , BasisInfo< InBasisType >::Coefficients > d = IntegrationDual< InBasisType >( tensor , EdgeMidpoints[(v+2)%3] , center ) + IntegrationDual< InBasisType >( tensor , center , EdgeMidpoints[(v+1)%3] );
				for( int i=0 ; i<BasisInfo< InBasisType >::Coefficients ; i++ ) D(i,v) = d[i];
			}

		}
		else Fail();
	}
	else if( InBasisType==BASIS_1_WHITNEY )
	{
		if( OutBasisType==BASIS_2_WHITNEY ) D(0,0) = D(1,0) = D(2,0) = 1;
		else Fail();
	}
	else Fail();
	return D;
}
template< class Real >
template< unsigned int BasisType >
typename FEM::BasisInfoSystem< Real , BasisType >::Matrix FEM::RightTriangle< Real >::GetMassMatrix( const SquareMatrix< Real , 2 >& tensor )
{
	SquareMatrix< Real , BasisInfo< BasisType >::Coefficients > mass;

	switch( BasisType )
	{
		case BASIS_0_WHITNEY:
		{
			{
				mass(0,0) = mass(1,1) = mass(2,2) = (Real)1./6;
				mass(1,0) = mass(0,1) = mass(1,2) = mass(2,1) = mass(0,2) = mass(2,0) = (Real)1./12;	
				mass *= (Real)( sqrt( tensor.determinant() ) / 2. );
			}
			break;
		}
		case BASIS_1_CONFORMING:
		{
			{
				SquareMatrix< Real , 2 > iTensor = tensor.inverse() * (Real)( sqrt( tensor.determinant() ) / 2. );
				for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) mass( 2*i , 2*j ) = mass( 2*i+1 , 2*j+1 ) = Point2D< Real >::Dot( CornerGradients[i] , iTensor * CornerGradients[j] );
			}
			break;
		}
		case BASIS_1_WHITNEY:
		{
			{
				SquareMatrix< Real , 3 > M = GetMassMatrix< BASIS_0_WHITNEY >( tensor ) , S;
				SquareMatrix< Real , 2 > iTensor = tensor.inverse();
				for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) S(i,j) = Point2D< Real >::Dot( CornerGradients[i] , iTensor * CornerGradients[j] );
		
				const int I[][2] = { {1,2} , {2,0} , {0,1} };
				for( int i=0 ; i<3;  i++ ) for( int j=0 ; j<3 ; j++ )
					mass(i,j) = M( I[i][0] , I[j][0] ) * S( I[i][1] , I[j][1] ) + M( I[i][1] , I[j][1] ) * S( I[i][0] , I[j][0] ) - M( I[i][0] , I[j][1] ) * S( I[i][1] , I[j][0] ) - M( I[i][1] , I[j][0] ) * S( I[i][0] , I[j][1] );
			}
			break;
		}
		case BASIS_1_TRIANGLE_CONSTANT:
		{
			SquareMatrix< Real , 2 > iTensor = tensor.inverse();
			for( int i=0 ; i<2 ; i++ ) for( int j=0 ; j<2 ; j++ ) mass(i,j) = iTensor(i,j);
			mass *= (Real)( sqrt( tensor.determinant() ) / 2. );
			break;
		}
		case BASIS_2_WHITNEY:
		{
			mass(0,0) = (Real)( 1./(Real)( sqrt( tensor.determinant() ) / 2. ) );
		}
		break;
		default: TestBasisType( BasisType , "FEM::RightTriangle::GetMassMatrix" , true );
	}

	return mass;
}
template< class Real >
template< unsigned int BasisType >
typename FEM::BasisInfoSystem< Real , BasisType >::Matrix FEM::RightTriangle< Real >::GetMassMatrix( const SquareMatrix< Real , 2 >& tensor , const SquareMatrix< Real , 2 >& newTensor )
{
	SquareMatrix< Real , BasisInfo< BasisType >::Coefficients > mass;
	SquareMatrix< Real , 2 > tensor_i = tensor.inverse() , newCotensor = tensor_i.transpose() * newTensor * tensor_i;

	switch( BasisType )
	{
		case BASIS_1_CONFORMING:
		{
			CotangentVector< Real > grads[] = { CornerGradients[0] , Rotate90( tensor , CornerGradients[0] ) , CornerGradients[1] , Rotate90( tensor , CornerGradients[1] ) , CornerGradients[2] , Rotate90( tensor , CornerGradients[2] ) };
			for( int i=0 ; i<6 ; i++ ) for( int j=0 ; j<6 ; j++ ) mass(i,j) = Point2D< Real >::Dot( grads[i]  , newCotensor * grads[j] );
			mass *= (Real)( sqrt( tensor.determinant() ) / 2. );
			break;
		}
		case BASIS_1_TRIANGLE_CONSTANT:
		{
			for( int i=0 ; i<2 ; i++ ) for( int j=0 ; j<2 ; j++ ) mass(i,j) = newCotensor(i,j);
			mass *= (Real)( sqrt( tensor.determinant() ) / 2. );
			break;
		}
		case BASIS_1_WHITNEY:
		{
			SquareMatrix< Real , 3 > M = GetMassMatrix< BASIS_0_WHITNEY >( tensor ) , S;
			for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) S(i,j) = Point2D< Real >::Dot( CornerGradients[i] , newCotensor * CornerGradients[j] );
		
			const int I[][2] = { {1,2} , {2,0} , {0,1} };
			for( int i=0 ; i<3;  i++ ) for( int j=0 ; j<3 ; j++ )
				mass(i,j) = M( I[i][0] , I[j][0] ) * S( I[i][1] , I[j][1] ) + M( I[i][1] , I[j][1] ) * S( I[i][0] , I[j][0] ) - M( I[i][0] , I[j][1] ) * S( I[i][1] , I[j][0] ) - M( I[i][1] , I[j][0] ) * S( I[i][0] , I[j][1] );
			break;
		}
		default: TestBasisType( BasisType , "FEM::RightTriangle::GetCovectorMassMatrix" , true );
	}

	return mass;
}
template< class Real >
template< unsigned int BasisType >
typename FEM::BasisInfoSystem< Real , BasisType >::Point FEM::RightTriangle< Real >::GetDiagonalMassMatrix( const SquareMatrix< Real , 2 >& tensor )
{
	Point< Real , BasisInfo< BasisType >::Coefficients > mass;

	switch( BasisType )
	{
		case BASIS_0_WHITNEY:
			mass[0] = mass[1] = mass[2] = (Real)( sqrt( tensor.determinant() ) / 6. );
			break;
		case BASIS_1_CONFORMING:
		{
			Point3D< Real > areas = CenterAreas( tensor , CENTER_CIRCUMCENTRIC )*2;
			mass[0] = mass[1] = areas[0] / SquareLength( tensor , EdgeDirections[0] );
			mass[2] = mass[3] = areas[1] / SquareLength( tensor , EdgeDirections[1] );
			mass[4] = mass[5] = areas[2] / SquareLength( tensor , EdgeDirections[2] );
			break;
		}
		case BASIS_1_WHITNEY:
		{
			Point3D< Real > areas = CenterAreas( tensor , CENTER_CIRCUMCENTRIC )*2;
			mass[0] = areas[0] / SquareLength( tensor , EdgeDirections[0] );
			mass[1] = areas[1] / SquareLength( tensor , EdgeDirections[1] );
			mass[2] = areas[2] / SquareLength( tensor , EdgeDirections[2] );
			break;
		}
		case BASIS_2_WHITNEY:
			mass[0] = (Real)( 1./( sqrt( tensor.determinant() ) / 2. ) );
			break;
		default: TestBasisType( BasisType , "FEM::RightTriangle::GetDiagonalMassMatrix" , true );
	}

	return mass;
}
template< class Real >
template< unsigned int BasisType >
typename FEM::BasisInfoSystem< Real , BasisType >::MaskMatrix FEM::RightTriangle< Real >::GetMassMask( bool useTensor )
{
	SquareMatrix< unsigned char , BasisInfo< BasisType >::Coefficients > M;
	M *= 0;

