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manifold_tracing_custom_function.cpp

# Nerve_GIC

#include <iostream>
#include <gudhi/Coxeter_triangulation.h>
#include <gudhi/Functions/Function_Sm_in_Rd.h>
#include <gudhi/Implicit_manifold_intersection_oracle.h>
#include <gudhi/Manifold_tracing.h>
#include <gudhi/Coxeter_triangulation/Cell_complex/Cell_complex.h>
#include <gudhi/Functions/Linear_transformation.h>
#include <gudhi/IO/build_mesh_from_cell_complex.h>
#include <gudhi/IO/output_meshes_to_medit.h>
using namespace Gudhi::coxeter_triangulation;
/* A definition of a function that defines a 2d surface embedded in R^4, but that normally
* lives on a complex projective plane.
* In terms of harmonic coordinates [x:y:z] of points on the complex projective plane,
* the equation of the manifold is x^3*y + y^3*z + z^3*x = 0.
* The embedding consists of restricting the manifold to the affine subspace z = 1.
*/
struct Function_surface_on_CP2_in_R4 {
Eigen::VectorXd operator()(const Eigen::VectorXd& p) const {
// The real and imaginary parts of the variables x and y
double xr = p(0), xi = p(1), yr = p(2), yi = p(3);
Eigen::VectorXd result(cod_d());
// Squares and cubes of real and imaginary parts used in the computations
double xr2 = xr * xr, xi2 = xi * xi, yr2 = yr * yr, yi2 = yi * yi, xr3 = xr2 * xr, xi3 = xi2 * xi, yr3 = yr2 * yr,
yi3 = yi2 * yi;
// The first coordinate of the output is Re(x^3*y + y^3 + x)
result(0) = xr3 * yr - 3 * xr * xi2 * yr - 3 * xr2 * xi * yi + xi3 * yi + yr3 - 3 * yr * yi2 + xr;
// The second coordinate of the output is Im(x^3*y + y^3 + x)
result(1) = 3 * xr2 * xi * yr + xr3 * yi - 3 * xr * xi2 * yi - xi3 * yr + 3 * yr2 * yi - yi3 + xi;
return result;
}
std::size_t amb_d() const { return 4; };
std::size_t cod_d() const { return 2; };
Eigen::VectorXd seed() const {
Eigen::VectorXd result = Eigen::VectorXd::Zero(4);
return result;
}
Function_surface_on_CP2_in_R4() {}
};
int main(int argc, char** argv) {
// The function for the (non-compact) manifold
Function_surface_on_CP2_in_R4 fun;
// Seed of the function
Eigen::VectorXd seed = fun.seed();
// Creating the function that defines the boundary of a compact region on the manifold
double radius = 3.0;
Function_Sm_in_Rd fun_sph(radius, 3, seed);
// Defining the intersection oracle
auto oracle = make_oracle(fun, fun_sph);
// Define a Coxeter triangulation scaled by a factor lambda.
// The triangulation is translated by a random vector to avoid violating the genericity hypothesis.
double lambda = 0.2;
Coxeter_triangulation<> cox_tr(oracle.amb_d());
cox_tr.change_offset(Eigen::VectorXd::Random(oracle.amb_d()));
cox_tr.change_matrix(lambda * cox_tr.matrix());
// Manifold tracing algorithm
using MT = Manifold_tracing<Coxeter_triangulation<> >;
using Out_simplex_map = typename MT::Out_simplex_map;
std::vector<Eigen::VectorXd> seed_points(1, seed);
Out_simplex_map interior_simplex_map, boundary_simplex_map;
manifold_tracing_algorithm(seed_points, cox_tr, oracle, interior_simplex_map, boundary_simplex_map);
// Constructing the cell complex
std::size_t intr_d = oracle.amb_d() - oracle.cod_d();
Cell_complex<Out_simplex_map> cell_complex(intr_d);
cell_complex.construct_complex(interior_simplex_map, boundary_simplex_map);
// Output the cell complex to a file readable by medit
output_meshes_to_medit(3, "manifold_on_CP2_with_boundary",
build_mesh_from_cell_complex(cell_complex, Configuration(true, true, true, 1, 5, 3),
Configuration(true, true, true, 2, 13, 14)));
return 0;
}
Gudhi::coxeter_triangulation::Cell_complex
A class that constructs the cell complex from the output provided by the class Gudhi::coxeter_triangu...
Definition Cell_complex.h:38
Gudhi::coxeter_triangulation::Coxeter_triangulation
A class that stores Coxeter triangulation of type . This triangulation has the greatest simplex quali...
Definition Coxeter_triangulation.h:45
Gudhi::coxeter_triangulation::Manifold_tracing
A class that assembles methods for manifold tracing algorithm.
Definition Manifold_tracing.h:39
Gudhi::coxeter_triangulation::manifold_tracing_algorithm
void manifold_tracing_algorithm(const Point_range &seed_points, const Triangulation &triangulation, const Intersection_oracle &oracle, Out_simplex_map &out_simplex_map)
Static method for Manifold_tracing<Triangulation_>::manifold_tracing_algorithm that computes the set ...
Definition Manifold_tracing.h:223
Gudhi::coxeter_triangulation::output_meshes_to_medit
void output_meshes_to_medit(std::size_t amb_d, std::string file_name, const Meshes &... meshes)
Outputs a text file with specified meshes that can be visualized in Medit.
Definition output_meshes_to_medit.h:81
Gudhi::coxeter_triangulation::build_mesh_from_cell_complex
Mesh_medit build_mesh_from_cell_complex(const Cell_complex &cell_complex, Configuration i_configuration=Configuration(), Configuration b_configuration=Configuration())
Builds a Gudhi::coxeter_triangulation::Mesh_medit from a Gudhi::coxeter_triangulation::Cell_complex.
Definition build_mesh_from_cell_complex.h:122
Gudhi::coxeter_triangulation::make_oracle
Implicit_manifold_intersection_oracle< Function_, Domain_function_ > make_oracle(const Function_ &function, const Domain_function_ &domain_function)
Static constructor of an intersection oracle from a function with a domain.
Definition Implicit_manifold_intersection_oracle.h:240
Gudhi::coxeter_triangulation::Function_Sm_in_Rd
A class for the function that defines an m-dimensional implicit sphere embedded in the d-dimensional ...
Definition Function_Sm_in_Rd.h:27