songsenand
/
cgal
mirror of https://github.com/CGAL/cgal
1
0
Fork 0
Code Issues Packages Projects Releases Wiki Activity
cgal/Jet_fitting_3/include/CGAL/Monge_via_jet_fitting.h

783 lines
28 KiB
C++
Raw Blame History

#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/jet_fitting_3_assertions.h>
#include <CGAL/Lapack/Linear_algebra_lapack.h>
#include <CGAL/NT_converter.h>
//#include <CGAL/eigen.h> //for ALTERNATIVE with CGAL eigen code
#include <math.h>
CGAL_BEGIN_NAMESPACE
// int fact(int n)
// {
// if (n == 0)
// return(1);
// return(n * fact(n-1));
// }
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 DFT;
typedef typename DataKernel::Point_3 DPoint;
typedef typename DataKernel::Vector_3 DVector;
protected:
//point on the fitted surface where diff quantities are computed
DPoint m_origin_pt;
//the monge trihedron (d1,d2,n) is orthonormal direct
DVector m_d1; //maximal ppal dir
DVector m_d2; //minimal ppal dir
DVector 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<DFT> m_coefficients;
public:
//constructor
Monge_form() {
m_origin_pt = DPoint(0.,0.,0.);
m_d1 = DVector(0.,0.,0.);
m_d2 = DVector(0.,0.,0.);
m_n = DVector(0.,0.,0.);
m_coefficients = std::vector<DFT>();
}
~Monge_form() {}
//access
const DPoint origin_pt() const { return m_origin_pt; }
DPoint& origin_pt() { return m_origin_pt; }
const DVector d1() const { return m_d1; }
DVector& d1() { return m_d1; }
const DVector d2() const { return m_d2; }
DVector& d2() { return m_d2; }
const DVector n() const { return m_n; }
DVector& n() { return m_n; }
const std::vector<DFT> coefficients() const { return m_coefficients; }
std::vector<DFT>& coefficients() { return m_coefficients; }
//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 DVector given_normal);
void dump_verbose(std::ostream& out_stream);
void dump_4ogl(std::ostream& out_stream, const DFT 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 DVector given_normal)
{
if ( given_normal*this->n() < 0 )
{
n() = -n();
std::swap(d1(), d2());
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<DFT>::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_pt() << std::endl
<< "n : " << n() << std::endl;
if ( coefficients().size() >= 2)
out_stream << "d1 : " << d1() << std::endl
<< "d2 : " << d2() << 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
<< "P1 : " <<
3*coefficients()[3]*coefficients()[3]
+(coefficients()[0]-coefficients()[1])
*(coefficients()[6]
-3*coefficients()[0]*coefficients()[0]*coefficients()[0]) << std::endl
//= 3*b2^2+(k2-k1)(c4-3k2^3)
<< "P2 : " <<
3*coefficients()[4]*coefficients()[4]
-(coefficients()[0]-coefficients()[1])
*(coefficients()[10]
-3*coefficients()[1]*coefficients()[1]*coefficients()[1] )
<< std::endl;
}
template <class DataKernel>
void Monge_form<DataKernel>::
dump_4ogl(std::ostream& out_stream, const DFT scale)
{
CGAL_precondition( coefficients().size() >= 2 );
out_stream << origin_pt() << " "
<< d1() * scale << " "
<< d2() * scale << " "
<< coefficients()[0] << " "
<< coefficients()[1] << " "
<< std::endl;
}
////////////////////// CLASS Monge_form_condition_numbers ////////////////////////
template <class LocalKernel>
class Monge_form_condition_numbers {
public:
typedef typename LocalKernel::FT LFT;
typedef typename LocalKernel::Vector_3 LVector;
protected:
LFT m_pca_eigen_vals[3];
LVector m_pca_eigen_vecs[3];
LFT m_cond_nb;//of the least square system
public:
//constructor
Monge_form_condition_numbers() {
m_cond_nb = 0.;
std::fill_n(m_pca_eigen_vals, 3, 0.);
std::fill_n(m_pca_eigen_vecs, 3, LVector());
}
//access
const LFT* pca_eigen_vals() const { return m_pca_eigen_vals; }
LFT* pca_eigen_vals() { return m_pca_eigen_vals; }
const LVector* pca_eigen_vecs() const { return m_pca_eigen_vecs; }
LVector* pca_eigen_vecs() { return m_pca_eigen_vecs; }
const LFT cond_nb() const { return m_cond_nb; }
LFT& cond_nb() { return m_cond_nb; }
