void draw_empty_grid(INT q, INT type, INT param_a, INT xmax, INT ymax, INT verbose_level)
{
{
BYTE fname[1000];
{
mp_graphics *G;
sprintf(fname, "grid_%ld_%ld", q, type);
draw_beginning(fname, G, xmax, ymax, verbose_level);
draw_grid_(*G, xmax, ymax, q, type, param_a);
draw_end(fname, G, xmax, ymax, verbose_level);
}
prepare_latex_simple(fname, verbose_level);
}
}
void draw_empty_grid(INT q, INT f_include_line_at_infinity, INT verbose_level)
{
{
BYTE fname[1000];
{
INT xmax = 300;
INT ymax = 300;
mp_graphics *G;
sprintf(fname, "grid_%ld_%ld", q, f_include_line_at_infinity);
draw_beginning(fname, G, xmax, ymax, verbose_level);
draw_grid_(*G, q, f_include_line_at_infinity, verbose_level);
draw_end(fname, G, xmax, ymax, verbose_level);
}
prepare_latex_simple(fname, verbose_level);
}
}
void conic(INT q, INT *six_coeffs, INT xmax, INT ymax, INT f_do_stabilizer, INT verbose_level)
{
const BYTE *override_poly = NULL;
finite_field F;
projective_space *P;
action *A;
INT n = 3;
//INT f_with_group = TRUE;
INT v[3];
//INT w[3];
//F.init(q), verbose_level - 2);
F.init_override_polynomial(q, override_poly, verbose_level);
P = new projective_space;
cout << "before P->init" << endl;
P->init(n - 1, &F,
FALSE /* f_init_incidence_structure */,
verbose_level/*MINIMUM(2, verbose_level)*/);
cout << "after P->init" << endl;
A = new action;
A->init_general_linear_group(n, &F, FALSE /* f_semilinear */, TRUE /* f_basis */, verbose_level - 2);
INT variety[100];
INT variety_size = 0;
INT a, i, j, h;
for (i = 0; i < P->N_points; i++) {
P->unrank_point(v, i);
a = F.evaluate_conic_form(six_coeffs, v);
cout << i << " : ";
INT_vec_print(cout, v, 3);
cout << " : " << a << endl;
if (a == 0) {
variety[variety_size++] = i;
}
}
cout << "the size of the variety is " << variety_size << endl;
cout << "the variety is:" << endl;
INT_vec_print(cout, variety, variety_size);
cout << endl;
for (i = 0; i < variety_size; i++) {
P->unrank_point(v, variety[i]);
a = F.evaluate_conic_form(six_coeffs, v);
cout << i << " : ";
INT_vec_print(cout, v, 3);
cout << " : " << a << endl;
}
#if 0
INT pts[] = {2, 22, 6, 18, 10};
find_collinear_triple(P, pts, 5);
INT five_pts[] = {3, 6, 7, 28, 30};
#endif
//INT six_coeffs[6];
INT five_pts[5];
five_pts[0] = variety[0];
five_pts[1] = variety[1];
five_pts[2] = variety[2];
five_pts[3] = variety[3];
five_pts[4] = variety[4];
P->determine_conic_in_plane(five_pts, 5, six_coeffs, verbose_level);
INT points[1000];
INT tangents[1000];
INT *exterior_points;
INT *secants;
INT nb_secants, nb_exterior_points;
INT nb_points;
//INT v[3];
P->conic_points(five_pts, six_coeffs, points, nb_points, verbose_level);
cout << "the " << nb_points << " conic points are: ";
INT_vec_print(cout, points, nb_points);
cout << endl;
for (i = 0; i < nb_points; i++) {
P->unrank_point(v, points[i]);
cout << i << " : " << points[i] << " : ";
INT_vec_print(cout, v, 3);
cout << endl;
}
strong_generators *Aut_gens;
if (f_do_stabilizer) {
// computing stabilizer:
cout << "computing stabilizer" << endl;
set_stabilizer_compute STAB;
sims *Stab;
INT nb_backtrack_nodes;
cout << "computing stabilizer of conic:" << endl;
STAB.init(A, variety, variety_size /* points, nb_points*/ , verbose_level);
STAB.compute_set_stabilizer(t0, nb_backtrack_nodes, Aut_gens, verbose_level + 10);
longinteger_object go, go2;
Stab = Aut_gens->create_sims(verbose_level - 1);
Stab->group_order(go);
cout << "computing stabilizer of conic done, found a group of order " << go << endl;
delete Stab;
}
else {
Aut_gens = NULL;
}
{
BYTE fname[1000];
{
mp_graphics *G;
INT f_include_line_at_infinity = TRUE;
sprintf(fname, "conic_%ld", q);
draw_beginning(fname, G, xmax, ymax, verbose_level);
//variety_size = 0;
draw_grid_(*G, q, f_include_line_at_infinity, verbose_level);
draw_points(*G, P, /*variety + 3, 5*/ variety, variety_size /*points, nb_points*/, verbose_level);
draw_end(fname, G, xmax, ymax, verbose_level);
}
if (f_do_stabilizer) {
prepare_latex(fname, P, points, nb_points, Aut_gens, verbose_level);
}
}
delete Aut_gens;
return;
P->find_tangent_lines_to_conic(six_coeffs,
points, nb_points,
tangents, verbose_level);
cout << "the " << nb_points << " tangent lines are: ";
INT_vec_print(cout, tangents, nb_points);
cout << endl;
nb_exterior_points = (nb_points * (nb_points - 1)) >> 1;
