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cgal/Jet_fitting_3/include/CGAL/Monge_via_jet_fitting.h

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#ifndef _MONGE_VIA_JET_FITTING_H_
#define _MONGE_VIA_JET_FITTING_H_
#include <CGAL/Cartesian.h>
#include <CGAL/circulator.h>
#include <CGAL/Linear_algebraCd.h>
#include <CGAL/eigen.h>
#include <CGAL/Cartesian_converter.h>
#include <CGAL/Lapack/Linear_algebra_lapack.h>
#include <CGAL/jet_fitting_3_assertions.h>
#include <math.h>
#include <utility>
CGAL_BEGIN_NAMESPACE
unsigned int fact(unsigned int n){
unsigned int i, p=1;
for(i=2; i<=n; i++) p *= i;
return p;
}
////////////////////// CLASS Monge_form ////////////////////////
template <class DataKernel>
class Monge_form {
public:
typedef typename DataKernel::FT FT;
typedef typename DataKernel::Point_3 Point_3;
typedef typename DataKernel::Vector_3 Vector_3;
protected:
//point on the fitted surface where diff quantities are computed
Point_3 m_origin_pt;
//the monge trihedron (d1,d2,n) is orthonormal direct
Vector_3 m_d1; //maximal ppal dir
Vector_3 m_d2; //minimal ppal dir
Vector_3 m_n; //normal direction
//coeff = (k1, k2, //ppal curv
// b0, b1, b2, b3, //third order
// c0, c1, c2, c3, c4) //fourth order
// if (degree==1) no coeff needed
std::vector<FT> m_coefficients;
public:
//constructor
Monge_form() {
m_origin_pt = Point_3(0.,0.,0.);
m_d1 = Vector_3(0.,0.,0.);
m_d2 = Vector_3(0.,0.,0.);
m_n = Vector_3(0.,0.,0.);
m_coefficients = std::vector<FT>();
}
~Monge_form() {}
//access
const Point_3 origin() const { return m_origin_pt; }
Point_3& origin() { return m_origin_pt; }
const Vector_3 maximal_principal_direction() const { return m_d1; }
Vector_3& maximal_principal_direction() { return m_d1; }
const Vector_3 minimal_principal_direction() const { return m_d2; }
Vector_3& minimal_principal_direction() { return m_d2; }
const Vector_3 normal_direction() const { return m_n; }
Vector_3& normal_direction() { return m_n; }
const std::vector<FT> coefficients() const { return m_coefficients; }
std::vector<FT>& coefficients() { return m_coefficients; }
const FT principal_curvatures(size_t i) const {
CGAL_precondition( (i == 0 || i == 1) && coefficients().size() >=2 );
return coefficients()[i]; }
const FT third_order_coefficients(size_t i) const {
CGAL_precondition( i >= 0 && i <= 3 && coefficients().size() >=6 );
return coefficients()[i+2]; }
const FT fourth_order_coefficients(size_t i) const {
CGAL_precondition( i >= 0 && i <= 4 && coefficients().size() >=11 );
return coefficients()[i+6]; }
//if d>=2, number of coeffs = (d+1)(d+2)/2 -4.
//we remove cst, linear and the xy coeff which vanish
void set_up(int degree);
//switch min-max ppal curv/dir wrt a given normal orientation.
