vcglib/vcg/complex/algorithms/inertia.h

/****************************************************************************
* VCGLib                                                            o o     *
* Visual and Computer Graphics Library                            o     o   *
*                                                                _   O  _   *
* Copyright(C) 2005                                                \/)\/    *
* Visual Computing Lab                                            /\/|      *
* ISTI - Italian National Research Council                           |      *
*                                                                    \      *
* All rights reserved.                                                      *
*                                                                           *
* This program is free software; you can redistribute it and/or modify      *
* it under the terms of the GNU General Public License as published by      *
* the Free Software Foundation; either version 2 of the License, or         *
* (at your option) any later version.                                       *
*                                                                           *
* This program is distributed in the hope that it will be useful,           *
* but WITHOUT ANY WARRANTY; without even the implied warranty of            *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the             *
* GNU General Public License (http://www.gnu.org/licenses/gpl.txt)          *
* for more details.                                                         *
*                                                                           *
****************************************************************************/
#ifndef _VCG_INERTIA_
#define _VCG_INERTIA_

/*
The algorithm is based on a three step reduction of the volume integrals
to successively simpler integrals. The algorithm is designed to minimize
the numerical errors that can result from poorly conditioned alignment of
polyhedral faces. It is also designed for efficiency. All required volume
integrals of a polyhedron are computed together during a single walk over
the boundary of the polyhedron; exploiting common subexpressions reduces
floating point operations.

For more information, check out:

Brian Mirtich,
``Fast and Accurate Computation of Polyhedral Mass Properties,''
journal of graphics tools, volume 1, number 2, 1996

*/

#include <eigenlib/Eigen/Core>
#include <eigenlib/Eigen/Eigenvalues>
#include <vcg/complex/algorithms/update/normal.h>

namespace vcg
{
  namespace tri
  {
template <class InertiaMeshType>
class Inertia
{
  typedef InertiaMeshType MeshType;
	typedef typename MeshType::VertexType     VertexType;
	typedef typename MeshType::VertexPointer  VertexPointer;
	typedef typename MeshType::VertexIterator VertexIterator;
	typedef	typename MeshType::ScalarType			ScalarType;
	typedef typename MeshType::FaceType       FaceType;
	typedef typename MeshType::FacePointer    FacePointer;
	typedef typename MeshType::FaceIterator   FaceIterator;
	typedef typename MeshType::ConstFaceIterator   ConstFaceIterator;
	typedef typename MeshType::FaceContainer  FaceContainer;
	typedef typename MeshType::CoordType  CoordType;

private :
	enum {X=0,Y=1,Z=2};
	inline ScalarType SQR(ScalarType &x) const { return x*x;}
	inline ScalarType CUBE(ScalarType &x) const { return x*x*x;}

 int A;   /* alpha */
 int B;   /* beta */
 int C;   /* gamma */

/* projection integrals */
 double P1, Pa, Pb, Paa, Pab, Pbb, Paaa, Paab, Pabb, Pbbb;

/* face integrals */
 double Fa, Fb, Fc, Faa, Fbb, Fcc, Faaa, Fbbb, Fccc, Faab, Fbbc, Fcca;

/* volume integrals */
 double T0, T1[3], T2[3], TP[3];

public:

/* compute various integrations over projection of face */
 void compProjectionIntegrals(FaceType &f)
{
  double a0, a1, da;
  double b0, b1, db;
  double a0_2, a0_3, a0_4, b0_2, b0_3, b0_4;
  double a1_2, a1_3, b1_2, b1_3;
  double C1, Ca, Caa, Caaa, Cb, Cbb, Cbbb;
  double Cab, Kab, Caab, Kaab, Cabb, Kabb;
  int i;

  P1 = Pa = Pb = Paa = Pab = Pbb = Paaa = Paab = Pabb = Pbbb = 0.0;

  for (i = 0; i < 3; i++) {
    a0 = f.V(i)->P()[A];
    b0 = f.V(i)->P()[B];
    a1 = f.V1(i)->P()[A];
    b1 = f.V1(i)->P()[B];
    da = a1 - a0;
    db = b1 - b0;
    a0_2 = a0 * a0; a0_3 = a0_2 * a0; a0_4 = a0_3 * a0;
    b0_2 = b0 * b0; b0_3 = b0_2 * b0; b0_4 = b0_3 * b0;
    a1_2 = a1 * a1; a1_3 = a1_2 * a1;
    b1_2 = b1 * b1; b1_3 = b1_2 * b1;

    C1 = a1 + a0;
    Ca = a1*C1 + a0_2; Caa = a1*Ca + a0_3; Caaa = a1*Caa + a0_4;
    Cb = b1*(b1 + b0) + b0_2; Cbb = b1*Cb + b0_3; Cbbb = b1*Cbb + b0_4;
    Cab = 3*a1_2 + 2*a1*a0 + a0_2; Kab = a1_2 + 2*a1*a0 + 3*a0_2;
    Caab = a0*Cab + 4*a1_3; Kaab = a1*Kab + 4*a0_3;
    Cabb = 4*b1_3 + 3*b1_2*b0 + 2*b1*b0_2 + b0_3;
    Kabb = b1_3 + 2*b1_2*b0 + 3*b1*b0_2 + 4*b0_3;

