1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
|
#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;
}
|