void dump_verbose(std::ostream& out_stream);
};
template <class LocalKernel>
void Monge_form_condition_numbers<LocalKernel>::
dump_verbose(std::ostream& out_stream)
{
out_stream << "cond_nb : " << cond_nb() << std::endl
<< "pca_eigen_vals " << pca_eigen_vals()[0]
<< " " << pca_eigen_vals()[1]
<< " " << pca_eigen_vals()[2] << std::endl
<< "pca_eigen_vecs : " << std::endl
<< pca_eigen_vecs()[0] << std::endl
<< pca_eigen_vecs()[1] << std::endl
<< pca_eigen_vecs()[2] << std::endl;
}
////////////////////// CLASS Monge_via_jet_fitting ////////////////////////
template < class DataKernel, class LocalKernel = Cartesian<double>, class LinAlgTraits = Lapack>
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> Monge_form;
typedef Monge_form_condition_numbers<Local_Kernel> Monge_form_condition_numbers;
//used to convert number types back and forth
//TODO: perform conversion b = D2L_converter()(a). cf also Cartesian_converter
typedef NT_converter<Data_Kernel::FT, Local_Kernel::FT> D2L_converter;
typedef NT_converter<Local_Kernel::FT, Data_Kernel::FT> L2D_converter;
public:
Monge_via_jet_fitting(Range_Iterator begin, Range_Iterator end,
int d, int dprime,
Monge_form &monge_form, Monge_form_condition_numbers &monge_form_condition_numbers);
protected:
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 LinAlgTraits::Matrix LAMatrix;
protected:
int deg;
int deg_monge;
int nb_d_jet_coeff;
int nb_input_pts;
LFT preconditionning;
CGAL::Sqrt<LFT> Lsqrt;
//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 monge_form_condition_numbers,
// change_world2fitting is computed
void compute_PCA(Range_Iterator begin, Range_Iterator end,
Monge_form_condition_numbers &monge_form_condition_numbers);
//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, LFT* Z);
//A is computed, solving MA=Z in the ls sense, the solution A is stored in Z
//Preconditionning is needed
//the condition number of the matrix M is stored in monge_form_condition_numbers
void solve_linear_system(LAMatrix &M, LFT* Z, Monge_form_condition_numbers& monge_form_condition_numbers);
//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, Monge_form& monge_form);
//if deg_monge >=3 then 3rd (and 4th) order info are computed
void compute_Monge_coefficients(LFT* A, int dprime,
Monge_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 LinAlgTraits>
Monge_via_jet_fitting<DataKernel, LocalKernel, LinAlgTraits>::
Monge_via_jet_fitting(Range_Iterator begin, Range_Iterator end,
int d, int dprime,
Monge_form& monge_form,
Monge_form_condition_numbers& monge_form_condition_numbers)
{
// 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
monge_form.set_up(dprime);
//for the system MA=Z
LAMatrix M(nb_input_pts, nb_d_jet_coeff);
LFT* Z = (LFT*) malloc(nb_input_pts*sizeof(LFT));
compute_PCA(begin, end, monge_form_condition_numbers);
fill_matrix(begin, end, d, M, Z);//with precond
solve_linear_system(M, Z, monge_form_condition_numbers); //correct with precond
compute_Monge_basis(Z, monge_form);
if ( dprime >= 3) compute_Monge_coefficients(Z, dprime, monge_form);
}
template < class DataKernel, class LocalKernel, class LinAlgTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, LinAlgTraits>::
compute_PCA(Range_Iterator begin, Range_Iterator end,
Monge_form_condition_numbers &monge_form_condition_numbers)
{
LAMatrix Cov(3,3);
LFT* eval = (LFT*) malloc(3*sizeof(LFT));
LAMatrix evec(3,3);
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++)
{
x = (*begin).x();
y = (*begin).y();
z = (*begin).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;
Cov.set_elt(0,0,xx);
Cov.set_elt(0,1,xy);
Cov.set_elt(0,2,xz);
Cov.set_elt(1,0,xy);
Cov.set_elt(1,1,yy);
Cov.set_elt(1,2,yz);
Cov.set_elt(2,0,xz);
Cov.set_elt(2,1,yz);
Cov.set_elt(2,2,zz);
// solve for eigenvalues and eigenvectors.
// eigen values are sorted in ascending order,
// eigen vectors are sorted in accordance.