// if given_normal.monge_normal < 0 then change the orientation
// if z=g(x,y) in the basis (d1,d2,n) then in the basis (d2,d1,-n)
// z=h(x,y)=-g(y,x)
void comply_wrt_given_normal(const Vector_3 given_normal);
void dump_verbose(std::ostream& out_stream);
void dump_4ogl(std::ostream& out_stream, const FT scale);
};
template <class DataKernel>
void Monge_form<DataKernel>::
set_up(int degree) {
if ( degree >= 2 ) std::fill_n(back_inserter(m_coefficients),
(degree+1)*(degree+2)/2-4, 0.);
}
template <class DataKernel>
void Monge_form<DataKernel>::
comply_wrt_given_normal(const Vector_3 given_normal)
{
if ( given_normal*this->normal_direction() < 0 )
{
normal_direction() = -normal_direction();
std::swap(maximal_principal_direction(), minimal_principal_direction());
if ( coefficients().size() >= 2)
std::swap(coefficients()[0],coefficients()[1]);
if ( coefficients().size() >= 6) {
std::swap(coefficients()[2],coefficients()[5]);
std::swap(coefficients()[3],coefficients()[4]);}
if ( coefficients().size() >= 11) {
std::swap(coefficients()[6],coefficients()[10]);
std::swap(coefficients()[7],coefficients()[9]);}
typename std::vector<FT>::iterator itb = coefficients().begin(),
ite = coefficients().end();
for (;itb!=ite;itb++) { *itb = -(*itb); }
}
}
template <class DataKernel>
void Monge_form<DataKernel>::
dump_verbose(std::ostream& out_stream)
{
out_stream << "origin : " << origin() << std::endl
<< "n : " << normal_direction() << std::endl;
if ( coefficients().size() >= 2)
out_stream << "d1 : " << maximal_principal_direction() << std::endl
<< "d2 : " << minimal_principal_direction() << std::endl
<< "k1 : " << coefficients()[0] << std::endl
<< "k2 : " << coefficients()[1] << std::endl;
if ( coefficients().size() >= 6)
out_stream << "b0 : " << coefficients()[2] << std::endl
<< "b1 : " << coefficients()[3] << std::endl
<< "b2 : " << coefficients()[4] << std::endl
<< "b3 : " << coefficients()[5] << std::endl;
if ( coefficients().size() >= 11)
out_stream << "c0 : " << coefficients()[6] << std::endl
<< "c1 : " << coefficients()[7] << std::endl
<< "c2 : " << coefficients()[8] << std::endl
<< "c3 : " << coefficients()[9] << std::endl
<< "c4 : " << coefficients()[10] << std::endl
<< std::endl;
}
template <class DataKernel>
void Monge_form<DataKernel>::
dump_4ogl(std::ostream& out_stream, const FT scale)
{
CGAL_precondition( coefficients().size() >= 2 );
out_stream << origin() << " "
<< maximal_principal_direction() * scale << " "
<< minimal_principal_direction() * scale << " "
<< coefficients()[0] << " "
<< coefficients()[1] << " "
<< std::endl;
}
template <class DataKernel>
std::ostream&
operator<<(std::ostream& out_stream, Monge_form<DataKernel>& monge)
{
monge.dump_verbose(out_stream);
return out_stream;
}
////////////////////// CLASS Monge_via_jet_fitting ////////////////////////
template < class DataKernel, class LocalKernel = Cartesian<double>, class SvdTraits = Lapack_svd>
class Monge_via_jet_fitting {
public:
typedef DataKernel Data_Kernel;
typedef LocalKernel Local_Kernel;
typedef typename std::vector<typename Data_Kernel::Point_3>::iterator Range_Iterator;
typedef Monge_form<Data_Kernel> LMonge_form;
//used to convert number types, points and vectors back and forth
typedef NT_converter<typename Local_Kernel::FT, typename Data_Kernel::FT> L2D_NTconverter;
Cartesian_converter<Data_Kernel, Local_Kernel> D2L_converter;
Cartesian_converter<Local_Kernel, Data_Kernel> L2D_converter;
typedef typename Local_Kernel::FT LFT;
typedef typename Local_Kernel::Point_3 LPoint;
typedef typename Local_Kernel::Vector_3 LVector;
typedef CGAL::Aff_transformation_3<Local_Kernel> Aff_transformation;
typedef typename Data_Kernel::FT DFT;
typedef typename Data_Kernel::Point_3 DPoint;
typedef typename SvdTraits::Vector LAVector;
typedef typename SvdTraits::Matrix LAMatrix;
public:
Monge_via_jet_fitting();
Monge_via_jet_fitting(Range_Iterator begin, Range_Iterator end,
size_t d, size_t dprime,
LMonge_form &monge_form);
LMonge_form operator()(Range_Iterator begin, Range_Iterator end,
size_t d, size_t dprime);
const LFT condition_number() const {return condition_nb;}
const std::pair<LFT, LVector> pca_basis(size_t i) const {return m_pca_basis[i];}
protected:
int deg;
int deg_monge;
int nb_d_jet_coeff;
int nb_input_pts;
LFT preconditionning;
CGAL::Sqrt<LFT> Lsqrt;
LFT condition_nb;
std::vector< std::pair<LFT, LVector> > m_pca_basis;
//translate_p0 changes the origin of the world to p0 the first point
// of the input data points
//change_world2fitting (coord of a vector in world) = coord of this
// vector in fitting. The matrix tranform has as lines the coord of
// the basis vectors of fitting in the world coord.