	if( useTensor )
		switch( BasisType )
		{
			case BASIS_0_WHITNEY:           for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) M(i,j) = 1 ; break;
			case BASIS_1_CONFORMING:        for( int i=0 ; i<6 ; i++ ) for( int j=0 ; j<6 ; j++ ) M(i,j) = 1 ; break;
			case BASIS_1_WHITNEY:           for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) M(i,j) = 1 ; break;
			case BASIS_1_TRIANGLE_CONSTANT: for( int i=0 ; i<2 ; i++ ) for( int j=0 ; j<2 ; j++ ) M(i,j) = 1 ; break;
			case BASIS_2_WHITNEY:                                                                 M(0,0) = 1 ; break;
			default: TestBasisType( BasisType , "FEM::RightTriangle::GetMassMask" , true );
		}
	else
		switch( BasisType )
		{
			case BASIS_0_WHITNEY:           for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) M(i,j) = 1 ; break;
			case BASIS_1_CONFORMING:        for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) M(2*i,2*j) = M(2*i+1,2*j+1) = 1 ; break;
			case BASIS_1_WHITNEY:           for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) M(i,j) = 1 ; break;
			case BASIS_1_TRIANGLE_CONSTANT: for( int i=0 ; i<2 ; i++ ) for( int j=0 ; j<2 ; j++ ) M(i,j) = 1 ; break;
			case BASIS_2_WHITNEY:                                                                 M(0,0) = 1 ; break;
			default: TestBasisType( BasisType , "FEM::RightTriangle::GetMassMask" , true );
		}
	return M;
}
template< class Real >
template< unsigned int BasisType >
Real FEM::RightTriangle< Real >::Integrate( const SquareMatrix< Real , 2 >& tensor , ConstPointer( Real ) linear )
{
	Real integral = 0;
	switch( BasisType )
	{
		case BASIS_0_WHITNEY:
		{
			SquareMatrix< Real , 3 > mass = GetMassMatrix< BASIS_0_WHITNEY >( tensor );
			for( int i=0 ; i<3 ; i++ ) integral += linear[i] * ( mass(i,0) + mass(i,1) + mass(i,2) );
			break;
		}
		default: TestBasisType( BasisType , "FEM::RightTriangle::Integrate" , true );
	}
	return integral;
}
template< class Real >
template< unsigned int BasisType >
typename FEM::BasisInfoSystem< Real , BasisType >::Point FEM::RightTriangle< Real >::IntegrationDual( const SquareMatrix< Real , 2 >& tensor , ConstPointer( Real ) linear )
{
	Point< Real , BasisInfo< BasisType >::Coefficients > integralDual;
	switch( BasisType )
	{
		case BASIS_0_WHITNEY:
		{
			SquareMatrix< Real , 3 > mass = GetMassMatrix< BASIS_0_WHITNEY >( tensor );
			for( int i=0 ; i<3 ; i++ ) for( int j=0 ; j<3 ; j++ ) integralDual[i] += linear[j] * mass(i,j);
			break;
		}
		default: TestBasisType( BasisType , "FEM::RightTriangle::IntegrationDual" , true );
	}
	return integralDual;
}
template< class Real >
template< unsigned int BasisType >
typename FEM::BasisInfoSystem< Real , BasisType >::Point FEM::RightTriangle< Real >::IntegrationDual( const SquareMatrix< Real , 2 >& tensor , ConstPointer( CotangentVector< Real > ) linear )
{
	Point< Real , BasisInfo< BasisType >::Coefficients > integralD;
	switch( BasisType )
	{
		case BASIS_1_TRIANGLE_CONSTANT:
		{
			SquareMatrix< Real , 2 > tensor_i = tensor.inverse();
			Point2D< Real > dualBasis[] = { tensor_i * Point2D< Real >( 1 , 0 ) , tensor_i * Point2D< Real >( 0 , 1 ) };
			Real _linear[3];
			for( int i=0 ; i<2 ; i++ )
			{
				for( int j=0 ; j<3 ; j++ ) _linear[j] = Point2D< Real >::Dot( dualBasis[i] , linear[j] );
				integralD[i] = Integrate< BASIS_0_WHITNEY >( tensor , ( ConstPointer( Real ) )GetPointer( _linear , 3 ) );
			}
			break;
		}
		case BASIS_1_CONFORMING:
		{
			SquareMatrix< Real , 2 > tensor_i = tensor.inverse();
			Point2D< Real > dualBasis[] =
			{
				tensor_i * CornerGradients[0] , tensor_i * Rotate90( tensor , CornerGradients[0] ) ,
				tensor_i * CornerGradients[1] , tensor_i * Rotate90( tensor , CornerGradients[1] ) ,
				tensor_i * CornerGradients[2] , tensor_i * Rotate90( tensor , CornerGradients[2] )
			};
			Real _linear[3];
			for( int i=0 ; i<6 ; i++ )
			{
				for( int j=0 ; j<3 ; j++ ) _linear[j] = Point2D< Real >::Dot( dualBasis[i] , linear[j] );
				integralD[i] = Integrate< BASIS_0_WHITNEY >( tensor , ( ConstPointer( Real ) )GetPointer( _linear , 3 ) );
			}
			break;
		}
		case BASIS_1_WHITNEY:
		{
			// \phi_{ij}(p) = \phi_i(p) \nabla \phi_j - \phi_j(p) \nabla \phi_i
			// phi[i](p) = hat[i+1](p) * CornerGradients[i+2] - hat[i+2](p) * CornerGradients[i+1]
			// < phi[i](p) , L(p) > = \sum_j < phi[i](p) , linear[j] * hat[j](p) >
			//                      = \sum_j < hat[i+1](p) * CornerGradients[i+2] - hat[i+2](p) * CornerGradients[i+1] , linear[j] * hat[j](p) >
			//                      = \sum_j < CornerGradients[i+2] , linear[j] > * Integral{ hat[i+1](p) * hat[j](p) } - < CornerGradients[i+1] , linear[j] >  * Integral{ hat[i+2](p) * hat[j](p) }
			SquareMatrix< Real , 3 > mass = GetMassMatrix< BASIS_0_WHITNEY >( tensor );
			SquareMatrix< Real , 2 > tensor_i = tensor.inverse();
			for( int j=0 ; j<3 ; j++ )
			{
				CotangentVector< Real > linearD = tensor_i * linear[j];
				for( int i=0 ; i<3 ; i++ ) 
				{
					integralD[i] += mass( (i+1)%3 , j ) * Point2D< Real >::Dot( CornerGradients[(i+2)%3] , linearD );
					integralD[i] -= mass( (i+2)%3 , j ) * Point2D< Real >::Dot( CornerGradients[(i+1)%3] , linearD );
				}
			}
			break;
		}
		default: TestBasisType( BasisType , "FEM::RightTriangle::IntegrationDual" , true );
	}
	return integralD;
}
template< class Real >
template< unsigned int BasisType >
typename FEM::BasisInfoSystem< Real , BasisType >::Point FEM::RightTriangle< Real >::IntegrationDual( const SquareMatrix< Real , 2 >& tensor , Point2D< Real > p , Point2D< Real > q )
{
	Point< Real , BasisInfo< BasisType >::Coefficients > integralDual , coefficients;
	switch( BasisType )
	{
		case BASIS_1_WHITNEY:
		case BASIS_1_CONFORMING:
		case BASIS_1_TRIANGLE_CONSTANT:
		{
			TangentVector< Real >dir = q - p;
			for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ )
			{
				coefficients[i] = 1;
				CotangentVector< Real > _p = EvaluateCovectorField< BasisType >( tensor , ( ConstPointer( Real ) )GetPointer( &coefficients[0] , BasisInfo< BasisType >::Coefficients ) , p );
				CotangentVector< Real > _q = EvaluateCovectorField< BasisType >( tensor , ( ConstPointer( Real ) )GetPointer( &coefficients[0] , BasisInfo< BasisType >::Coefficients ) , q );
				integralDual[i] = ( Point2D< Real >::Dot( dir , _p ) + Point2D< Real >::Dot( dir , _q ) ) / 2;
				coefficients[i] = 0;
			}
			break;
		}
		default: TestBasisType( BasisType , "FEM::RightTriangle::IntegrationDual" , true );
	}
	return integralDual;
}

////////////////////////////////////////////////////
// FEM::RiemannianMesh::TangentVectorFieldWrapper //
////////////////////////////////////////////////////
template< class Real , unsigned int BasisType >
FEM::TangentVectorFieldWrapper< Real , BasisType >::TangentVectorFieldWrapper( const RiemannianMesh< Real >* mesh , ConstPointer( Real ) coefficients , bool precomputeInverses ) : _mesh(mesh) , _coefficients(coefficients)
{
	if( precomputeInverses )
	{
		_gInverse = AllocPointer< SquareMatrix< Real , 2 > >( _mesh->tCount() );
#pragma omp parallel for
		for( int t=0 ; t<_mesh->tCount() ; t++ ) _gInverse[t] = _mesh->g(t).inverse();
	}
	else _gInverse = NullPointer< SquareMatrix< Real , 2 > >();
}
template< class Real , unsigned int BasisType >
FEM::TangentVectorFieldWrapper< Real , BasisType >::~TangentVectorFieldWrapper( void ){ FreePointer( _gInverse ); }
template< class Real , unsigned int BasisType >
FEM::TangentVector< Real > FEM::TangentVectorFieldWrapper< Real , BasisType >::operator() ( const FEM::SamplePoint< Real >& p ) const
{
	bool isAligned;
	Real c[BasisInfo< BasisType >::Coefficients];
	for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ )
	{
		int idx =_mesh->template index< BasisType >( p.tIdx , i , isAligned );
		c[i] = isAligned ? _coefficients[ idx ] : -_coefficients[ idx ] ;
	}
	if( _gInverse ) return _gInverse[ p.tIdx ]          * RightTriangle< Real >::template EvaluateCovectorField< BasisType >( _mesh->g(p.tIdx) , ( ConstPointer( Real ) )GetPointer( &c[0] , BasisInfo< BasisType >::Coefficients ) , p.p );
	else            return _mesh->g( p.tIdx ).inverse() * RightTriangle< Real >::template EvaluateCovectorField< BasisType >( _mesh->g(p.tIdx) , ( ConstPointer( Real ) )GetPointer( &c[0] , BasisInfo< BasisType >::Coefficients ) , p.p );
}