    P1 += db*C1;
    Pa += db*Ca;
    Paa += db*Caa;
    Paaa += db*Caaa;
    Pb += da*Cb;
    Pbb += da*Cbb;
    Pbbb += da*Cbbb;
    Pab += db*(b1*Cab + b0*Kab);
    Paab += db*(b1*Caab + b0*Kaab);
    Pabb += da*(a1*Cabb + a0*Kabb);
  }

  P1 /= 2.0;
  Pa /= 6.0;
  Paa /= 12.0;
  Paaa /= 20.0;
  Pb /= -6.0;
  Pbb /= -12.0;
  Pbbb /= -20.0;
  Pab /= 24.0;
  Paab /= 60.0;
  Pabb /= -60.0;
}


void CompFaceIntegrals(FaceType &f)
{
	Point3<ScalarType>  n;
	ScalarType w;
  double k1, k2, k3, k4;

  compProjectionIntegrals(f);

  n = f.N();
  w = -f.V(0)->P()*n;
  k1 = 1 / n[C]; k2 = k1 * k1; k3 = k2 * k1; k4 = k3 * k1;

  Fa = k1 * Pa;
  Fb = k1 * Pb;
  Fc = -k2 * (n[A]*Pa + n[B]*Pb + w*P1);

  Faa = k1 * Paa;
  Fbb = k1 * Pbb;
  Fcc = k3 * (SQR(n[A])*Paa + 2*n[A]*n[B]*Pab + SQR(n[B])*Pbb
	 + w*(2*(n[A]*Pa + n[B]*Pb) + w*P1));

  Faaa = k1 * Paaa;
  Fbbb = k1 * Pbbb;
  Fccc = -k4 * (CUBE(n[A])*Paaa + 3*SQR(n[A])*n[B]*Paab
	   + 3*n[A]*SQR(n[B])*Pabb + CUBE(n[B])*Pbbb
	   + 3*w*(SQR(n[A])*Paa + 2*n[A]*n[B]*Pab + SQR(n[B])*Pbb)
	   + w*w*(3*(n[A]*Pa + n[B]*Pb) + w*P1));

  Faab = k1 * Paab;
  Fbbc = -k2 * (n[A]*Pabb + n[B]*Pbbb + w*Pbb);
  Fcca = k3 * (SQR(n[A])*Paaa + 2*n[A]*n[B]*Paab + SQR(n[B])*Pabb
	 + w*(2*(n[A]*Paa + n[B]*Pab) + w*Pa));
}


void Compute(MeshType &m)
{
  tri::UpdateNormal<MeshType>::PerFaceNormalized(m);
  double nx, ny, nz;

  T0 = T1[X] = T1[Y] = T1[Z]
     = T2[X] = T2[Y] = T2[Z]
     = TP[X] = TP[Y] = TP[Z] = 0;
	FaceIterator fi;
  for (fi=m.face.begin(); fi!=m.face.end();++fi) if(!(*fi).IsD()) {
		FaceType &f=(*fi);

    nx = fabs(f.N()[0]);
    ny = fabs(f.N()[1]);
    nz = fabs(f.N()[2]);
    if (nx > ny && nx > nz) C = X;
    else C = (ny > nz) ? Y : Z;
    A = (C + 1) % 3;
    B = (A + 1) % 3;

    CompFaceIntegrals(f);

    T0 += f.N()[X] * ((A == X) ? Fa : ((B == X) ? Fb : Fc));

    T1[A] += f.N()[A] * Faa;
    T1[B] += f.N()[B] * Fbb;
    T1[C] += f.N()[C] * Fcc;
    T2[A] += f.N()[A] * Faaa;
    T2[B] += f.N()[B] * Fbbb;
    T2[C] += f.N()[C] * Fccc;
    TP[A] += f.N()[A] * Faab;
    TP[B] += f.N()[B] * Fbbc;
    TP[C] += f.N()[C] * Fcca;
  }

  T1[X] /= 2; T1[Y] /= 2; T1[Z] /= 2;
  T2[X] /= 3; T2[Y] /= 3; T2[Z] /= 3;
  TP[X] /= 2; TP[Y] /= 2; TP[Z] /= 2;
}

ScalarType Mass()
{
	return static_cast<ScalarType>(T0);
}

Point3<ScalarType>  CenterOfMass()
{
	Point3<ScalarType>  r;
  r[X] = T1[X] / T0;
  r[Y] = T1[Y] / T0;
  r[Z] = T1[Z] / T0;
	return r;
}
void InertiaTensor(Matrix33<ScalarType> &J ){
	Point3<ScalarType>  r;
  r[X] = T1[X] / T0;
  r[Y] = T1[Y] / T0;
  r[Z] = T1[Z] / T0;
  /* compute inertia tensor */
  J[X][X] = (T2[Y] + T2[Z]);
  J[Y][Y] = (T2[Z] + T2[X]);
  J[Z][Z] = (T2[X] + T2[Y]);
  J[X][Y] = J[Y][X] = - TP[X];
  J[Y][Z] = J[Z][Y] = - TP[Y];
  J[Z][X] = J[X][Z] = - TP[Z];