//idem for change_fitting2monge
Aff_transformation translate_p0, change_world2fitting,
change_fitting2monge;
//eigen val and vect stored in m_pca_basis
// change_world2fitting is computed
void compute_PCA(Range_Iterator begin, Range_Iterator end);
//Coordinates of input points are computed in the fitting basis with
// p0 as origin.
//Preconditionning is computed, M and Z are filled
void fill_matrix(Range_Iterator begin, Range_Iterator end,
int d, LAMatrix& M, LAVector& Z);
//A is computed, solving MA=Z in the ls sense, the solution A is stored in Z
//Preconditionning is needed
void solve_linear_system(LAMatrix &M, LAVector& Z);
//Classical differential geometric calculus
//change_fitting2monge is computed
//if deg_monge =1 only 1st order info
//if deg_monge >= 2 2nd order info are computed
void compute_Monge_basis(const LFT* A, LMonge_form& monge_form);
//if deg_monge >=3 then 3rd (and 4th) order info are computed
void compute_Monge_coefficients(LFT* A, int dprime,
LMonge_form& monge_form);
//for a trihedron (v1,v2,v3) switches v1 to -v1 if det(v1,v2,v3) < 0
void switch_to_direct_orientation(LVector& v1, const LVector& v2,
const LVector& v3);
};
//-------------------------------------------------------------------
// Implementation
//------------------------------------------------------------------
template < class DataKernel, class LocalKernel, class SvdTraits>
Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
Monge_via_jet_fitting()
{
m_pca_basis = std::vector< std::pair<LFT, LVector> >(3);
}
template < class DataKernel, class LocalKernel, class SvdTraits>
Monge_form<DataKernel>
Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
operator()(Range_Iterator begin, Range_Iterator end,
size_t d, size_t dprime)
{
// precondition: on the degrees, jet and monge
CGAL_precondition( (d >=1) && (dprime >= 1)
&& (dprime <= 4) && (dprime <= d) );
this->deg = d;
this->deg_monge = dprime;
this->nb_d_jet_coeff = (d+1)*(d+2)/2;
this->nb_input_pts = end - begin;
// precondition: solvable ls system
CGAL_precondition( nb_input_pts >= nb_d_jet_coeff );
//Initialize
LMonge_form monge_form;
monge_form.set_up(dprime);
//for the system MA=Z
LAMatrix M(nb_input_pts, nb_d_jet_coeff);
LAVector Z(nb_input_pts);
compute_PCA(begin, end);
fill_matrix(begin, end, d, M, Z);//with precond
solve_linear_system(M, Z); //correct with precond
compute_Monge_basis(Z.vector(), monge_form);
if ( dprime >= 3) compute_Monge_coefficients(Z.vector(), dprime, monge_form);
return monge_form;
}
template < class DataKernel, class LocalKernel, class SvdTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
compute_PCA(Range_Iterator begin, Range_Iterator end)
{
int n = this->nb_input_pts;
LFT x, y, z,
sumX = 0., sumY = 0., sumZ = 0.,
sumX2 = 0., sumY2 = 0., sumZ2 = 0.,
sumXY = 0., sumXZ = 0., sumYZ = 0.,
xx, yy, zz, xy, xz, yz;
for (; begin != end; begin++)
{
LPoint lp = D2L_converter(*begin);
x = lp.x();
y = lp.y();
z = lp.z();
sumX += x / n;
sumY += y / n;
sumZ += z / n;
sumX2 += x * x / n;
sumY2 += y * y / n;
sumZ2 += z * z / n;
sumXY += x * y / n;
sumXZ += x * z / n;
sumYZ += y * z / n;
}
xx = sumX2 - sumX * sumX;
yy = sumY2 - sumY * sumY;
zz = sumZ2 - sumZ * sumZ;
xy = sumXY - sumX * sumY;
xz = sumXZ - sumX * sumZ;
yz = sumYZ - sumY * sumZ;
// assemble covariance matrix as a
// semi-definite matrix.
// Matrix numbering:
// 0
// 1 2
// 3 4 5
LFT covariance[6] = {xx,xy,yy,xz,yz,zz};
LFT eigen_values[3];
LFT eigen_vectors[9];
// solve for eigenvalues and eigenvectors.
// eigen values are sorted in descending order,
// eigen vectors are sorted in accordance.