/////////////////////////
// FEM::RiemannianMesh //
/////////////////////////
template< class Real >
FEM::RiemannianMesh< Real >::RiemannianMesh( Pointer( TriangleIndex ) t , size_t tC ) : _triangles(t) , _tCount(tC) , _edgeMap( t , tC )
{
	_g = AllocPointer< SquareMatrix< Real , 2 > >( _tCount );
	for( int t=0 ; t<_tCount ; t++ ) _g[t] = SquareMatrix< Real , 2 >::Identity();
	_vCount = 0;
	for( size_t i=0 ; i<_tCount ; i++ ) for( int j=0 ; j<3 ; j++ ) _vCount = std::max< size_t >( _vCount , _triangles[i][j]+1 );
}
template< class Real >
FEM::RiemannianMesh< Real >::~RiemannianMesh( void ){ FreePointer( _g ); }
template< class Real >
template< unsigned int BasisType >
size_t FEM::RiemannianMesh< Real >::dimension( void ) const
{
	size_t dim;
	switch( BasisInfo< BasisType >::ElementType )
	{
		case ELEMENT_VERTEX:   dim = vCount() * BasisInfo< BasisType >::CoefficientsPerElement ; break;
		case ELEMENT_EDGE:     dim = eCount() * BasisInfo< BasisType >::CoefficientsPerElement ; break;
		case ELEMENT_TRIANGLE: dim = tCount() * BasisInfo< BasisType >::CoefficientsPerElement ; break;
		default:
			TestBasisType( BasisType , "FEM::RiemannianMesh::dimension" , false );
			TestElementType( BasisInfo< BasisType >::ElementType , "FEM::RiemannianMesh::dimension" , true );
	}
	return dim;
}
template< class Real >
template< unsigned int BasisType >
int FEM::RiemannianMesh< Real >::index( int t , int idx ) const
{
	bool isAligned;
	return index< BasisType >( t , idx , isAligned );
}
template< class Real >
template< unsigned int BasisType >
int FEM::RiemannianMesh< Real >::index( int t , int idx , bool& isAligned ) const
{
	isAligned = true;
	int i , e = idx / BasisInfo< BasisType >::CoefficientsPerElement , c = idx % BasisInfo< BasisType >::CoefficientsPerElement;
	switch( BasisInfo< BasisType >::ElementType )
	{
		case ELEMENT_VERTEX:   i = _triangles[t][e]                   ; break;
		case ELEMENT_EDGE:     i = _edgeMap.edge( t*3+e , isAligned ) ; break;
		case ELEMENT_TRIANGLE: i = t                                  ; break;
		default: fprintf( stderr , "[ERROR] FEM::RiemannianMesh::index: unrecognized element type: %d\n" , BasisInfo< BasisType >::ElementType ) , exit( 0 );
	}
	return i * BasisInfo< BasisType >::CoefficientsPerElement + c;
}

template< class Real >
Pointer( FEM::CoordinateXForm< Real > ) FEM::RiemannianMesh< Real >::getCoordinateXForms( void ) const
{
	Pointer( CoordinateXForm< Real > ) xForms = NewPointer< CoordinateXForm< Real > >( _tCount*3 );
	setCoordinateXForms( xForms );
	return xForms;
}
template< class Real >
FEM::CoordinateXForm< Real > FEM::RiemannianMesh< Real >::xForm( int he ) const
{
	CoordinateXForm< Real > xForm;
	int ohe = _edgeMap.opposite(he);
	if( ohe==-1 ) fprintf( stderr , "[ERROR] FEM::RiemannianMesh::xForm: Boundary edge\n" ) , exit( 0 );

	// The two triangles on this edge
	int tIdx[] = { he/3 , ohe/3 };

	// The end-points of the edge
	int  _v =  he%3 ,  v[] = { ( he+1)%3 , ( he+2)%3 };
	int _ov = ohe%3 , ov[] = { (ohe+1)%3 , (ohe+2)%3 };

	// The direction of the edge
	TangentVector< Real > edgeDir = RightTriangle< Real >::EdgeDirections[_v] , oEdgeDir = -RightTriangle< Real >::EdgeDirections[_ov];
	// The perpendicular direction to the edge
	TangentVector< Real > edgePerpDir = Rotate90( _g[ tIdx[0] ] , edgeDir ) , oEdgePerpDir = Rotate90( _g[ tIdx[1] ] , oEdgeDir );
		
	// The linear part of the transformation should map ( edgeDir , edgePerpDir ) -> ( oEdgeDir , oEdgePerpDir )
	{
		SquareMatrix< Real , 2 > M , oM;
		M( 0 , 0 ) =     edgeDir[0] , M( 0 , 1 ) =     edgeDir[1];
		M( 1 , 0 ) = edgePerpDir[0] , M( 1 , 1 ) = edgePerpDir[1];
		oM( 0 , 0 ) =     oEdgeDir[0] , oM( 0 , 1 ) =     oEdgeDir[1];
		oM( 1 , 0 ) = oEdgePerpDir[0] , oM( 1 , 1 ) = oEdgePerpDir[1];
		xForm.linear = oM * M.inverse();
	}

	// The transformation should also take ( v[0] + v[1] ) / 2 to ( ov[0] + ov[1] ) / 2
	xForm.constant = RightTriangle< Real >::EdgeMidpoints[_ov] - xForm.linear * ( RightTriangle< Real >::EdgeMidpoints[_v] );
	return xForm;
}
template< class Real >
void FEM::RiemannianMesh< Real >::setCoordinateXForms( Pointer( CoordinateXForm< Real > ) xForms ) const
{
#pragma omp parallel for
	for( int e=0 ; e<_edgeMap.size() ; e++ )
	{
		const int* he = _edgeMap[e];
		if( he[1]!=-1 )
		{
			xForms[ he[0] ] = xForm( he[0] );
			xForms[ he[1] ] = xForms[ he[0] ].inverse();
		}
		else xForms[ he[0] ] = CoordinateXForm< Real >();
	}
}
template< class Real >
void FEM::RiemannianMesh< Real >::edgeVertices( int edge , int& v1 , int& v2 ) const
{
	int he = _edgeMap[edge][0];
	int  t =  he / 3 ,  v =  he % 3;
	v1 = _triangles[t][(v+1)%3] , v2 = _triangles[t][(v+2)%3];
}
template< class Real >
bool FEM::RiemannianMesh< Real >::oppositeEdgeVertices( int edge , int& v1 , int& v2 ) const
{
	int he = _edgeMap[edge][0] , ohe = _edgeMap[edge][1];
	int  t =  he / 3 ,  v =  he % 3;
	int ot = ohe / 3 , ov = ohe % 3;
	if( ohe==-1 ) return false;
	v1 = _triangles[t][v] , v2 = _triangles[ot][ov];
	return true;
}
template< class Real >
bool FEM::RiemannianMesh< Real >::edgeFlip( int edge , Real eps )
{
	int he = _edgeMap[edge][0] , ohe = _edgeMap[edge][1];
	int  t =  he / 3 ,  v =  he % 3;
	int ot = ohe / 3 , ov = ohe % 3;

	// First test that the edge is not on the boundary
	if( ohe==-1 ) return false;

	// Get the coordinates of the old and new edges and test that if we can flip
	Point2D< Real > newEdge[] = { RightTriangle< Real >::Corners[v] , xForm( ohe )( RightTriangle< Real >::Corners[ ov ] ) };
	Point2D< Real > oldEdge[] = { RightTriangle< Real >::Corners[(v+1)%3] , RightTriangle< Real >::Corners[(v+2)%3] };
	{

		// Solve for the point of intersection between the two corresponding line segments
		//  => Solve for s and t such that:
		//		newEdge[0] + s * ( newEdge[1]-newEdge[0]) = oldEdge[0] + t * ( oldEdge[1] - oldEdge[0] )
		// <=>	( newEdge[1]-newEdge[0] | -oldEdge[1]+oldEdge[0] ) * (s,t)^t = oldEdge[0] - newEdge[0]
		SquareMatrix< Real , 2 > M;
		for( int i=0 ; i<2 ; i++ ) M(0,i) = newEdge[1][i]-newEdge[0][i] , M(1,i) = -oldEdge[1][i]+oldEdge[0][i];
		Point2D< Real > st = M.inverse() * ( oldEdge[0] - newEdge[0] );

		// Test that the intersection point is in bounds
		if( st[0]<=eps || st[0]>=1-eps || st[1]<=eps || st[1]>=1-eps ) return false;
	}

	//     o              2,1
	//    / \             /|\
	//   /   \           / | \
	//  /     \         /  |  \
	// |-------|  =>  0| o | t |0
	//  \  e  /         \  |  /
	//   \   /           \ | /
	//    \ /             \|/
	//     v              1,2

	//  t*3 + ( v+1) -> ot*3+2
	//  t*3 + ( v+2) ->  t*3+1
	// ot*3 + (ov+1) ->  t*3+2
	// ot*3 + (ov+2) -> ot*3+1

	// The new triangles
	TriangleIndex tris[] = { TriangleIndex( _triangles[t][(v+1)%3] , _triangles[ot][ov] , _triangles[t][v] ) , TriangleIndex( _triangles[t][(v+2)%3] , _triangles[t][v] , _triangles[ot][ov] ) };

	// The new metrics
	SquareMatrix< Real , 2 > tensors[2];
	tensors[0](0,0) = Point2D< Real >::Dot( RightTriangle< Real >::Corners[ov] - RightTriangle< Real >::Corners[(ov+2)%3] , _g[ot] * ( RightTriangle< Real >::Corners[ov] - RightTriangle< Real >::Corners[(ov+2)%3] ) );
	tensors[0](1,1) = Point2D< Real >::Dot( RightTriangle< Real >::Corners[ v] - RightTriangle< Real >::Corners[( v+1)%3] , _g[ t] * ( RightTriangle< Real >::Corners[ v] - RightTriangle< Real >::Corners[( v+1)%3] ) );
	tensors[0](1,0) = tensors[0](0,1) = ( tensors[0](0,0) + tensors[0](1,1) - Point2D< Real >::Dot( newEdge[1]-newEdge[0] , _g[t] * ( newEdge[1]-newEdge[0] ) ) ) / (Real)2.;

	tensors[1](0,0) = Point2D< Real >::Dot( RightTriangle< Real >::Corners[ v] - RightTriangle< Real >::Corners[( v+2)%3] , _g[ t] * ( RightTriangle< Real >::Corners[ v] - RightTriangle< Real >::Corners[( v+2)%3] ) );
	tensors[1](1,1) = Point2D< Real >::Dot( RightTriangle< Real >::Corners[ov] - RightTriangle< Real >::Corners[(ov+1)%3] , _g[ot] * ( RightTriangle< Real >::Corners[ov] - RightTriangle< Real >::Corners[(ov+1)%3] ) );
	tensors[1](1,0) = tensors[1](0,1) = ( tensors[1](0,0) + tensors[1](1,1) - Point2D< Real >::Dot( newEdge[1]-newEdge[0] , _g[t] * ( newEdge[1]-newEdge[0] ) ) ) / (Real)2.;

