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/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2025, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* As an exception, when this program is distributed through (i) the *
* App Store by Apple Inc.; (ii) the Mac App Store by Apple Inc.; or *
* (iii) Google Play by Google Inc., then that store may impose any *
* digital rights management, device limits and/or redistribution *
* restrictions that are required by its terms of service. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public License *
* along with this program. If not, see <https://www.gnu.org/licenses/>. *
* *
**************************************************************************/
#include "triangulation/dim2.h"
#include "triangulation/example2.h"
#include <sstream>
namespace regina {
Triangulation<2> Example<2>::orientable(unsigned genus, unsigned punctures) {
if (genus == 0 && punctures == 0)
return sphere();
Triangulation<2> ans;
if (genus == 0) {
// Fact: punctures >= 1.
unsigned n = 3 * punctures - 2;
unsigned i;
ans.newTriangles(n);
for (i = 0; i < n - 1; ++i)
ans.triangle(i)->join(1, ans.triangle(i + 1), Perm<3>(1, 2));
ans.triangle(0)->join(0, ans.triangle(n - 1), Perm<3>(0, 1));
for (i = 1; i < punctures; ++i)
ans.triangle(3 * i - 2)->join(0, ans.triangle(3 * i),
Perm<3>(1, 2));
} else {
unsigned n = 4 * genus + 3 * punctures - 2;
unsigned i;
ans.newTriangles(n);
for (i = 0; i < n - 1; ++i)
ans.triangle(i)->join(1, ans.triangle(i + 1), Perm<3>(1, 2));
ans.triangle(0)->join(2, ans.triangle(n - 1), Perm<3>(0, 2));
ans.triangle(0)->join(0, ans.triangle(n - 1), Perm<3>(0, 1));
for (i = 1; i < genus; ++i) {
ans.triangle(4 * i - 3)->join(0, ans.triangle(4 * i - 1),
Perm<3>(1, 2));
ans.triangle(4 * i - 2)->join(0, ans.triangle(4 * i),
Perm<3>(1, 2));
}
for (i = 0; i < punctures; ++i)
ans.triangle(4 * genus + 3 * i - 3)->join(
0, ans.triangle(4 * genus + 3 * i - 1), Perm<3>(1, 2));
}
return ans;
}
Triangulation<2> Example<2>::nonOrientable(unsigned genus, unsigned punctures) {
if (genus == 0)
return orientable(0, punctures); // Just in case. *shrug*
if (genus == 1 && punctures == 0)
return rp2(); // Avoid 2-gons.
Triangulation<2> ans;
// The generic code below will create one internal vertex, and one for
// each puncture. This is minimal for zero punctures, but non-minimal
// otherwise. For now, we use a different triangulation for the
// once-punctured case so at least that gets to be minimal also;
// ideally these should be minimal for all values of punctures.
if (punctures == 1) {
// Thanks to Alex He for this code.
// Let g denote the given genus. We use g-1 "inner" triangles and g
// "outer" triangles, for a total of 2*g-1 triangles. We start by using
// the g-1 "inner" triangles to build a (g+1)-sided polygon P. We then
// form each of the g "outer" triangles into a one-triangle Mobius band,
// and attach the boundary of each of these Mobius bands to one of the
// sides of P. It is clear that the resulting surface is once-punctured
// and one-vertex, and has non-orientable genus g.
unsigned n = 2*genus - 1;
unsigned i;
ans.newTriangles(n);
// Form "outer" triangles into Mobius bands.
for ( i = genus - 1; i < n; ++i ) {
ans.triangle(i)->join(
0, ans.triangle(i), Perm<3>(1, 2, 0) );
}
// Glue everything together.
for ( i = 1; i < n; ++i ) {
ans.triangle(i)->join(
2, ans.triangle( (i-1)/2 ), Perm<3>( 2, i%2 ) );
}
} else {
unsigned n = 2 * genus + 3 * punctures - 2;
unsigned i;
ans.newTriangles(n);
for (i = 0; i < n - 1; ++i)
ans.triangle(i)->join(1, ans.triangle(i + 1), Perm<3>(1, 2));
ans.triangle(0)->join(2, ans.triangle(n - 1), Perm<3>(2, 0, 1));
for (i = 1; i < genus; ++i)
ans.triangle(2 * i - 2)->join(0, ans.triangle(2 * i - 1), Perm<3>());
for (i = 0; i < punctures; ++i)
ans.triangle(2 * genus + 3 * i - 2)->join(
0, ans.triangle(2 * genus + 3 * i), Perm<3>(1, 2));
}
return ans;
}
Triangulation<2> Example<2>::sphereOctahedron() {
Triangulation<2> ans;
auto [r, s, t, u, v, w, x, y] = ans.newTriangles<8>();
r->join(1, s, Perm<3>(1, 2));
s->join(1, t, Perm<3>(1, 2));
t->join(1, u, Perm<3>(1, 2));
u->join(1, r, Perm<3>(1, 2));
v->join(2, w, Perm<3>(1, 2));
w->join(2, x, Perm<3>(1, 2));
x->join(2, y, Perm<3>(1, 2));
y->join(2, v, Perm<3>(1, 2));
r->join(0, v, Perm<3>(1, 2));
s->join(0, w, Perm<3>(1, 2));
t->join(0, x, Perm<3>(1, 2));
u->join(0, y, Perm<3>(1, 2));
return ans;
}
Triangulation<2> Example<2>::rp2() {
Triangulation<2> ans;
auto [r, s] = ans.newTriangles<2>();
r->join(0, s, Perm<3>(1, 2));
r->join(1, s, Perm<3>());
r->join(2, s, Perm<3>());
return ans;
}
} // namespace regina
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