CGAL::CGALi::eigen_symmetric<LFT>(covariance,3,eigen_vectors,eigen_values);
//store in m_pca_basis
for (int i=0; i<3; i++)
{
m_pca_basis[i].first = eigen_values[i];
}
LVector v1(eigen_vectors[0],eigen_vectors[1],eigen_vectors[2]);
m_pca_basis[0].second = v1;
LVector v2(eigen_vectors[3],eigen_vectors[4],eigen_vectors[5]);
m_pca_basis[1].second = v2;
LVector v3(eigen_vectors[6],eigen_vectors[7],eigen_vectors[8]);
m_pca_basis[2].second = v3;
switch_to_direct_orientation(m_pca_basis[0].second,
m_pca_basis[1].second,
m_pca_basis[2].second);
//Store the change of basis W->F
Aff_transformation
change_basis (m_pca_basis[0].second[0], m_pca_basis[0].second[1], m_pca_basis[0].second[2],
m_pca_basis[1].second[0], m_pca_basis[1].second[1], m_pca_basis[1].second[2],
m_pca_basis[2].second[0], m_pca_basis[2].second[1], m_pca_basis[2].second[2]);
this->change_world2fitting = change_basis;
}
template < class DataKernel, class LocalKernel, class SvdTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
fill_matrix(Range_Iterator begin, Range_Iterator end,
int d, LAMatrix &M, LAVector& Z)
{
//origin of fitting coord system = first input data point
LPoint point0 = D2L_converter(*begin);
//transform coordinates of sample points with a
//translation ($-p$) and multiplication by $ P_{W\rightarrow F}$.
LPoint orig(0.,0.,0.);
LVector v_point0_orig(orig - point0);
Aff_transformation transl(CGAL::TRANSLATION, v_point0_orig);
this->translate_p0 = transl;
Aff_transformation transf_points = this->change_world2fitting *
this->translate_p0;
//compute and store transformed points
std::vector<LPoint> pts_in_fitting_basis;
CGAL_For_all(begin,end){
LPoint cur_pt = transf_points(D2L_converter(*begin));
pts_in_fitting_basis.push_back(cur_pt);
}
//Compute preconditionning
LFT precond = 0.;
typename std::vector<LPoint>::iterator itb = pts_in_fitting_basis.begin(),
ite = pts_in_fitting_basis.end();
CGAL_For_all(itb,ite) precond += std::fabs(itb->x()) + std::fabs(itb->y());
precond /= 2*this->nb_input_pts;
this->preconditionning = precond;
//fill matrices M and Z
itb = pts_in_fitting_basis.begin();
int line_count = 0;
LFT x, y;
CGAL_For_all(itb,ite) {
x = itb->x();
y = itb->y();
// Z[line_count] = itb->z();
Z.set(line_count,itb->z());
for (int k=0; k <= d; k++) for (int i=0; i<=k; i++)
M.set(line_count, k*(k+1)/2+i, std::pow(x,k-i)*std::pow(y,i)
/(fact(i)*fact(k-i)*std::pow(this->preconditionning,k)));
line_count++;
}
}
template < class DataKernel, class LocalKernel, class SvdTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
solve_linear_system(LAMatrix &M, LAVector& Z)
{
SvdTraits::solve(M, Z, condition_nb);
for (int k=0; k <= this->deg; k++) for (int i=0; i<=k; i++)
// Z[k*(k+1)/2+i] /= std::pow(this->preconditionning,k);
Z.set( k*(k+1)/2+i, Z(k*(k+1)/2+i) / std::pow(this->preconditionning,k) );
}
template < class DataKernel, class LocalKernel, class SvdTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
compute_Monge_basis(const LFT* A, LMonge_form& monge_form)
{
// only 1st order info.
