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#ifndef _VCG_INERTIA_
#define _VCG_INERTIA_
#include <Eigen/Core>
#include <Eigen/Eigenvalues>
#include <vcg/complex/algorithms/update/normal.h>
namespace vcg
{
namespace tri
{
/*! \brief Methods for computing Polyhedral Mass properties (like inertia tensor, volume, etc)
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:
<b>Brian Mirtich,</b>
``Fast and Accurate Computation of Polyhedral Mass Properties,''
journal of graphics tools, volume 1, number 2, 1996
*/
template <class MeshType>
class Inertia
{
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:
/*! \brief Basic constructor
When you create a Inertia object, you have to specify the mesh that it refers to.
The properties are computed at that moment. Subsequent modification of the mesh does not affect these values.
*/
Inertia(MeshType &m) {Compute(m);}
/* 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));
}
/*! main function to be called.
It requires a watertight mesh with per face normals.
*/
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() && vcg::DoubleArea(*fi)>std::numeric_limits<float>::min()) {
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;
}
/*! \brief Return the Volume (or mass) of the mesh.
Meaningful only if the mesh is watertight.
*/
ScalarType Mass()
{
return static_cast<ScalarType>(T0);
}
/*! \brief Return the Center of Mass (or barycenter) of the mesh.
Meaningful only if the mesh is watertight.
*/
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];
}
/*! \brief Return the Inertia tensor the mesh.
The result is factored as eigenvalues and eigenvectors (as ROWS).
*/
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(); // eigenvector are stored as columns.
EV.FromEigenMatrix(c_vec);
EV.transposeInPlace();
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).cP(0);
const CoordType &P1 = (*fi).cP(1);
const CoordType &P2 = (*fi).cP(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