22:
152:) parallel redrawings of a structural framework is dual to the existence of an infinitesimal motion, one that preserves its edge lengths but not their orientations. Thus, a framework has one kind of motion if it has the other kind, but detecting the existence of a parallel redrawing may be easier than detecting the existence of an infinitesimal motion.
136:
to determine the existence of a morph for three or more slopes. Any parallel morph can be parameterized so that the each point moves with constant speed along a line. The graphs that remain planar throughout such a motion can be derived from
46:
128:, there may exist parallel redrawings that cannot be connected by a parallel morph. For two-dimensional planar drawings, with parallel edges required to preserve their orientation, a morph always exists when the
113:, a parallel drawing can be obtained by translating the plane of one of the polyhedron's face, and adjusting the positions of the vertices and edges that border that face. A polyhedron is said to be
121:, the cube and dodecahedron are not tight (because of the possibility of translating one face while keeping the others fixed), but the tetrahedron, octahedron, and icosahedron are tight.
94:
is another drawing of the same graph such that all edges of the second drawing are parallel to their corresponding edges in the first drawing. A
109:, and other modifications of the drawing that change it more locally. For instance, for graphs drawn as the vertices or edges of a
36:
464:
260:
454:
124:
In three dimensions, even for drawings where all edges are axis-parallel and the drawing forms the boundary of a
392:
Graph
Drawing: 13th International Symposium, GD 2005, Limerick, Ireland, September 12-14, 2005, Revised Papers
306:
Graph
Drawing: 13th International Symposium, GD 2005, Limerick, Ireland, September 12-14, 2005, Revised Papers
206:, IMS Lecture Notes Monogr. Ser., vol. 48, Inst. Math. Statist., Beachwood, OH, pp. 169–175,
117:
if its only parallel redrawings are similarities (combinations of translation and scaling); among the
459:
102:
71:
409:
365:
323:
283:
229:
184:
390:(2006), "Parallel-redrawing mechanisms, pseudo-triangulations and kinetic planar graphs",
8:
394:, Lecture Notes in Computer Science, vol. 3843, Berlin: Springer, pp. 421–433,
304:; Spriggs, Michael J. (2006), "Morphing planar graphs while preserving edge directions",
258:; Spriggs, Michael J. (2009), "Morphing polyhedra with parallel faces: counterexamples",
145:
75:
51:
188:
423:
369:
344:; Petrick, Mark; Spriggs, Michael (2013), "Morphing orthogonal planar graph drawings",
308:, Lecture Notes in Computer Science, vol. 3843, Berlin: Springer, pp. 13–24,
233:
207:
174:
106:
98:
of a graph is a continuous family of drawings, all parallel redrawings of each other.
373:
110:
237:
395:
353:
309:
269:
217:
32:
405:
361:
319:
279:
274:
225:
91:
87:
387:
221:
138:
118:
448:
337:
297:
251:
149:
83:
129:
427:
400:
341:
314:
301:
255:
125:
212:
179:
41:
357:
133:
21:
435:
Newsletter of the SIAM Activity Group on
Discrete Mathematics
336:
169:
Bolker, Ethan; Guillemin, Victor; Holm, Tara (2002),
202:Ward, Thomas (2006), "Mixing and tight polyhedra",
168:
296:
250:
446:
428:"Combinatorics and the rigidity of frameworks"
422:
399:
313:
273:
211:
178:
386:
447:
201:
15:
13:
14:
476:
20:
171:How is a graph like a manifold?
416:
380:
346:ACM Transactions on Algorithms
330:
290:
244:
195:
162:
1:
155:
275:10.1016/j.comgeo.2008.09.006
101:Parallel redrawings include
7:
86:with straight edges in the
10:
481:
222:10.1214/074921706000000194
204:Dynamics & stochastics
465:Mathematics of rigidity
35:, as no other articles
455:Geometric graph theory
261:Computational Geometry
90:or higher-dimensional
72:geometric graph theory
148:, the existence of (
139:pseudotriangulations
74:, and the theory of
424:Servatius, Brigitte
401:10.1007/11618058_38
189:2002math......6103B
146:structural rigidity
76:structural rigidity
352:(4): Art. 29, 24,
315:10.1007/11618058_2
132:is two, but it is
80:parallel redrawing
54:for suggestions.
44:to this page from
111:simple polyhedron
68:
67:
472:
439:
437:
432:
420:
414:
412:
403:
384:
378:
376:
334:
328:
326:
317:
294:
288:
286:
277:
248:
242:
240:
215:
199:
193:
191:
182:
166:
63:
60:
49:
47:related articles
24:
16:
480:
479:
475:
474:
473:
471:
470:
469:
445:
444:
443:
442:
430:
421:
417:
388:Streinu, Ileana
385:
381:
358:10.1145/2500118
335:
331:
295:
291:
249:
245:
200:
196:
167:
163:
158:
119:Platonic solids
92:Euclidean space
88:Euclidean plane
64:
58:
55:
45:
42:introduce links
25:
12:
11:
5:
478:
468:
467:
462:
457:
441:
440:
415:
379:
338:Biedl, Therese
329:
298:Biedl, Therese
289:
268:(5): 395–402,
252:Biedl, Therese
243:
194:
160:
159:
157:
154:
96:parallel morph
66:
65:
52:Find link tool
28:
26:
19:
9:
6:
4:
3:
2:
477:
466:
463:
461:
460:Graph drawing
458:
456:
453:
452:
450:
436:
429:
425:
419:
411:
407:
402:
397:
393:
389:
383:
375:
371:
367:
363:
359:
355:
351:
347:
343:
339:
333:
325:
321:
316:
311:
307:
303:
299:
293:
285:
281:
276:
271:
267:
263:
262:
257:
253:
247:
239:
235:
231:
227:
223:
219:
214:
209:
205:
198:
190:
186:
181:
176:
172:
165:
161:
153:
151:
150:infinitesimal
147:
142:
140:
135:
131:
127:
122:
120:
116:
112:
108:
104:
99:
97:
93:
89:
85:
84:graph drawing
81:
77:
73:
62:
53:
48:
43:
39:
38:
34:
29:This article
27:
23:
18:
17:
434:
418:
391:
382:
349:
345:
332:
305:
292:
265:
259:
246:
213:math/0503569
203:
197:
180:math/0206103
170:
164:
143:
130:slope number
123:
114:
103:translations
100:
95:
79:
69:
56:
30:
342:Lubiw, Anna
302:Lubiw, Anna
256:Lubiw, Anna
59:August 2023
449:Categories
156:References
126:polyhedron
50:; try the
37:link to it
374:122564585
40:. Please
426:(1993),
238:16378875
410:2244531
366:3119787
324:2244496
284:2512668
230:2306198
185:Bibcode
134:NP-hard
107:scaling
408:
372:
364:
322:
282:
236:
228:
33:orphan
31:is an
431:(PDF)
370:S2CID
234:S2CID
208:arXiv
175:arXiv
115:tight
82:of a
78:, a
396:doi
354:doi
310:doi
270:doi
218:doi
144:In
70:In
451::
433:,
406:MR
404:,
368:,
362:MR
360:,
348:,
340:;
320:MR
318:,
300:;
280:MR
278:,
266:42
264:,
254:;
232:,
226:MR
224:,
216:,
183:,
173:,
141:.
105:,
438:.
413:.
398::
377:.
356::
350:9
327:.
312::
287:.
272::
241:.
220::
210::
192:.
187::
177::
61:)
57:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