nb_secants = nb_exterior_points;
exterior_points = NEW_INT(nb_exterior_points);
h = 0;
for (i = 0; i < nb_points; i++) {
for (j = i + 1; j < nb_points; j++) {
exterior_points[h++] = P->Line_intersection[
tangents[i] * P->N_lines + tangents[j]];
}
}
INT_vec_heapsort(exterior_points, nb_exterior_points);
cout << "the " << nb_exterior_points << " exterior points are: ";
INT_vec_print(cout, exterior_points, nb_exterior_points);
cout << endl;
secants = NEW_INT(nb_secants);
h = 0;
for (i = 0; i < nb_points; i++) {
for (j = i + 1; j < nb_points; j++) {
secants[h++] = P->Line_through_two_points[
points[i] * P->N_points + points[j]];
}
}
INT_vec_heapsort(secants, nb_secants);
cout << "the " << nb_secants << " secants are: ";
INT_vec_print(cout, secants, nb_secants);
cout << endl;
INT *external_lines;
INT nb_external_lines;
external_lines = NEW_INT(P->N_lines);
for (i = 0; i < P->N_lines; i++) {
external_lines[i] = i;
}
nb_external_lines = P->N_lines;
set_delete_elements(external_lines, nb_external_lines, tangents, nb_points);
set_delete_elements(external_lines, nb_external_lines, secants, nb_secants);
cout << "the " << nb_external_lines << " external lines are: ";
INT_vec_print(cout, external_lines, nb_external_lines);
cout << endl;
INT *adjacency;
INT idx;
adjacency = NEW_INT(nb_external_lines * nb_external_lines);
for (i = 0; i < nb_external_lines; i++) {
adjacency[i * nb_external_lines + i] = 0;
for (j = i + 1; j < nb_external_lines; j++) {
a = P->Line_intersection[
external_lines[i] * P->N_lines + external_lines[j]];
if (INT_vec_search(exterior_points, nb_exterior_points, a, idx)) {
adjacency[i * nb_external_lines + j] = 1;
adjacency[j * nb_external_lines + i] = 1;
}
else {
adjacency[i * nb_external_lines + j] = 0;
adjacency[j * nb_external_lines + i] = 0;
}
}
}
cout << "adjacency matrix:" << endl;
print_integer_matrix_width(cout, adjacency,
nb_external_lines, nb_external_lines, nb_external_lines,
1);
INT *Edges1;
INT *Edges2;
INT *Incidence;
INT nb_e = 0;
for (i = 0; i < nb_external_lines; i++) {
for (j = i + 1; j < nb_external_lines; j++) {
if (adjacency[i * nb_external_lines + j])
nb_e++;
}
}
Edges1 = NEW_INT(nb_e);
Edges2 = NEW_INT(nb_e);
Incidence = NEW_INT(nb_external_lines * nb_e);
for (i = 0; i < nb_external_lines * nb_e; i++) {
Incidence[i] = 0;
}
nb_e = 0;
for (i = 0; i < nb_external_lines; i++) {
for (j = i + 1; j < nb_external_lines; j++) {
if (adjacency[i * nb_external_lines + j]) {
Edges1[nb_e] = i;
Edges2[nb_e] = j;
nb_e ++;
}
}
}
for (j = 0; j < nb_e; j++) {
Incidence[Edges1[j] * nb_e + j] = 1;
Incidence[Edges2[j] * nb_e + j] = 1;
}
BYTE fname[1000];
sprintf(fname, "ext_lines_%ld.inc", P->F->q);
{
ofstream f(fname);
f << nb_external_lines << " " << nb_e << " " << nb_e * 2 << endl;
for (i = 0; i < nb_external_lines * nb_e; i++) {
if (Incidence[i]) {
f << i << " ";
}
}
f << endl;
f << -1 << endl;
}
cout << "written file " << fname << " of size " << file_size(fname) << endl;
INT colors[] = {0,1};
INT C = 2;
INT *Pijk;
Pijk = NEW_INT(C * C * C);
if (analyze_color_graph(C, colors, nb_external_lines, adjacency, Pijk, verbose_level)) {
cout << "is association scheme" << endl;
}
else {
cout << "not an association scheme" << endl;
}
//exit(1);
{
set_stabilizer_compute STAB;
strong_generators *Aut_gens;
sims *Stab;
//vector_ge gens;
//INT *tl;
INT nb_backtrack_nodes;
cout << "computing stabilizer of conic:" << endl;
STAB.init(A, points, nb_points, verbose_level);
STAB.compute_set_stabilizer(t0, nb_backtrack_nodes, Aut_gens, verbose_level + - 2);
Stab = Aut_gens->create_sims(verbose_level - 1);
longinteger_object go, go2;
Stab->group_order(go);
cout << "computing stabilizer of conic done, found a group of order " << go << endl;
action *A2;
action A2r;
INT f_induce_action = TRUE;
projective_space_init_line_action(P, A, A2, verbose_level);
A2r.induced_action_by_restriction(*A2,
f_induce_action, Stab,
nb_external_lines, external_lines, verbose_level);
A2r.group_order(go2);
cout << "induced action on external lines has order " << go2 << endl;
for (i = 0; i < Aut_gens->gens->len; i++) {
A2r.element_print_quick(Aut_gens->gens->ith(i), cout);
A2->element_print_as_permutation(Aut_gens->gens->ith(i), cout);
A2r.element_print_as_permutation(Aut_gens->gens->ith(i), cout);
}
delete Aut_gens;
delete Stab;
delete A2;
}
}
