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
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
|
/**************************************************************************
* *
* 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 "enumerate/validityconstraints.h"
#include "surface/normalsurface.h"
#include "maths/matrix.h"
#include "snappea/snappeatriangulation.h"
#include "triangulation/dim3.h"
namespace regina {
ValidityConstraints makeEmbeddedConstraints(
const Triangulation<3>& triangulation, NormalCoords coords) {
const NormalEncoding enc(coords);
if (enc.storesOctagons()) {
// At most one quad/oct per tetrahedron.
// Also at most one oct type overall.
ValidityConstraints ans(enc.block(), triangulation.size(), 1, 1);
if (enc.storesTriangles()) {
ans.addLocal({ 4, 5, 6, 7, 8, 9 });
ans.addGlobal({ 7, 8, 9 });
} else {
ans.addLocal({ 0, 1, 2, 3, 4, 5 });
ans.addGlobal({ 3, 4, 5 });
}
return ans;
} else {
// No octagon constraints.
ValidityConstraints ans(enc.block(), triangulation.size(), 1);
if (enc.storesTriangles())
ans.addLocal({ 4, 5, 6 });
else
ans.addLocal({ 0, 1, 2 });
return ans;
}
}
MatrixInt makeMatchingEquations(const Triangulation<3>& triangulation,
NormalCoords coords) {
switch (coords) {
case NormalCoords::Standard:
case NormalCoords::AlmostNormal:
{
const size_t block = (coords == NormalCoords::Standard ? 7 : 10);
const size_t nCoords = block * triangulation.size();
// Three equations per non-boundary triangle.
// F_boundary + 2 F_internal = 4 T
const size_t nEquations = 3 * (4 * triangulation.size() -
triangulation.countTriangles());
MatrixInt ans(nEquations, nCoords);
// Run through each internal triangle and add the corresponding
// three equations.
size_t row = 0;
for (Triangle<3>* t : triangulation.triangles()) {
if (! t->isBoundary()) {
size_t pos0 = block * t->embedding(0).tetrahedron()->index();
size_t pos1 = block * t->embedding(1).tetrahedron()->index();
Perm<4> perm0 = t->embedding(0).vertices();
Perm<4> perm1 = t->embedding(1).vertices();
for (int i=0; i<3; i++) {
// Triangles:
++ans.entry(row, pos0 + perm0[i]);
--ans.entry(row, pos1 + perm1[i]);
// Quads:
++ans.entry(row, pos0 + 4 +
quadSeparating[perm0[i]][perm0[3]]);
--ans.entry(row, pos1 + 4 +
quadSeparating[perm1[i]][perm1[3]]);
// Octagons:
if (coords == NormalCoords::AlmostNormal) {
++ans.entry(row, pos0 + 7 +
quadMeeting[perm0[i]][perm0[3]][0]);
--ans.entry(row, pos1 + 7 +
quadMeeting[perm1[i]][perm1[3]][0]);
++ans.entry(row, pos0 + 7 +
quadMeeting[perm0[i]][perm0[3]][1]);
--ans.entry(row, pos1 + 7 +
quadMeeting[perm1[i]][perm1[3]][1]);
}
++row;
}
}
}
return ans;
}
case NormalCoords::Quad:
case NormalCoords::QuadOct:
{
const size_t block = (coords == NormalCoords::Quad ? 3 : 6);
const size_t nCoords = block * triangulation.size();
// One equation per non-boundary edge.
size_t nEquations = triangulation.countEdges();
for (BoundaryComponent<3>* bc : triangulation.boundaryComponents())
nEquations -= bc->countEdges();
MatrixInt ans(nEquations, nCoords);
// Run through each internal edge and add the corresponding
// equation.