	int old_he[] = { t*3+(v+1)%3 , ot*3+(ov+1)%3 , t*3+(v+2)%3 , ot*3+(ov+2)%3 } , new_he[] = { 3*ot+2 , 3*t+2 , 3*t+1 , 3*ot+1 };
	int edges[4];
	for( int i=0 ; i<4 ; i++ ) edges[i] = _edgeMap.edge( old_he[i] );
	for( int i=0 ; i<4 ; i++ )
	{
		if     ( _edgeMap._e2he[2*edges[i]+0]==old_he[i] ) _edgeMap._e2he[2*edges[i]+0] = new_he[i] , _edgeMap._he2e[ new_he[i] ] =  edges[i]+1;
		else if( _edgeMap._e2he[2*edges[i]+1]==old_he[i] ) _edgeMap._e2he[2*edges[i]+1] = new_he[i] , _edgeMap._he2e[ new_he[i] ] = -edges[i]-1;
		else fprintf( stderr , "[ERROR] FEM::RiemannianMesh::edgeFlip: Unmatched half-edge\n" ) , exit( 0 );
	}
	_edgeMap._e2he[2*edge+0] = 3* t , _edgeMap._he2e[3* t] =  edge+1;
	_edgeMap._e2he[2*edge+1] = 3*ot , _edgeMap._he2e[3*ot] = -edge-1;

	// Set the triangles and metrics
	_triangles[t] = tris[0] , _triangles[ot] = tris[1];
	_g[t] = tensors[0] , _g[ot] = tensors[1];
	return true;
}
template< class Real >
void FEM::RiemannianMesh< Real >::sanityCheck( ConstPointer( CoordinateXForm< Real > ) xForms , Real eps ) const
{
#if 0
	static const Point2D< Real > Corners[] = { Point2D< Real >(0,0) , Point2D< Real >(1,0) , Point2D< Real >(0,1) };
	for( int t=0 ; t<_tCount ; t++ )
	{
		bool success = true;

		// Check that the metric tensor is symmetric
		if( fabs( _g[t](1,0) - _g[t](0,1) )>eps ) fprintf( stderr , "[ERROR] Metric tensor is not symetric: %g != %g\n" , _g[t](1,0) , _g[t](0,1) ) , success = false;

		// The characteristic polynomial of the metric tensor is:
		// P(x) = x^2 - Tr * x + det
		double a = 1. , b = -_g[t].trace() , c = _g[t].determinant();
		double disc = b*b - 4.*a*c;
		if( disc<0 ) fprintf( stderr , "[ERROR] Vanishing discriminant: %g\n" , disc ) , success = false;
		else
		{
			double x[] = { ( -b - sqrt( disc ) ) / (2.*a) , ( -b + sqrt( disc ) ) / (2.*a) };
			if( x[0]<=0 ) fprintf( stderr , "[ERROR] Metric tensor is not positive definite: %g %g\n" , x[0] , x[1] ) , success = false;
		}

		if( !success ) fprintf( stderr , "[ERROR] Failure on triangle: %d\n" , t ) , exit( 0 );
	}
	for( int e=0 ; e<_tCount*3 ; e++ ) if( _edgeMap.opposite( e )!=-1 )
	{
		bool success = true;
		int oe = _edgeMap.opposite( e );
		int t = e/3 , v = e%3 , ot = oe/3 , ov = oe%3;

		// Test the opposite of the opposite is itself
		if( e!=_edgeMap.opposite( oe ) ) fprintf( stderr , "[ERROR] edge is not the opposite of its opposite: %d != %d\n" , e , _edgeMap.opposite( oe ) ) , success = false;

		// Test that the shared vertices are the same (assumes oriented)
		if( _triangles[t][(v+1)%3]!=_triangles[ot][(ov+2)%3] || _triangles[t][(v+2)%3]!=_triangles[ot][(ov+1)%3] )
			fprintf( stderr , "[ERROR] Vertices don't match: %d %d != %d %d\n" , _triangles[t][(v+1)%3] , _triangles[t][(v+2)%3] , _triangles[ot][(ov+2)%3] , _triangles[ot][(ov+1)%3] ) , success = false;

		// Test that transforming across the edge and then back gives the identity
		{
			FEM::CoordinateXForm< Real > xForm = edges[oe] * edges[e];
			Real cError = ( xForm.constant ).squareNorm() , lError = ( xForm.linear - SquareMatrix< Real , 2 >::Identity() ).squareNorm();
			if( cError > eps*eps || lError > eps*eps ) fprintf( stderr , "[ERROR] edge transformations are not inverses: %g %g\n" , cError , lError ) , success = false;
		}

		// Test that the lengths of the shared edge agree
		{
			Real l1 = (Real)sqrt( Point2D< Real >::Dot( Corners[( v+1)%3] - Corners[( v+2)%3] , _g[ t] * ( Corners[( v+1)%3] - Corners[( v+2)%3] ) ) );
			Real l2 = (Real)sqrt( Point2D< Real >::Dot( Corners[(ov+1)%3] - Corners[(ov+2)%3] , _g[ot] * ( Corners[(ov+1)%3] - Corners[(ov+2)%3] ) ) );
			if( fabs( l1-l2 )>eps ) fprintf( stderr , "[ERROR] edge lengths don't agree: %g != %g\n" , l1 , l2 ) , success = false;
		}

		// Test that the edges map into each other
		{
			Point2D< Real > e1 = Corners[(v+1)%3] - Corners[(v+2)%3];
			Point2D< Real > e2 = -edges[oe].linear * ( Corners[(ov+1)%3] - Corners[(ov+2)%3]  );
			if( Point2D< Real >::Dot( e1-e2 , _g[t]*(e1-e2) )>eps*eps ) fprintf( stderr , "[ERROR] edges don't match: %g %g %g\n" , e1[0]-e2[0] , e1[1]-e2[1] , e1[2]-e2[2] ) , success = false;
		}

		// Test that the perpendiculars map into each other
		{
			Point2D< Real > p1 = Rotate90( _g[t] , Corners[(v+1)%3] - Corners[(v+2)%3] );
			Point2D< Real > p2 = -edges[oe].linear * Rotate90( _g[ot] , Corners[(ov+1)%3] - Corners[(ov+2)%3] );
			if( Point2D< Real >::Dot( p1-p2 , _g[t]*(p1-p2) )>eps*eps ) fprintf( stderr , "[ERROR] edge perps don't match: %g %g %g\n" , p1[0]-p2[0] , p1[1]-p2[1] , p1[2]-p2[2] ) , success = false;
		}

		if( !success ) fprintf( stderr , "[ERROR] Failure on edge between triangles: %d %d\n" , t , ot ) , exit( 0 );
	}
#endif
}

template< class Real >
bool FEM::RiemannianMesh< Real >::isVoronoiEdge( int e , Real eps ) const
{
	int he = _edgeMap[e][0] , ohe = _edgeMap[e][1];
	int  t =  he / 3 ,  v =  he % 3;
	int ot = ohe / 3 , ov = ohe % 3;
	if( ohe==-1 ) return true;
	Point2D< Real > center = RightTriangle< Real >::Center( _g[t] , RightTriangle< Real >::CENTER_CIRCUMCENTRIC ) , oVertex = xForm( ohe )( RightTriangle< Real >::Corners[ov] );
	return Point2D< Real >::Dot( center - oVertex , _g[t] * ( center - oVertex ) )+eps > Point2D< Real >::Dot( center - RightTriangle< Real >::Corners[0] , _g[t] * ( center - RightTriangle< Real >::Corners[0] ) );
}

template< class Real >
std::vector< FEM::SamplePoint< Real > > FEM::RiemannianMesh< Real >::randomSamples( unsigned int count ) const
{
	std::vector< FEM::SamplePoint< Real > > samples( count );
	Real* cumAreas = new Real[ tCount() ];
	cumAreas[0] = area(0);
	for( int i=1 ; i<tCount() ; i++ ) cumAreas[i] = cumAreas[i-1] + area(i);

#pragma omp parallel for
	for( int i=0 ; i<(int)count ; i++ )
	{
		Real r1 = Random< Real >() , r2 = Random< Real >() , r3 = Random< Real >();