if ( this->deg_monge == 1 ) {
LPoint orig_monge(0., 0., A[0]);
LVector normal(-A[1], -A[2], 1.);
LFT norm2 = normal * normal;
normal = normal / Lsqrt(norm2);
monge_form.origin() = L2D_converter(
(this->translate_p0.inverse() *
this->change_world2fitting.inverse()) (orig_monge) );
monge_form.normal_direction() = L2D_converter(this->change_world2fitting.inverse()(normal));
}
// else (deg_monge >= 2) then 2nd order info are computed
else {
//bi-index to uni-index conversion : A(i,j)=A[(i+j)(i+j+1)/2+j]
LPoint orig_monge(0., 0., A[0]);
//normal = Xu crossprod Xv
LVector Xu(1.,0.,A[1]), Xv(0.,1.,A[2]), normal(-A[1], -A[2], 1.);
LFT norm2 = normal * normal;
normal = normal / Lsqrt(norm2);
//Surface in fitting_basis : X(u,v)=(u,v,J_A(u,v))
//in the basis Xu=(1,0,A[1]), Xv=(0,1,A[2]), Weingarten=-I^{-1}II
//first fond form I=(e,f,f,g)
// =(Xu.Xu, Xu.Xv, Xu.Xv, Xv.Xv)
//second fond form II=(l,m,m,n)/norm2^(1/2)
// =(n.Xuu, n.Xuv, n.Xuv, n.Xvv)
//ppal curv are the opposite of the eigenvalues of Weingarten or the
// eigenvalues of weingarten = -Weingarten = I^{-1}II
typedef typename CGAL::Linear_algebraCd<LFT>::Matrix Matrix;
LFT e = 1+A[1]*A[1], f = A[1]*A[2], g = 1+A[2]*A[2],
l = A[3], m = A[4], n = A[5];
Matrix weingarten(2,2,0.);
weingarten(0,0) = (g*l-f*m)/ (Lsqrt(norm2)*norm2);
weingarten(0,1) = (g*m-f*n)/ (Lsqrt(norm2)*norm2);
weingarten(1,0) = (e*m-f*l)/ (Lsqrt(norm2)*norm2);
weingarten(1,1) = (e*n-f*m)/ (Lsqrt(norm2)*norm2);
// Y, Z are normalized GramSchmidt of Xu, Xv
// Xu->Y=Xu/||Xu||;
// Xv->Z=Xv-(Xu.Xv)Xu/||Xu||^2;
// Z-> Z/||Z||
LVector Y, Z;
LFT normXu = Lsqrt( Xu*Xu );
Y = Xu / normXu;
LFT XudotXv = Xu * Xv;
Z = Xv - XudotXv * Xu / (normXu*normXu);
LFT normZ = Lsqrt( Z*Z );
Z = Z / normZ;
Matrix change_XuXv2YZ(2,2,0.);
change_XuXv2YZ(0,0) = 1 / normXu;
change_XuXv2YZ(0,1) = -XudotXv / (normXu * normXu * normZ);
change_XuXv2YZ(1,0) = 0;
change_XuXv2YZ(1,1) = 1 / normZ;
LFT det = 0.;
Matrix inv = CGAL::Linear_algebraCd<LFT>::inverse ( change_XuXv2YZ, det );
//in the new orthonormal basis (Y,Z) of the tangent plane :
weingarten = inv *(1/det) * weingarten * change_XuXv2YZ;
//switch to eigen_symmetric algo for diagonalization of weingarten
LFT W[3] = {weingarten(0,0), weingarten(1,0), weingarten(1,1)};
LFT eval[2];
LFT evec[4];
//eval in decreasing order
CGAL::CGALi::eigen_symmetric<LFT>(W,2,evec,eval);
LVector d_max = evec[0]*Y + evec[1]*Z,
d_min = evec[2]*Y + evec[3]*Z;
switch_to_direct_orientation(d_max, d_min, normal);
Aff_transformation change_basis (d_max[0], d_max[1], d_max[2],
d_min[0], d_min[1], d_min[2],
normal[0], normal[1], normal[2]);
this->change_fitting2monge = change_basis;
//store the monge basis origin and vectors with their world coord
//store ppal curv
monge_form.origin() = L2D_converter(
(this->translate_p0.inverse() *
this->change_world2fitting.inverse()) (orig_monge ));
monge_form.maximal_principal_direction() = L2D_converter(this->change_world2fitting.inverse()(d_max));
monge_form.minimal_principal_direction() = L2D_converter(this->change_world2fitting.inverse()(d_min));
monge_form.normal_direction() = L2D_converter(this->change_world2fitting.inverse()(normal));
monge_form.coefficients()[0] = L2D_NTconverter()(eval[0]);
monge_form.coefficients()[1] = L2D_NTconverter()(eval[1]);
}
//end else
}
template < class DataKernel, class LocalKernel, class SvdTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
compute_Monge_coefficients(LFT* A, int dprime,
LMonge_form& monge_form)
{
//One has the equation w=J_A(u,v) of the fitted surface S
// in the fitting_basis
//Substituing (u,v,w)=change_fitting2monge^{-1}(x,y,z)
//One has the equation f(x,y,z)=0 on this surface S in the monge
// basis
//The monge form of the surface at the origin is the bivariate fct
// g(x,y) s.t. f(x,y,g(x,y))=0
//voir les calculs Maple dans monge.mws
//Notations are f123= d^3f/dxdydz
// g(x,y)=sum (gij x^i y^j/ i!j!) with
// g00=g10=g01=g11=0, g20=kmax, g02=kmin
//
//g(x,y)= 1/2*(k1x^2 +k2y^2)
// +1/6*(b0x^3 +3b1x^2y +3b2xy^2 +b3y^3)
// +1/24*(c0x^4 +4c1x^3y +6c2x^2y^2 +4c3xy^3 +c4y^4)
// +...