size_t row = 0;
for (Edge<3>* e : triangulation.edges()) {
if (! e->isBoundary()) {
for (const auto& emb : *e) {
size_t pos = block * emb.tetrahedron()->index();
Perm<4> perm = emb.vertices();
++ans.entry(row, pos +
quadSeparating[perm[0]][perm[2]]);
--ans.entry(row, pos +
quadSeparating[perm[0]][perm[3]]);
if (coords == NormalCoords::QuadOct) {
--ans.entry(row, pos + 3 +
quadSeparating[perm[0]][perm[2]]);
++ans.entry(row, pos + 3 +
quadSeparating[perm[0]][perm[3]]);
}
}
++row;
}
}
return ans;
}
case NormalCoords::QuadClosed:
case NormalCoords::QuadOctClosed:
{
// Enforce our basic preconditions.
if (! (triangulation.isOriented() && triangulation.isIdeal() &&
triangulation.countBoundaryComponents() == 1 &&
triangulation.countVertices() == 1 &&
triangulation.vertex(0)->linkType() ==
Vertex<3>::Link::Torus))
throw InvalidArgument(
"NormalCoords::QuadClosed and NormalCoords::QuadOctClosed "
"require an oriented ideal triangulation with "
"precisely one torus cusp and no other vertices");
// We will use SnapPea to build the additional constraint that
// enforces closed surfaces.
SnapPeaTriangulation snapPea(triangulation, false);
if (snapPea.isNull())
throw UnsolvedCase("SnapPea produced a null triangulation "
"when attempting to build the matching equations");
MatrixInt coeffs = snapPea.slopeEquations();
// Use a static_cast to ensure we are using the Triangulation<3>
// equality test. Otherwise C++20 complains about ambiguity.
if (static_cast<const Triangulation<3>&>(snapPea) != triangulation)
throw UnsolvedCase("SnapPea retriangulated "
"when attempting to build the matching equations");
const size_t block = (coords == NormalCoords::QuadClosed ? 3 : 6);
const size_t nCoords = block * triangulation.size();
// One equation per edge, plus two per ideal vertex.
// (This code is written a little more generically, in order to
// support multiple ideal vertices at some later date.)
const size_t nEquations = triangulation.countEdges() +
2 * triangulation.countBoundaryComponents();
MatrixInt ans(nEquations, nCoords);
size_t row = 0;
// Run through each internal edge and add the corresponding
// equation.
for (Edge<3>* e : triangulation.edges()) {
for (auto& emb : *e) {
size_t pos = block * emb.tetrahedron()->index();
Perm<4> perm = emb.vertices();
++ans.entry(row, pos +
quadSeparating[perm[0]][perm[2]]);
--ans.entry(row, pos +
quadSeparating[perm[0]][perm[3]]);
if (coords == NormalCoords::QuadOctClosed) {
--ans.entry(row, pos + 3 +
quadSeparating[perm[0]][perm[2]]);
++ans.entry(row, pos + 3 +
quadSeparating[perm[0]][perm[3]]);
}
}
++row;
}
// Run through each ideal vertex and add the corresponding meridian
// and longitude equations.
//
// Note: from preconditions, #vertices == #boundaries.
for (size_t i = 0; i < triangulation.countVertices(); ++i) {
// These two branches could be merged a bit better.
// Note: the cusp equations are always expressed in terms of
// quad coordinates.
if (coords == NormalCoords::QuadClosed) {
for (size_t j = 0; j < 3 * triangulation.size(); ++j) {
ans.entry(row, j) = coeffs.entry(2 * i, j);
ans.entry(row + 1, j) = coeffs.entry(2 * i + 1, j);
}
} else {
for (size_t j = 0; j < triangulation.size(); ++j) {
for (int k = 0; k < 3; ++k){
// Quad contributions
ans.entry(row, 6*j + k) =
coeffs.entry(2 * i, 3*j + k);
ans.entry(row + 1, 6*j + k) =
coeffs.entry(2 * i + 1, 3*j + k);
// Oct contributions; signs are opposite of those
// for the quads as with the edge equations.
ans.entry(row, 6*j + 3 + k) =
-coeffs.entry(2 * i, 3*j + k);
ans.entry(row + 1, 6*j + 3 + k) =
-coeffs.entry(2 * i + 1, 3*j + k);
}
}
}
row += 2;
}
return ans;
}
default:
throw InvalidArgument("makeMatchingEquations() was given "
"an invalid coordinate system");
}
}
} // namespace regina
|