		// Choose a random triangle
		{
			Real r = r1 * cumAreas[ tCount()-1 ];
			Real *a , *b;
			a = cumAreas - 1;
			b = cumAreas + tCount() - 1;

			int d;
			while( (d=(int)(b-a))>1 )
			{
				Real *i = a + d/2;
				if( r<*i ) b=i;
				else       a=i;
			}
			samples[i].tIdx = (int)( b - cumAreas );
		}
		// Choose a random point in the triangle
		{
			if( r2+r3>1 ) r2 = 1-r2 , r3 = 1-r3;
			samples[i].p = Point2D< Real >( (Real)r2 , (Real) r3 );
		}
	}
	delete[] cumAreas;

	return samples;
}

template< class Real >
FEM::CoordinateXForm< Real > FEM::RiemannianMesh< Real >::getVertexCoordinateXForm( ConstPointer( CoordinateXForm< Real > ) xForms , int t , int v ) const
{
	const int VertexToEdgeMap[] = { 1 , 2 , 0 };
	const int EdgeToVertexMap[] = { 1 , 2 , 0 };
	// Assume that the mesh is oriented
	FEM::CoordinateXForm< Real > xForm;
	int currentT = t , currentV = v;
	do
	{
		int edge = currentT*3 + VertexToEdgeMap[ currentV ] , oEdge = _edgeMap.opposite( edge );
		xForm = xForms[ edge ] * xForm;
		if( oEdge==-1 ) fprintf( stderr , "[ERROR] Boundary vertex\n" ) , exit( 0 );
		currentT = oEdge / 3;
		currentV = EdgeToVertexMap[ oEdge%3 ];
	}
	while( currentT!=t );
	return xForm;
}

template< class Real >
std::vector< int > FEM::RiemannianMesh< Real >::getVertexCorners( int t , int v ) const
{
	// Circulate CCW
	const int VertexToEdgeMap[] = { 1 , 2 , 0 };
	const int EdgeToVertexMap[] = { 1 , 2 , 0 };
	// Assume that the mesh is oriented
	std::vector< int > neighbors;
	int currentT = t , currentV = v;
	do
	{
		int he = currentT*3 + VertexToEdgeMap[ currentV ] , ohe = _edgeMap.opposite( he );
		neighbors.push_back( currentT*3 + currentV );
		if( ohe==-1 ) fprintf( stderr , "[ERROR] Boundary vertex\n" ) , exit( 0 );
		currentT = ohe / 3;
		currentV = EdgeToVertexMap[ ohe%3 ];
	}
	while( currentT!=t );
	return neighbors;
}
template< class Real >
Real FEM::RiemannianMesh< Real >::getVertexConeAngle( int t , int v ) const
{
	const int VertexToEdgeMap[] = { 1 , 2 , 0 };
	const int EdgeToVertexMap[] = { 1 , 2 , 0 };
	// Assume that the mesh is oriented
	Real angle = (Real)0;
	int currentT = t , currentV = v;
	do
	{
		int he = currentT*3 + VertexToEdgeMap[ currentV ] , ohe = _edgeMap.opposite( he );
		angle += RightTriangle< Real >::Angle( _g[currentT] , currentV );
		if( ohe==-1 ) fprintf( stderr , "[ERROR] Boundary vertex\n" ) , exit( 0 );
		currentT = ohe / 3;
		currentV = EdgeToVertexMap[ ohe%3 ];
	}
	while( currentT!=t );
	return angle;
}

template< class Real >
FEM::CoordinateXForm< Real > FEM::RiemannianMesh< Real >::exp( ConstPointer( CoordinateXForm< Real > ) xForms , HermiteSamplePoint< Real >& p , Real eps , bool noWarning ) const
{
	HermiteSamplePoint< Real > startP = p;
	CoordinateXForm< Real > xForm = CoordinateXForm< Real >::Identity();
	if( !Point2D< Real >::SquareNorm( p.v ) ) return xForm;
	const int MAX_ITERS = 10000;
	int count = 0;
	int inEdge = -1;
#if 1
	// If the starting point happens to be on an edge
	{
		int idx = -1;
		if     ( p.p[0]<=0 && p.v[0]<0 ) idx = 1;
		else if( p.p[1]<=0 && p.v[1]<0 ) idx = 2;
		else if( p.p[0]+p.p[1]>=1 && p.v[0]+p.v[1]>0 ) idx = 0;
		if( idx!=-1 )
		{
			int he = p.tIdx * 3 + idx , ohe = _edgeMap.opposite( he );
			const CoordinateXForm< Real >& edge = xForms[he];
			p.tIdx = ohe/3;
			p.p = edge( p.p ) ; p.v = edge.linear * p.v;
			inEdge = ohe%3;
			xForm = edge * xForm;
		}
	}
#endif
	while( count<MAX_ITERS )
	{
#pragma message( "[WARNING] Misha made a fix here" )
		if( !( Point2D< Real >::SquareNorm(p.v)<1e-24 ) ) break;

		// Intersect the ray p + s*v with each of the three edges
		// Bottom edge:   p[1] + s * v[1] = 0                         => s = -p[1]/v[1]
		// Left edge:     p[0] + s * v[0] = 0                         => s = -p[0]/v[0]
		// Diagonal edge: p[1] + s * v[1] = 1 - ( p[0] + s * v[0] )   => s = ( 1 - p[0]  - p[1] ) / ( v[1] + v[0] )
		Real maxS = 0;
		int idx = -1;
		{
			Real s[] = { -p.p[1] / p.v[1]  , -p.p[0] / p.v[0] , ( Real(1.) - p.p[0]  - p.p[1] ) / ( p.v[1] + p.v[0] ) };
			if( inEdge!=2 && s[0]>0 ){ Real foo = p.p[0] + p.v[0] * s[0] ; if( foo>=-eps && foo<=1+eps ) if( s[0]>maxS ) idx = 2 , maxS = s[0]; }
			if( inEdge!=1 && s[1]>0 ){ Real foo = p.p[1] + p.v[1] * s[1] ; if( foo>=-eps && foo<=1+eps ) if( s[1]>maxS ) idx = 1 , maxS = s[1]; }
			if( inEdge!=0 && s[2]>0 ){ Real foo = p.p[0] + p.v[0] * s[2] ; if( foo>=-eps && foo<=1+eps ) if( s[2]>maxS ) idx = 0 , maxS = s[2]; }
		}
		if( idx==-1 )
		{
			fprintf( stderr , "[ERROR] FEM::Mesh::exp:\n" );
			fprintf( stderr , "        Ray does not intersect triangle[%d]: p=(%f %f) v=(%g %g) [%g/%g]\n" , count , p.p[0] , p.p[1] , p.v[0] , p.v[1] , Point2D< Real >::SquareNorm(p.v) , eps*eps );
			fprintf( stderr , "                             Started at[%d]: p=(%f %f) v=(%g %g)\n" , 0 , startP.p[0] , startP.p[1] , startP.v[0] , startP.v[1] );
			exit( 0 );
		}
		if( maxS>1 ) // The end-point is within the triangle
		{
			p.p += p.v , p.v -= p.v;
			return xForm;
		}
		else // The end-point is outside the triangle
		{
			int he = p.tIdx*3 + idx , ohe = _edgeMap.opposite( he );
			const CoordinateXForm< Real >& edge = xForms[he];
			p.p += p.v*maxS ; p.v -= p.v*maxS ; p.tIdx = ohe/3;
			p.p = edge( p.p ) ; p.v = edge.linear * p.v;
			inEdge = ohe%3;
			xForm = edge * xForm;
		}
		count++;
	}
	if( !noWarning ) fprintf( stderr , "[WARNING] Failed to converge exp after %d iterations\n" , MAX_ITERS );
	return xForm;
}

template< class Real >
FEM::CoordinateXForm< Real > FEM::RiemannianMesh< Real >::flow( ConstPointer( CoordinateXForm< Real > ) xForms , const TangentVectorField< Real >& vf , Real flowTime , SamplePoint< Real >& p , Real minStepSize , Real eps , std::vector< SamplePoint< Real > >* path , bool noWarning ) const
{
	CoordinateXForm< Real > xForm = CoordinateXForm< Real >::Identity();
	int MAX_ITERS = 1000000;
	int count = 0;
	int inEdge = -1;
	Real direction = (flowTime<0) ? (Real)-1. : (Real)1.;
	Real stepSizeLeft = minStepSize;
	TangentVector< Real > v = vf( p ) * direction;
	Real vLength = (Real)sqrt( Point2D< Real >::Dot( v , _g[p.tIdx]*v ) );
	flowTime *= direction;
	if( path ) path->push_back( p );
	while( count<MAX_ITERS )
	{
		if( !Point2D< Real >::SquareNorm( v ) ) return xForm;
		// Intersect the ray p + s * v with each of the three edges
		// Bottom edge:   p[1] + s * v[1] = 0                         => s = -p[1]/v[1]
		// Left edge:     p[0] + s * v[0] = 0                         => s = -p[0]/v[0]
		// Diagonal edge: p[1] + s * v[1] = 1 - ( p[0] + s * v[0] )   => s = ( 1 - p[0]  - p[1] ) / ( v[1] + v[0] )
		Real s = 0;
		int idx = -1;
		{
#if 1
			Real _s[] = { -p.p[1] / v[1]  , -p.p[0] / v[0] , ( Real(1.) - p.p[0]  - p.p[1] ) / ( v[1] + v[0] ) };
			s = 1e10;
			if( inEdge!=2 && _s[0]>0 && _s[0]<s ) idx = 2 , s = _s[0];
			if( inEdge!=1 && _s[1]>0 && _s[1]<s ) idx = 1 , s = _s[1];
			if( inEdge!=0 && _s[2]>0 && _s[2]<s ) idx = 0 , s = _s[2];
#else
			if( inEdge!=2 && _s[0]>0 ){ Real foo = p.p[0] + v[0] * _s[0] ; if( foo>=-eps && foo<=1+eps ) if( _s[0]>s ) idx = 2 , s = _s[0]; }
			if( inEdge!=1 && _s[1]>0 ){ Real foo = p.p[1] + v[1] * _s[1] ; if( foo>=-eps && foo<=1+eps ) if( _s[1]>s ) idx = 1 , s = _s[1]; }
			if( inEdge!=0 && _s[2]>0 ){ Real foo = p.p[0] + v[0] * _s[2] ; if( foo>=-eps && foo<=1+eps ) if( _s[2]>s ) idx = 0 , s = _s[2]; }
#endif
		}
#if 0
		if( idx==-1 )
		{
			fprintf( stderr , "[ERROR] Ray does not intersect triangle[%d]: (%f %f) (%g %g) [%g/%g]\n" , count , p.p[0] , p.p[1] , v[0] , v[1] , Point2D< Real >::SquareNorm(v) , eps*eps );
			Real s[] = { -p.p[1] / v[1]  , -p.p[0] / v[0] , ( Real(1.) - p.p[0]  - p.p[1] ) / ( v[1] + v[0] ) };
			if( inEdge!=2 ) { Real foo = p.p[0] + v[0] * s[0] ; printf( "\t0] %g -> %f\n" , s[0] , foo ); }
			if( inEdge!=1 ) { Real foo = p.p[1] + v[1] * s[1] ; printf( "\t1] %g -> %f\n" , s[1] , foo ); }
			if( inEdge!=0 ) { Real foo = p.p[0] + v[0] * s[2] ; printf( "\t2] %g -> %f\n" , s[2] , foo ); }
			exit( 0 );
		}
#else
		if( idx==-1 ) return xForm;
#endif
		CotangentVector< Real > gv = _g[p.tIdx] * v;
		Real stepSize = vLength * s;
		bool updateVector = false;
		if( minStepSize>0 && stepSize>stepSizeLeft )
		{
			s = stepSizeLeft / vLength;
			updateVector = true;
		}