// p stores change_fitting2monge^{-1}=change_fitting2monge^{T}
LFT p[3][3];
p[0][0] = this->change_fitting2monge.m(0,0);
p[1][0] = this->change_fitting2monge.m(0,1);
p[2][0] = this->change_fitting2monge.m(0,2);
p[0][1] = this->change_fitting2monge.m(1,0);
p[1][1] = this->change_fitting2monge.m(1,1);
p[2][1] = this->change_fitting2monge.m(1,2);
p[0][2] = this->change_fitting2monge.m(2,0);
p[1][2] = this->change_fitting2monge.m(2,1);
p[2][2] = this->change_fitting2monge.m(2,2);
// formula are designed for w=sum( Aij ui vj), but we have J_A = sum( Aij/i!j! ui vj)
for (int k=0; k <= this->deg; k++) for (int i=0; i<=k; i++)
A[k*(k+1)/2+i] /= fact(k-i)*fact(i);//this is A(k-i;i)
/* //debug */
/* std::cout << "coeff of A" << std::endl */
/* << A[0] << " "<< A[1] << " "<< A[2] << std::endl */
/* << A[3] << " "<< A[4] << " "<< A[5] << std::endl */
/* << A[6] << " "<< A[7] << " "<< A[8] << " "<< A[9]<< std::endl */
/* << A[10] << " "<< A[11] << " "<< A[12] << " "<< A[13]<< " " << A[14] << std::endl; */
// note f1 = f2 = f12 = 0
// LFT f1 = A[1] * p[0][0] + A[2] * p[1][0] - p[2][0];
// LFT f2 = A[2] * p[1][1] + A[1] * p[0][1] - p[2][1];
// LFT f12 =
// 2 * A[3] * p[0][0] * p[0][1]
// + 2 * A[5] * p[1][0] * p[1][1]
// + A[4] * p[0][1] * p[1][0]
// + A[4] * p[0][0] * p[1][1];
// -f11 / f3 = kmax
// -f22 / f3 = kmin
LFT f3 = A[1] * p[0][2] + A[2] * p[1][2] - p[2][2];
LFT f11 =
2 * A[4] * p[0][0] * p[1][0]
+ 2 * A[5] * p[1][0] * p[1][0]
+ 2 * A[3] * p[0][0] * p[0][0];
LFT f13 =
A[4] * p[0][0] * p[1][2]
+ A[4] * p[0][2] * p[1][0]
+ 2 * A[5] * p[1][0] * p[1][2]
+ 2 * A[3] * p[0][0] * p[0][2];
LFT f22 =
2 * A[4] * p[0][1] * p[1][1]
+ 2 * A[5] * p[1][1] * p[1][1]
+ 2 * A[3] * p[0][1] * p[0][1];
LFT f23 =
A[4] * p[0][1] * p[1][2]
+ 2 * A[5] * p[1][1] * p[1][2]
+ A[4] * p[0][2] * p[1][1]
+ 2 * A[3] * p[0][1] * p[0][2];
LFT f33 =
2 * A[5] * p[1][2] * p[1][2]
+ 2 * A[3] * p[0][2] * p[0][2]
+ 2 * A[4] * p[0][2] * p[1][2];
LFT f111 =
6 * A[8] * p[0][0] * p[1][0] * p[1][0]
+ 6 * A[7] * p[0][0] * p[0][0] * p[1][0]
+ 6 * A[6] * p[0][0] * p[0][0] * p[0][0]
+ 6 * A[9] * p[1][0] * p[1][0] * p[1][0];
LFT f222 =
6 * A[7] * p[0][1] * p[0][1] * p[1][1]
+ 6 * A[8] * p[0][1] * p[1][1] * p[1][1]