LinAlgTraits::eigen_symm_algo(Cov, eval, evec);
//store in monge_form_condition_numbers, pca eigenvalues are stored in descending order
monge_form_condition_numbers.pca_eigen_vals()[0] = eval[2];//implicit cast LAFT->LFT
LVector temp_vectn(evec.get_elt(0,2),evec.get_elt(1,2),evec.get_elt(2,2));
monge_form_condition_numbers.pca_eigen_vecs()[0] = temp_vectn;
monge_form_condition_numbers.pca_eigen_vals()[1] = eval[1];
LVector temp_vect1(evec.get_elt(0,1),evec.get_elt(1,1),evec.get_elt(2,1));
monge_form_condition_numbers.pca_eigen_vecs()[1] = temp_vect1;
monge_form_condition_numbers.pca_eigen_vals()[2] = eval[0];
LVector temp_vect2(evec.get_elt(0,0),evec.get_elt(1,0),evec.get_elt(2,0));
monge_form_condition_numbers.pca_eigen_vecs()[2] = temp_vect2;
switch_to_direct_orientation(monge_form_condition_numbers.pca_eigen_vecs()[0],
monge_form_condition_numbers.pca_eigen_vecs()[1],
monge_form_condition_numbers.pca_eigen_vecs()[2]);
//Store the change of basis W->F
const LVector* pca_vecs = monge_form_condition_numbers.pca_eigen_vecs();
Aff_transformation
change_basis (pca_vecs[0][0], pca_vecs[0][1], pca_vecs[0][2],
pca_vecs[1][0], pca_vecs[1][1], pca_vecs[1][2],
pca_vecs[2][0], pca_vecs[2][1], pca_vecs[2][2]);
this->change_world2fitting = change_basis;
/* //debug //test the old method, fitting basis is a permutation of the world basis */
/* const LVector* pca_vecs = monge_form_condition_numbers.pca_eigen_vecs(); */
/* const LVector n_pca = pca_vecs[2]; */
/* int index_max=0; */
/* x = std::fabs(n_pca[0]); y = std::fabs(n_pca[1]); z = std::fabs(n_pca[2]); */
/* if (x>y) if (x>z) index_max = 0; else index_max = 2; */
/* else if (y>z) index_max = 1; else index_max = 2; */
/* Aff_transformation change_basis; */
/* if (index_max == 0) change_basis = Aff_transformation(0,1,0, */
/* 0,0,1, */
/* 1,0,0); */
/* if (index_max == 1) change_basis = Aff_transformation(0,0,1, */
/* 1,0,0, */
/* 0,1,0); */
/* if (index_max == 2) change_basis = Aff_transformation(1,0,0, */
/* 0,1,0, */
/* 0,0,1); */
/* this->change_world2fitting = change_basis; */
//test the old method END
}
template < class DataKernel, class LocalKernel, class LinAlgTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, LinAlgTraits>::
fill_matrix(Range_Iterator begin, Range_Iterator end,
int d, LAMatrix &M, LFT* Z)
{
//origin of fitting coord system = first input data point
LPoint point0 = *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){//implicit cast DPoint->LPoint
LPoint cur_pt = transf_points(*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();
for (int k=0; k <= d; k++) for (int i=0; i<=k; i++)
M.set_elt(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 LinAlgTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, LinAlgTraits>::
solve_linear_system(LAMatrix &M, LFT* Z, Monge_form_condition_numbers& monge_form_condition_numbers)
{
LinAlgTraits::solve_ls_svd_algo(M, Z, monge_form_condition_numbers.cond_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);
}
template < class DataKernel, class LocalKernel, class LinAlgTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, LinAlgTraits>::
compute_Monge_basis(const LFT* A, Monge_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_pt() =
(this->translate_p0.inverse() *
this->change_world2fitting.inverse()) (orig_monge );
monge_form.n() = 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 LinAlgTraits for diagonalization of weingarten
LAMatrix W(2,2);
for (int i=0; i<=1; i++) for (int j=0; j<=1; j++)
W.set_elt(i, j, weingarten(i,j));
LFT* eval = (LFT*) malloc(2*sizeof(LFT));
LAMatrix evec(2,2);
//eval in increasing order
LinAlgTraits::eigen_symm_algo(W, eval, evec);
LVector d_min = evec.get_elt(0,0)*Y + evec.get_elt(1,0)*Z,
d_max = evec.get_elt(0,1)*Y + evec.get_elt(1,1)*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_pt() =
(this->translate_p0.inverse() *
this->change_world2fitting.inverse()) (orig_monge );
monge_form.d1() = this->change_world2fitting.inverse()(d_max);
monge_form.d2() = this->change_world2fitting.inverse()(d_min);
monge_form.n() = this->change_world2fitting.inverse()(normal);
monge_form.coefficients()[0] = eval[1];
monge_form.coefficients()[1] = eval[0];
}
//end else
}
template < class DataKernel, class LocalKernel, class LinAlgTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, LinAlgTraits>::
compute_Monge_coefficients(LFT* A, int dprime,
Monge_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] = b0;
monge_form.coefficients()[3] = b1;
monge_form.coefficients()[4] = b2;
monge_form.coefficients()[5] = 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] = c0;
monge_form.coefficients()[7] = c1;
monge_form.coefficients()[8] = c2;
monge_form.coefficients()[9] = c3;
monge_form.coefficients()[10] = c4;
}
}
template < class DataKernel, class LocalKernel, class LinAlgTraits>
void Monge_via_jet_fitting<DataKernel, LocalKernel, LinAlgTraits>::
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_
Reference in New Issue View Git Blame Copy Permalink