		// If we can finish the flow
		if( flowTime<s )
		{
			p.p += v * flowTime;
			if( path ) path->push_back( p );
			return xForm;
		}
		else
		{
			p.p += v * s , flowTime -= s;

			// If we do not cross a boundary, change direction
			if( updateVector )
			{
				// If the the vectors are oppositely oriented, terminate the flow
				if( Point2D< Real >::Dot( gv , vf( p ) )*direction < 0 ) return xForm;

				v = vf( p ) * direction;
				vLength = (Real)sqrt( Point2D< Real >::Dot( v , _g[p.tIdx]*v ) );
				stepSizeLeft = minStepSize;
				inEdge = -1;
			}
			// If we cross the boundary, transport the direction of the flow
			else // The end-point is outside the triangle
			{
				// Advance along the flow until you hit the edge	
				int he = p.tIdx*3 + idx , ohe = _edgeMap.opposite( he );
				const CoordinateXForm< Real >& edge = xForms[he];
				// Switch into the next triangle
				p.tIdx = ohe/3;
				p.p = edge( p.p );
				v = edge.linear * v;

				// Mark the edge we came in on
				inEdge = ohe%3;

				// Accumulate the transformations
				xForm = edge * xForm;

				stepSizeLeft -= stepSize;
			}

			if( path ) path->push_back( p );
		}
		count++;
	}
	if( !noWarning ) fprintf( stderr , "[WARNING] Failed to converge flow after %d iterations\n" , MAX_ITERS );
	return xForm;
#undef NEW_CODE
}

/////////////////////////
// FEM::RiemannianMesh //
/////////////////////////
template< class Real >
inline void FEM::RiemannianMesh< Real >::makeArea( Real area )
{
	double scale = 0;
#pragma omp parallel for reduction( + : scale )
	for( int i=0 ; i<_tCount ; i++ ) scale += sqrt( _g[i].determinant() );
	scale = 2. / scale;
	scale *= area;
#pragma omp parallel for
	for( int i=0 ; i<_tCount ; i++ ) _g[i] *= (Real)scale;
}
template< class Real >
inline Real FEM::RiemannianMesh< Real >::area( void ) const
{
	Real area = 0;
#pragma omp parallel for reduction( + : area )
	for( int i=0 ; i<_tCount ; i++ ) area += (Real)sqrt( _g[i].determinant() );
	return area / (Real)2.;
}
template< class Real >
inline Real FEM::RiemannianMesh< Real >::area( int idx ) const { return (Real)sqrt( _g[idx].determinant() ) / (Real)2.; }

template< class Real >
inline Real FEM::RiemannianMesh< Real >::squareEdgeLength( int heIdx ) const { return SquareLength( _g[heIdx/3] , RightTriangle< Real >::EdgeDirections[heIdx%3] ); }

template< class Real >
template< class Vertex >
void FEM::RiemannianMesh< Real >::setMetricFromEmbedding( ConstPointer( Vertex ) vertices , bool warnSingular )
{

#pragma omp parallel for
	for( int i=0 ; i<_tCount ; i++ )
	{
		Point3D< Real > e[] = { Point3D< Real >( vertices[ _triangles[i][1] ] ) - Point3D< Real >( vertices[ _triangles[i][0] ] ) , Point3D< Real >( vertices[ _triangles[i][2] ] ) - Point3D< Real >( vertices[ _triangles[i][0] ] ) };
		for( int j=0 ; j<2 ; j++ ) for( int k=0 ; k<2 ; k++ ) _g[i](j,k) = Point3D< Real >::Dot( e[j] , e[k] );
		_g[i](0,1) = _g[i](1,0) = (Real)( ( _g[i](0,1) + _g[i](1,0) )/2 );

		if( warnSingular && !_g[i].determinant() )
		{
			fprintf( stderr , "[WARNING] Vanishing metric tensor determinant\n" );
			printf( "%g %g %g\t%g %g %g\n" , e[0][0] , e[0][1] , e[0][2] , e[1][0] , e[1][1] , e[1][2] );
		}
	}
}
template< class Real >
void FEM::RiemannianMesh< Real >::setMetricFromEdgeLengths( ConstPointer( Real ) edgeLengths )
{
#pragma omp parallel for
	for( int i=0 ; i<_tCount ; i++ )
	{
		// The conditions the matrix needs to satisfy are:
		// -- < g[i] * (1,0) , (1,0) > = edgeLengths[i*3+2]^2
		//		=> g[i](0,0) = edgeLengths[i*3+2]^2
		// -- < g[i] * (0,1) , (0,1) > = edgeLengths[i*3+1]^2
		//		=> g[i](1,1) = edgeLengths[i*3+1]^2
		// -- < g[i] * (-1,1) , (-1,1) > = edgeLengths[i*3+0]^2
		//		=> g[i](0,0) + g[i](1,1) - g[i](0,1) - g[i](1,0) = edgeLengths[i*3+0]^2
		//		=> - g[i](0,1) - g[i](1,0) = edgeLengths[i*3+0]^2 - edgeLengths[i*3+2]^2 - edgeLengths[i*3+1]^2
		//		=>  g[i](0,1) = g[i](1,0) = ( edgeLengths[i*3+2]^2 + edgeLengths[i*3+1]^2 - edgeLengths[i*3+0]^2 ) / 2

		_g[i](0,0) = edgeLengths[i*3+2] * edgeLengths[i*3+2];
		_g[i](1,1) = edgeLengths[i*3+1] * edgeLengths[i*3+1];
		_g[i](0,1) = _g[i](1,0) = ( _g[i](0,0) + _g[i](1,1) - edgeLengths[i*3] * edgeLengths[i*3] ) / (Real)2.;
	}
}
template< class Real >
void FEM::RiemannianMesh< Real >::setMetricFromSquareEdgeLengths( ConstPointer( Real ) squareEdgeLengths )
{
#pragma omp parallel for
	for( int i=0 ; i<tCount ; i++ )
	{
		_g[i](0,0) = squareEdgeLengths[i*3+2];
		_g[i](1,1) = squareEdgeLengths[i*3+1];
		_g[i](0,1) = _g[i](1,0) = ( _g[i](0,0) + _g[i](1,1) - squareEdgeLengths[i*3] ) / (Real)2.;
	}
}
template< class Real >
template< unsigned int BasisType , class V >
V FEM::RiemannianMesh< Real >::evaluateScalarField( ConstPointer( V ) coefficients , const SamplePoint< Real >& p ) const
{
	V v;
	TestBasisType( BasisType , "FEM::RiemannianMesh::evaluateScalarField" , false );
	if( BasisInfo< BasisType >::Dimension==0 )
	{
		V _coefficients[ BasisInfo< BasisType >::Coefficients ];
		for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ )
		{
			bool isAligned;
			int ii = index< BasisType >( p.tIdx , i , isAligned );
			_coefficients[i] = isAligned ? coefficients[ii] : -coefficients[ii];
		}
		v = RightTriangle< Real >::template EvaluateScalarField< BasisType >( ( ConstPointer( V ) )GetPointer( _coefficients , BasisInfo< BasisType >::Coefficients ) , p.p );
	}
	else TestBasisType( BasisType , "FEM::RiemannianMesh::evaluateScalarField" , false );
	return v;
}
template< class Real >
template< unsigned int BasisType , class V >
FEM::CotangentVector< V > FEM::RiemannianMesh< Real >::evaluateScalarFieldGradient( ConstPointer( V ) coefficients , const SamplePoint< Real >& p ) const
{
	CotangentVector< V > v;
	TestBasisType( BasisType , "FEM::RiemannianMesh::evaluateScalarField" , false );
	if( BasisInfo< BasisType >::Dimension==0 )
	{
		Real _coefficients[ BasisInfo< BasisType >::Coefficients ];
		for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ )
		{
			bool isAligned;
			int ii = index< BasisType >( p.tIdx , i , isAligned );
			_coefficients[i] = isAligned ? coefficients[ii] : -coefficients[ii];
		}
		v = RightTriangle< Real >::template EvaluateScalarFieldGradient< BasisType >( GetPointer( _coefficients , BasisInfo< BasisType >::Coefficients ) , p.p );
	}
	else TestBasisType( BasisType , "FEM::RiemannianMesh::evaluateScalarFieldGradient" , false );
	return v;
}
template< class Real >
template< unsigned int BasisType , class V >
FEM::CotangentVector< V > FEM::RiemannianMesh< Real >::evaluateCovectorField( ConstPointer( V ) coefficients , const SamplePoint< Real >& p ) const
{
	CotangentVector< V > v;
	TestBasisType( BasisType , "FEM::RiemannianMesh::evaluateCovectorField" , false );
	if( BasisInfo< BasisType >::Dimension==1 )
	{
		Real _coefficients[ BasisInfo< BasisType >::Coefficients ];
		for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ )
		{
			bool isAligned;
			int ii = index< BasisType >( p.tIdx , i , isAligned );
			_coefficients[i] = isAligned ? coefficients[ii] : -coefficients[ii];
		}
		v = RightTriangle< Real >::template EvaluateCovectorField< BasisType >( GetPointer( _coefficients , BasisInfo< BasisType >::Coefficients ) , p.p );
	}
	else TestBasisType( BasisType , "FEM::RiemannianMesh::evaluateScalarField" , false );
	return v;
}
template< class Real >
template< unsigned int BasisType >
SparseMatrix< Real , int > FEM::RiemannianMesh< Real >::massMatrix( bool lump , ConstPointer( SquareMatrix< Real , 2 > ) newTensors ) const
{
	SparseMatrix< Real , int > M;
	if( BasisType==BASIS_2_VERTEX_CONSTANT )
	{
		M.resize( _vCount );
#pragma omp parallel for
		for( int i=0 ; i<_vCount ; i++ ) M.SetRowSize( i , 1 ) , M[i][0] = MatrixEntry< Real , int >( i , 0 );
#pragma omp parallel for
		for( int i=0 ; i<_tCount ; i++ )
		{
			Real a = area( i ) / (Real)3.;
			if( newTensors ) a *= newTensors[i].determinant() / _g[i].determinant() / _g[i].determinant();
			for( int j=0 ; j<3 ; j++ )
#pragma omp atomic
				M[ _triangles[i][j] ][0].Value += a;
		}
#pragma omp parallel for
		for( int i=0 ; i<M.rows ; i++ ) M[i][0].Value = (Real)( 1. / M[i][0].Value );
		return M;
	}
	auto mask = RightTriangle< Real >::template GetMassMask< BasisType >( newTensors!=( ConstPointer( SquareMatrix< Real , 2 > ) )NullPointer< SquareMatrix< Real , 2 > >() );
	Point< int , BasisInfo< BasisType >::Coefficients > nonZeroCount;
	for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ ) for( int j=0 ; j<BasisInfo< BasisType >::Coefficients ; j++ ) if( mask(i,j) ) nonZeroCount[i]++;