+ 6 * A[9] * p[1][1] * p[1][1] * p[1][1]
+ 6 * A[6] * p[0][1] * p[0][1] * p[0][1];
LFT f112 =
2 * A[7] * p[0][0] * p[0][0] * p[1][1]
+ 6 * A[6] * p[0][0] * p[0][0] * p[0][1]
+ 2 * A[8] * p[0][1] * p[1][0] * p[1][0]
+ 4 * A[8] * p[0][0] * p[1][0] * p[1][1]
+ 6 * A[9] * p[1][0] * p[1][0] * p[1][1]
+ 4 * A[7] * p[0][0] * p[0][1] * p[1][0];
LFT f122 =
4 * A[8] * p[0][1] * p[1][0] * p[1][1]
+ 2 * A[8] * p[0][0] * p[1][1] * p[1][1]
+ 6 * A[6] * p[0][0] * p[0][1] * p[0][1]
+ 2 * A[7] * p[0][1] * p[0][1] * p[1][0]
+ 4 * A[7] * p[0][0] * p[0][1] * p[1][1]
+ 6 * A[9] * p[1][0] * p[1][1] * p[1][1];
LFT f113 =
6*A[6]*p[0][0]*p[0][0]*p[0][2]
+6*A[9]*p[1][0]*p[1][0]*p[1][2]
+2*A[7]*p[0][0]*p[0][0]*p[1][2]
+2*A[8]*p[0][2]*p[1][0]*p[1][0]
+4*A[7]*p[0][0]*p[0][2]*p[1][0]
+4*A[8]*p[0][0]*p[1][0]*p[1][2];
LFT f223 =
2*A[8]*p[0][2]*p[1][1]*p[1][1]
+6*A[6]*p[0][1]*p[0][1]*p[0][2]
+6*A[9]*p[1][1]*p[1][1]*p[1][2]
+2*A[7]*p[0][1]*p[0][1]*p[1][2]
+4*A[7]*p[0][1]*p[0][2]*p[1][1]
+4*A[8]*p[0][1]*p[1][1]*p[1][2];
LFT f123 =
2*A[8]*p[0][2]*p[1][0]*p[1][1]
+2*A[7]*p[0][0]*p[0][1]*p[1][2]
+2*A[7]*p[0][0]*p[0][2]*p[1][1]
+6*A[9]*p[1][0]*p[1][1]*p[1][2]
+2*A[7]*p[0][1]*p[0][2]*p[1][0]
+6*A[6]*p[0][0]*p[0][1]*p[0][2]
+2*A[8]*p[0][0]*p[1][1]*p[1][2]
+2*A[8]*p[0][1]*p[1][0]*p[1][2];
LFT b0 = 1/(f3*f3)*(-f111*f3+3*f13*f11);
LFT b1 = 1/(f3*f3)*(-f112*f3+f23*f11);
LFT b2 = 1/(f3*f3)*(-f122*f3+f13*f22);
LFT b3 = -1/(f3*f3)*(f222*f3-3*f23*f22);
monge_form.coefficients()[2] = L2D_NTconverter()(b0);
monge_form.coefficients()[3] = L2D_NTconverter()(b1);
monge_form.coefficients()[4] = L2D_NTconverter()(b2);
monge_form.coefficients()[5] = L2D_NTconverter()(b3);
if ( dprime == 4 )
{
LFT f1111 =
24*A[13]*p[0][0]*p[1][0]*p[1][0]*p[1][0]
+24*A[12]*p[0][0]*p[0][0]*p[1][0]*p[1][0]
+24*A[11]*p[0][0]*p[0][0]*p[0][0]*p[1][0]
+24*A[14]*p[1][0]*p[1][0]*p[1][0]*p[1][0]
+24*A[10]*p[0][0]*p[0][0]*p[0][0]*p[0][0];
LFT f1112 =
6*A[13]*p[0][1]*p[1][0]*p[1][0]*p[1][0]
+18*A[13]*p[0][0]*p[1][0]*p[1][0]*p[1][1]
+24*A[10]*p[0][0]*p[0][0]*p[0][0]*p[0][1]
+12*A[12]*p[0][0]*p[0][1]*p[1][0]*p[1][0]
+18*A[11]*p[0][0]*p[0][0]*p[0][1]*p[1][0]
+24*A[14]*p[1][0]*p[1][0]*p[1][0]*p[1][1]
+6*A[11]*p[0][0]*p[0][0]*p[0][0]*p[1][1]
+12*A[12]*p[0][0]*p[0][0]*p[1][0]*p[1][1];