  J[X][X] -= T0 * (r[Y]*r[Y] + r[Z]*r[Z]);
  J[Y][Y] -= T0 * (r[Z]*r[Z] + r[X]*r[X]);
  J[Z][Z] -= T0 * (r[X]*r[X] + r[Y]*r[Y]);
  J[X][Y] = J[Y][X] += T0 * r[X] * r[Y];
  J[Y][Z] = J[Z][Y] += T0 * r[Y] * r[Z];
  J[Z][X] = J[X][Z] += T0 * r[Z] * r[X];
}

//void InertiaTensor(Matrix44<ScalarType> &J )
void InertiaTensor(Eigen::Matrix3d &J )
{
  J=Eigen::Matrix3d::Identity();
  Point3d  r;
  r[X] = T1[X] / T0;
  r[Y] = T1[Y] / T0;
  r[Z] = T1[Z] / T0;
  /* compute inertia tensor */
  J(X,X) = (T2[Y] + T2[Z]);
  J(Y,Y) = (T2[Z] + T2[X]);
  J(Z,Z) = (T2[X] + T2[Y]);
  J(X,Y) = J(Y,X) = - TP[X];
  J(Y,Z) = J(Z,Y) = - TP[Y];
  J(Z,X) = J(X,Z) = - TP[Z];

  J(X,X) -= T0 * (r[Y]*r[Y] + r[Z]*r[Z]);
  J(Y,Y) -= T0 * (r[Z]*r[Z] + r[X]*r[X]);
  J(Z,Z) -= T0 * (r[X]*r[X] + r[Y]*r[Y]);
  J(X,Y) = J(Y,X) += T0 * r[X] * r[Y];
  J(Y,Z) = J(Z,Y) += T0 * r[Y] * r[Z];
  J(Z,X) = J(X,Z) += T0 * r[Z] * r[X];
}


/** Compute eigenvalues and eigenvectors of inertia tensor.
		The eigenvectors make a rotation matrix that aligns the mesh along the axes of min/max inertia
 */
void InertiaTensorEigen(Matrix33<ScalarType> &EV, Point3<ScalarType> &ev )
{
	Eigen::Matrix3d it;
	InertiaTensor(it);
	Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> eig(it);
	Eigen::Vector3d c_val = eig.eigenvalues();
	Eigen::Matrix3d c_vec = eig.eigenvectors();

	EV.FromEigenMatrix(c_vec);
	ev.FromEigenVector(c_val);
}

/** Compute covariance matrix of a mesh, i.e. the integral
		int_{M} { (x-b)(x-b)^T }dx  where b is the barycenter and x spans over the mesh M
 */
static void Covariance(const MeshType & m, vcg::Point3<ScalarType> & bary, vcg::Matrix33<ScalarType> &C)
{
	// find the barycenter
	ConstFaceIterator fi;
	ScalarType area = 0.0;
	bary.SetZero();
	for(fi = m.face.begin(); fi != m.face.end(); ++fi)
		if(!(*fi).IsD())
			{
				bary += vcg::Barycenter( *fi )* vcg::DoubleArea(*fi);
				area+=vcg::DoubleArea(*fi);
			}
	bary/=area;


	C.SetZero();
	// C as covariance of triangle (0,0,0)(1,0,0)(0,1,0)
	vcg::Matrix33<ScalarType> C0;
	C0.SetZero();
	C0[0][0] = C0[1][1] = 2.0;
	C0[0][1] = C0[1][0] = 1.0;
	C0*=1/24.0;

	// integral of (x,y,0) in the same triangle
	CoordType X(1/6.0,1/6.0,0);
	vcg::Matrix33<ScalarType> A, // matrix that bring the vertices to (v1-v0,v2-v0,n)
												    DC;
	for(fi = m.face.begin(); fi != m.face.end(); ++fi)
		if(!(*fi).IsD())
		{
			const CoordType &P0 = (*fi).P(0);
			const CoordType &P1 = (*fi).P(1);
			const CoordType &P2 = (*fi).P(2);
			CoordType  n = ((P1-P0)^(P2-P0));
			const float da = n.Norm();
			n/=da*da;

			A.SetColumn(0, P1-P0);
			A.SetColumn(1, P2-P0);
			A.SetColumn(2, n);
			CoordType delta = P0 - bary;

			/* DC is calculated as integral of (A*x+delta) * (A*x+delta)^T over the triangle,
				 where delta = v0-bary
			*/

			DC.SetZero();
			DC+= A*C0*A.transpose();
			vcg::Matrix33<ScalarType> tmp;
			tmp.OuterProduct(A*X,delta);
			DC += tmp + tmp.transpose();
			DC+= tmp;
			tmp.OuterProduct(delta,delta);
			DC+=tmp*0.5;
// 		DC*=fabs(A.Determinant());	// the determinant of A is the jacobian of the change of variables A*x+delta
 			DC*=da;											// the determinant of A is also the double area of *fi
			C+=DC;
	}

}
}; // end class Inertia

  } // end namespace tri
} // end namespace vcg


#endif