	M.resize( dimension< BasisType >() );
	Pointer( std::atomic< int > ) rowSizes = NewPointer< std::atomic< int > >( M.rows ); // need to support atomic increment and set, which is not supported with OpenMP
#pragma omp parallel for
	for( int i=0 ; i<M.rows ; i++ ) rowSizes[i] = 0;

	// First, set the row sizes
#pragma omp parallel for
	for( int t=0 ; t<_tCount ; t++ )
		if( lump ) for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ ) rowSizes[ index< BasisType >( t , i ) ]++;
		else for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ ) rowSizes[ index< BasisType >( t , i ) ] += nonZeroCount[i];
#pragma omp parallel for
	for( int i=0 ; i<M.rows ; i++ ) M.SetRowSize( i , rowSizes[i] ) , rowSizes[i] = 0;

	// Next, set the entries
#pragma omp parallel for
	for( int t=0 ; t<_tCount ; t++ )
	{
		if( lump )
		{
			auto m = RightTriangle< Real >::template GetDiagonalMassMatrix< BasisType >( _g[t] );
			for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ )
			{
				int ii = index< BasisType >( t , i );
				M[ ii ][ rowSizes[ii]++ ] = MatrixEntry< Real , int >( ii , m[i] );
			}
		}
		else
		{
			auto m = newTensors ? RightTriangle< Real >::template GetMassMatrix< BasisType >( _g[t] , newTensors[t] ) : RightTriangle< Real >::template GetMassMatrix< BasisType >( _g[t] );
			for( int i=0 ; i<BasisInfo< BasisType >::Coefficients ; i++ ) 
			{
				bool iAligned;
				int ii = index< BasisType >( t , i , iAligned );

				int idx = 0;
				for( int j=0 ; j<BasisInfo< BasisType >::Coefficients ; j++ ) if( mask(i,j) )
				{
					bool jAligned;
					int jj = index< BasisType >( t , j , jAligned );
					M[ ii ][ rowSizes[ii]++ ] = MatrixEntry< Real , int >( jj , iAligned==jAligned ? m(i,j) : - m(i,j) );
				}
			}
		}
	}
	DeletePointer( rowSizes );

	// Collapse the duplicate entries (and sort)
#pragma omp parallel for
	for( int i=0 ; i<M.rows ; i++ )
	{
		std::sort( M[i] , M[i] + M.rowSizes[i] , []( MatrixEntry< Real , int > e1 , MatrixEntry< Real , int > e2 ){ return e1.N<e2.N; } );
		int idx = 0;
		for( int j=1 ; j<M.rowSizes[i] ; j++ )
			if( M[i][j].N==M[i][idx].N ) M[i][idx].Value += M[i][j].Value;
			else M[i][++idx] = M[i][j];
		M.ResetRowSize( i , idx+1 );
		for( int j=0 ; j<M.rowSizes[i] ; j++ ) if( M[i][j].N==i ) std::swap( M[i][j] , M[i][0] );
	}
	return M;
}

template< class Real >
template< unsigned int InBasisType , unsigned int OutBasisType >
SparseMatrix< Real , int > FEM::RiemannianMesh< Real >::dMatrix( void ) const
{
	TestBasisType(  InBasisType , "FEM::RiemannianMesh::dMatrix" , false );
	TestBasisType( OutBasisType , "FEM::RiemannianMesh::dMatrix" , false );

	auto Fail = [&] ( void ){ fprintf( stderr , "[ERROR] FEM::RiemannianMesh::dMatrix: %s -> %s\n" , BasisNames[ InBasisType ] , BasisNames[ OutBasisType ] ) , exit( 0 ); };

	bool redundant;
	Matrix< unsigned char , BasisInfo< InBasisType >::Coefficients , BasisInfo< OutBasisType >::Coefficients > mask = RightTriangle< Real >::template GetDMask< InBasisType , OutBasisType >( redundant );
	Point< int , BasisInfo< OutBasisType >::Coefficients > nonZeroCount;
	for( int i=0 ; i<BasisInfo< InBasisType >::Coefficients ; i++ ) for( int j=0 ; j<BasisInfo< OutBasisType >::Coefficients ; j++ ) if( mask(i,j) ) nonZeroCount[j]++;

	SparseMatrix< Real , int > D;
	D.resize( dimension< OutBasisType >() );
	Pointer( std::atomic< int > ) rowSizes = NewPointer< std::atomic< int > >( D.rows ); // need to support atomic increment and set, which is not supported with OpenMP
#pragma omp parallel for
	for( int i=0 ; i<D.rows ; i++ ) rowSizes[i] = 0;

	// First, set the row sizes
#pragma omp parallel for
	for( int t=0 ; t<_tCount ; t++ ) for( int j=0 ; j<BasisInfo< OutBasisType >::Coefficients ; j++ ) rowSizes[ index< OutBasisType >( t , j ) ] += nonZeroCount[j];
#pragma omp parallel for
	for( int i=0 ; i<D.rows ; i++ ) D.SetRowSize( i , rowSizes[i] ) , rowSizes[i] = 0;

	// Next, set the entries
#pragma omp parallel for
	for( int t=0 ; t<_tCount ; t++ )
	{
		auto d = RightTriangle< Real >::template GetDMatrix< InBasisType , OutBasisType >( _g[t] );
		for( int j=0 ; j<BasisInfo< OutBasisType >::Coefficients ; j++ ) 
		{
			bool jAligned ; int jj = index< OutBasisType >( t , j , jAligned );
			for( int i=0 ; i<BasisInfo< InBasisType >::Coefficients ; i++ ) if( mask(i,j) )
			{
				bool iAligned ; int ii = index< InBasisType >( t , i , iAligned );
				D[ jj ][ rowSizes[jj]++ ] = MatrixEntry< Real , int >( ii , iAligned==jAligned ? d(i,j) : -d(i,j) );
			}
		}
	}
	DeletePointer( rowSizes );

	// Collapse the duplicate entries
#pragma omp parallel for
	for( int i=0 ; i<D.rows ; i++ )
	{
		std::sort( D[i] , D[i] + D.rowSizes[i] , []( MatrixEntry< Real , int > e1 , MatrixEntry< Real , int > e2 ){ return e1.N<e2.N; } );
		int idx = 0;
		if( D.rowSizes[i] )
		{
			for( int j=1 ; j<D.rowSizes[i] ; j++ )
				if( D[i][j].N==D[i][idx].N ) D[i][idx].Value += D[i][j].Value;
				else D[i][++idx] = D[i][j];
			D.ResetRowSize( i , idx+1 );
		}
	}

	if( redundant )
	{
		std::vector< int > valence;
		if( BasisInfo< OutBasisType >::ElementType==ELEMENT_VERTEX )
		{
			valence.resize( _vCount , 0 );
#pragma omp parallel for
			for( int t=0 ; t<_tCount ; t++ ) for( int v=0 ; v<3 ; v++ )
#pragma omp atomic
				valence[ _triangles[t][v] ]++;
		}
		else if( BasisInfo< OutBasisType >::ElementType==ELEMENT_EDGE )
		{
			valence.resize( _edgeMap.size() , 0 );
#pragma omp parallel for
			for( int t=0 ; t<_tCount ; t++ ) for( int e=0 ; e<3 ; e++ )
#pragma omp atomic
				valence[ _edgeMap.edge( t*3+e ) ]++;
		}
		else TestElementType( BasisInfo< OutBasisType >::ElementType , "FEM::RiemannianMesh::dMatrix" , true );
#pragma omp parallel for
		for( int i=0 ; i<D.rows ; i++ )
		{
			Real scale = (Real)( 1. / valence[i/BasisInfo< OutBasisType >::CoefficientsPerElement] );
			for( int j=0 ; j<D.rowSizes[i] ; j++ ) D[i][j].Value *= scale;
		}
	}
	return D;
}
template< class Real >
template< unsigned int BasisType , unsigned int PreBasisType , unsigned int PostBasisType >
SparseMatrix< Real , int > FEM::RiemannianMesh< Real >::stiffnessMatrix( ConstPointer( SquareMatrix< Real , 2 > ) newTensors ) const
{
	auto MassMatrixInverse = [] ( const SparseMatrix< Real , int >& M )
	{
		SparseMatrix< Real , int > M_inverse;
		M_inverse.resize( M.rows );
#pragma omp parallel for
		for( int i=0 ; i<M.rows ; i++ )
		{
			M_inverse.SetRowSize( i , 1 );
			Real sum = 0;
			for( int j=0 ; j<M.rowSizes[i] ; j++ ) sum += M[i][j].Value;
			M_inverse[i][0] = MatrixEntry< Real , int >( i , (Real)(1./sum) );
		}
		return M_inverse;
	};