LFT f1122 =
12*A[11]*p[0][0]*p[0][0]*p[0][1]*p[1][1]
+12*A[13]*p[0][0]*p[1][0]*p[1][1]*p[1][1]
+12*A[13]*p[0][1]*p[1][0]*p[1][0]*p[1][1]
+16*A[12]*p[0][0]*p[0][1]*p[1][0]*p[1][1]
+12*A[11]*p[0][0]*p[0][1]*p[0][1]*p[1][0]
+24*A[10]*p[0][0]*p[0][0]*p[0][1]*p[0][1]
+4*A[12]*p[0][1]*p[0][1]*p[1][0]*p[1][0]
+4*A[12]*p[0][0]*p[0][0]*p[1][1]*p[1][1]
+24*A[14]*p[1][0]*p[1][0]*p[1][1]*p[1][1];
LFT f1222 =
6*A[13]*p[0][0]*p[1][1]*p[1][1]*p[1][1]
+24*A[10]*p[0][0]*p[0][1]*p[0][1]*p[0][1]
+24*A[14]*p[1][0]*p[1][1]*p[1][1]*p[1][1]
+6*A[11]*p[0][1]*p[0][1]*p[0][1]*p[1][0]
+18*A[11]*p[0][0]*p[0][1]*p[0][1]*p[1][1]
+12*A[12]*p[0][0]*p[0][1]*p[1][1]*p[1][1]
+12*A[12]*p[0][1]*p[0][1]*p[1][0]*p[1][1]
+18*A[13]*p[0][1]*p[1][0]*p[1][1]*p[1][1];
LFT f2222 =
24*A[13]*p[0][1]*p[1][1]*p[1][1]*p[1][1]
+24*A[11]*p[0][1]*p[0][1]*p[0][1]*p[1][1]
+24*A[12]*p[0][1]*p[0][1]*p[1][1]*p[1][1]
+24*A[10]*p[0][1]*p[0][1]*p[0][1]*p[0][1]
+24*A[14]*p[1][1]*p[1][1]*p[1][1]*p[1][1];
LFT c0 =
-1/(f3*f3*f3)*(f1111*(f3*f3)-4*f13*f3*f111+12*f13*f13*f11-6*f113*f3*f11+3*f33*f11*f11);
LFT c1 =
1/(f3*f3*f3)*(f23*f3*f111+3*f3*f123*f11+3*f13*f3*f112-f1112*(f3*f3)-6*f13*f23*f11);
LFT c2 =
1/(f3*f3*f3)*(-f33*f22*f11+f113*f3*f22+2*f13*f3*f122-2*f13*f13*f22+f223*f3*f11+2*f23*f3*f112-2*f23*f23*f11-f1122*(f3*f3));
LFT c3 =
1/(f3*f3*f3)*(-f1222*(f3*f3)-6*f13*f23*f22+3*f123*f3*f22+f13*f3*f222+3*f23*f3*f122);
LFT c4 =
-1/(f3*f3*f3)*(f2222*(f3*f3)+3*f33*f22*f22-6*f223*f3*f22-4*f23*f3*f222+12*f23*f23*f22) ;
monge_form.coefficients()[6] = L2D_NTconverter()(c0);
monge_form.coefficients()[7] = L2D_NTconverter()(c1);
monge_form.coefficients()[8] = L2D_NTconverter()(c2);
monge_form.coefficients()[9] = L2D_NTconverter()(c3);
monge_form.coefficients()[10] = L2D_NTconverter()(c4);
}
}
template < class DataKernel, class LocalKernel, class SvdTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>::
switch_to_direct_orientation(LVector& v1, const LVector& v2,
const LVector& v3)
{
CGAL::Sign orientation = CGAL::sign_of_determinant3x3(v1[0], v2[0], v3[0],
v1[1], v2[1], v3[1],
v1[2], v2[2], v3[2]);
if (orientation == CGAL::NEGATIVE) v1 = -v1;
}
CGAL_END_NAMESPACE
#endif //_MONGE_VIA_JET_FITTING_H_
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