	SparseMatrix< Real , int > S;
	TestBasisType( BasisType , "FEM::RiemannianMesh::stiffnessMatrix" , false );
	switch( BasisInfo< BasisType >::Dimension )
	{
		case 0:
		{	
			TestBasisType( PostBasisType , "FEM::RiemannianMesh::stiffnessMatrix" , false );
			if( BasisInfo< PostBasisType >::Dimension!=BasisInfo< BasisType >::Dimension+1 ) fprintf( stderr , "[ERROR] FEM::RiemannianMesh::stiffnessMatrix: Incompatible basis type: %s -> %s\n" , BasisNames[BasisType] , BasisNames[PostBasisType] ) , exit( 0 );
			SparseMatrix< Real , int > M2 = massMatrix< PostBasisType >( BasisInfo< BasisType >::Lumpable , newTensors );
			SparseMatrix< Real , int > D1 = dMatrix< BasisType , PostBasisType >( );
			S = D1.transpose() * M2 * D1;
			break;
		}
		case 1:
		{
			TestBasisType(  PreBasisType , "FEM::RiemannianMesh::stiffnessMatrix" , false );
			TestBasisType( PostBasisType , "FEM::RiemannianMesh::stiffnessMatrix" , false );
			if( BasisInfo< PreBasisType >::Dimension!=BasisInfo< BasisType >::Dimension-1 ) fprintf( stderr , "[ERROR] FEM::RiemannianMesh::stiffnessMatrix: Incompatible basis type: %s -> %s\n" , BasisNames[PreBasisType] , BasisNames[BasisType] ) , exit( 0 );
			if( BasisInfo< PostBasisType >::Dimension!=BasisInfo< BasisType >::Dimension+1 ) fprintf( stderr , "[ERROR] FEM::RiemannianMesh::stiffnessMatrix: Incompatible basis type: %s -> %s\n" , BasisNames[BasisType] , BasisNames[PostBasisType] ) , exit( 0 );
			SparseMatrix< Real , int > M0 = massMatrix<  PreBasisType >( true , newTensors );
			SparseMatrix< Real , int > M1 = massMatrix<     BasisType >( BasisInfo< BasisType >::Lumpable , newTensors );
			SparseMatrix< Real , int > M2 = massMatrix< PostBasisType >( BasisInfo< BasisType >::Lumpable , newTensors );
			SparseMatrix< Real , int > D0 = dMatrix< PreBasisType ,     BasisType >( );
			SparseMatrix< Real , int > D1 = dMatrix<    BasisType , PostBasisType >( );
			S = M1 * D0 * MassMatrixInverse( M0 ) * D0.transpose() * M1 + D1.transpose() * M2 * D1;
			break;
		}
		case 2:
		{
			TestBasisType(  PreBasisType , "FEM::RiemannianMesh::stiffnessMatrix" , false );
			if( BasisInfo< PreBasisType >::Dimension!=BasisInfo< BasisType >::Dimension-1 ) fprintf( stderr , "[ERROR] FEM::RiemannianMesh::stiffnessMatrix: Incompatible basis type: %s -> %s\n" , BasisNames[PreBasisType] , BasisNames[BasisType] ) , exit( 0 );
			SparseMatrix< Real , int > M0 = massMatrix< PreBasisType >( true , newTensors );
			SparseMatrix< Real , int > M1 = massMatrix<    BasisType >( BasisInfo< BasisType >::Lumpable , newTensors );
			SparseMatrix< Real , int > D0 = dMatrix< PreBasisType , BasisType >( );
			S = M1 * D0 * MassMatrixInverse( M0 ) * D0.transpose() * M1;
			break;
		}
		default: TestBasisType( BasisType , "FEM::RiemannianMesh::stiffnessMatrix" , true );
	}
#pragma omp parallel for
	for( int i=0 ; i<S.rows ; i++ )
	{
		std::sort( S[i] , S[i] + S.rowSizes[i] , []( MatrixEntry< Real , int > e1 , MatrixEntry< Real , int > e2 ){ return e1.N<e2.N; } );
		int idx=0;
		for( int j=1 ; j<S.rowSizes[i] ; j++ )
			if( S[i][j].N==S[i][idx].N ) S[i][idx].Value += S[i][j].Value;
			else idx++ , S[i][idx] = S[i][j];
		S.ResetRowSize( i , idx+1 );
		for( int j=0 ; j<S.rowSizes[i] ; j++ ) if( S[i][j].N==i ) std::swap( S[i][j] , S[i][0] );
	}
	return S;
}
template< class Real >
template< unsigned int BasisType >
SparseMatrix< Real , int > FEM::RiemannianMesh< Real >::stiffnessMatrix( void ) const
{
	TestBasisType( BasisType , "FEM::RiemannianMesh::stiffnessMatrix" , false );
	SparseMatrix< Real , int > S;
	switch( BasisInfo< BasisType >::Dimension )
	{
	case 0: S = stiffnessMatrix< BasisType , BasisType       , BASIS_1_WHITNEY >( ) ; break;
	case 1:
		if( BasisType==BASIS_1_WHITNEY ) S = stiffnessMatrix< BasisType , BASIS_0_WHITNEY , BASIS_2_WHITNEY >( );
		else                             S = stiffnessMatrix< BasisType , BASIS_0_WHITNEY , BASIS_2_VERTEX_CONSTANT >( );
		break;
	case 2: S = stiffnessMatrix< BasisType , BASIS_1_WHITNEY , BasisType       >( ) ; break;
	default: TestBasisType( BasisType , "FEM::RiemannianMesh::stiffnessMatrix" , true );
	}
	return S;
}

template< class Real >
inline Real FEM::RiemannianMesh< Real >::getIntegral( ConstPointer( Real ) coefficients ) const
{
	Real integral = (Real)0;
#pragma omp parallel for reduction( + : integral )
	for( int i=0 ; i<_tCount ; i++ )
	{
		SquareMatrix< Real , 3 > mass = FEM::RightTriangle< Real >::template GetMassMatrix< FEM::BASIS_0_WHITNEY >( _g[i] );
		for( int j=0 ; j<3 ; j++ )
		{
			Real sum = (Real)0;
			for( int k=0 ; k<3 ; k++ ) sum += mass(j,k);
			integral += coefficients[ _triangles[i][j] ] * sum;
		}
	}
	return integral;
}
template< class Real >
inline Real FEM::RiemannianMesh< Real >::getDotProduct( ConstPointer( Real ) coefficients1 , ConstPointer( Real ) coefficients2 , bool lump ) const
{
	Real dotProduct = (Real)0;
#pragma omp parallel for reduction( + : dotProduct )
	for( int i=0 ; i<_tCount ; i++ )
	{
		if( lump )
		{
			Point< Real , 3 > mass = RightTriangle< Real >::GetDiagonalMassMatrix( _g[i] );
			for( int j=0 ; j<3 ; j++ ) dotProduct += mass[j] * coefficients1[ triangles[i][j] ] * coefficients2[ triangles[i][j] ];
		}
		else
		{
			SquareMatrix< Real , 3 > mass = RightTriangle< Real >::GetMassMatrix( _g[i] );
			for( int j=0 ; j<3 ; j++ ) for( int k=0 ; k<3 ; k++ ) dotProduct += mass(j,k) * coefficients1[ triangles[i][j] ] * coefficients2[ triangles[i][k] ];
		}
	}
	return dotProduct;
}

FEM::EdgeMap::EdgeMap( ConstPointer( TriangleIndex ) triangles , size_t tCount )
{
	_eCount = 0;
	std::unordered_map< long long , int > edgeMap;
	for( int t=0 ; t<tCount ; t++ ) for( int v=0 ; v<3 ; v++ )
	{
		long long key = EdgeKey( triangles[t][(v+1)%3] , triangles[t][(v+2)%3] );
		if( edgeMap.find(key)==edgeMap.end() ) edgeMap[key] = (int)_eCount++;
	}
	_he2e = AllocPointer< int >( tCount*3 );
	_e2he = AllocPointer< int >( 2*_eCount );
	memset( _e2he , -1 , sizeof(int) * 2 * _eCount );

	for( int t=0 ; t<tCount ; t++ ) for( int v=0 ; v<3 ; v++ )
	{
		long long key = EdgeKey( triangles[t][(v+1)%3] , triangles[t][(v+2)%3] );
		int heIdx = t*3+v , eIdx = edgeMap[key];
		if( _e2he[2*eIdx]==-1 ) _e2he[eIdx*2+0] = heIdx , _he2e[heIdx] =  (eIdx+1);
		else                    _e2he[eIdx*2+1] = heIdx , _he2e[heIdx] = -(eIdx+1);
	}
}

