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331:
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363:
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267:
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39:
30:
17:
111:
cases may be considered, but a well-formed holed-polygon must have no contact between exterior and interior boundaries, or between interior boundaries. Nondegenerate holes should have 3 or more sides, excluding internal point boundaries
128:
Area fill algorithms in computational lists the external boundary vertices can be listed in counter-clockwise order, and interior boundaries clockwise. This allows the interior area to be defined as
205:(vertices and edges) of a polyhedron with holed-faces is not a connected graph. Each set of connected edges will make a separate polyhedron if their edge-connected holes are replaced with faces.
456:
362:
394:
346:
330:
282:
476:
266:
378:
410:
314:
549:
298:
20:
Polygons with holes, with simply connected brown regions and interior boundaries, including degenerate cases of single vertices and edges, (a,b,f).
144:
can be transformed into an ordinary unicursal boundary path by adding (degenerate) connecting double-edges between boundaries, or by
568:
644:
618:
583:
578:, International Series of Monographs on Computer Science, vol. 3, Oxford University Press, pp. 125–145,
564:
446:
hole adds 2 vertices and 2 coinciding edges, where the two edges attach to two coplanar faces, as a
201:
can also be defined connecting a holed-face to a holed-faced on the opposite side (excavated). The
149:
108:
73:
with one external boundary and one or more interior boundaries (holes). Polygons with holes can be
526:
501:
598:
162:
145:
74:
48:
439:
435:
209:
8:
233:
198:
610:
534:
522:
639:
614:
579:
606:
81:
194:
179:
77:
into multiple polygons by adding new edges, so they are not frequently needed.
633:
157:
434:
hole, adding one vertex, and one edge, and can attached to a degenerate
202:
16:
183:
161:
Example conversion of a single-holed polygon by connecting edges, or
38:
447:
62:
29:
463:
431:
113:
70:
191:
504:— largest cube that can pass through a unit cube's hole.
483:
443:
368:
Toroid (genus 1) with one 2-holed-face, and one 1-holed-face.
117:
55:-sided boundaries with the same center, but different radius.
187:
537:
190:
with a smaller cube externally placed on one of its
197:(augmented), with their common surfaces removed. A
543:
272:(genus 0) with two 1-holed-faces (top and bottom).
601:(2000), "Art Gallery and Illumination Problems",
135:
631:
442:hole with zero radius. A face with a degenerate
563:
521:
430:A face with a point hole is considered a
336:Toroid (genus 5), with six 1-holed faces.
527:"IX.4: Polyhedra with ring-shaped faces"
462:(genus 1) with two (degenerate point or
416:Toroid (genus 1) with two 1-holed-faces.
400:Toroid (genus 1) with two 1-holed-faces.
352:Toroid (genus 2) with two 2-holed-faces.
288:Toroid (genus 1) with two 1-holed-faces.
156:
15:
597:
123:
632:
553:, Methuen & Co., pp. 144–145
576:Art Gallery Theorems and Algorithms
531:An Introduction To The Geometry Of
152:it into 2 or more simple polygons.
103:
13:
603:Handbook of Computational Geometry
320:(genus 0), with six 1-holed faces.
80:An ordinary polygon can be called
14:
656:
611:10.1016/b978-044482537-7/50023-1
475:
455:
409:
393:
384:(genus 0) with one 2-holed-face.
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361:
345:
329:
313:
304:(genus 0) with one 1-holed-face.
297:
281:
265:
212:of hole-faced polyhedron is χ =
84:, while a polygon-with-holes is
37:
28:
605:, Elsevier, pp. 973–1027,
482:(genus 1) with two (degenerate
170:
591:
557:
515:
426:Examples with degenerate holes
136:Conversion to ordinary polygon
116:) and single edge boundaries (
1:
508:
69:is an area-connected planar
7:
495:
51:can be approximated by two
10:
661:
645:Euclidean plane geometry
545:
418:V=32, E=48, F=18, H=2.
402:V=24, E=36, F=14, H=2.
386:V=24, E=36, F=16, H=2.
370:V=24, E=36, F=15, H=3.
354:V=24, E=36, F=14, H=4.
338:V=40, E=72, F=30, H=6.
322:V=32, E=36, F=12, H=6.
306:V=16, E=24, F=11, H=1.
290:V=16, E=24, F=10, H=2.
165:
21:
546:
488:V=12, E=18, F=8, H=2.
468:V=10, E=15, F=7, H=2.
274:V=16, E=20, F=8, H=2.
160:
19:
535:
523:Somerville, D. M. Y.
502:Prince Rupert's cube
436:monogonal hosohedron
255:holes in the faces.
210:Euler characteristic
124:Boundary orientation
199:toroidal polyhedron
176:Polygons with holes
142:polygons with holes
569:"Chapter 5: Holes"
541:
166:
92:-holed-polygon is
86:multiply-connected
67:polygon with holes
22:
544:{\displaystyle N}
652:
624:
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595:
589:
588:
573:
565:O'Rourke, Joseph
561:
555:
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548:
547:
542:
519:
486:) 1-holed-faces.
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466:) 1-holed-faces.
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365:
349:
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317:
301:
285:
269:
104:Degenerate holes
82:simply-connected
41:
32:
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178:can be seen as
173:
138:
126:
106:
59:
58:
57:
56:
44:
43:
42:
34:
33:
12:
11:
5:
658:
648:
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642:
626:
625:
619:
599:Urrutia, Jorge
590:
584:
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540:
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172:
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137:
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132:of each edge.
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46:
45:
36:
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27:
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9:
6:
4:
3:
2:
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622:
620:9780444825377
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585:0-19-503965-3
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453:
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451:
449:
445:
441:
438:hole, like a
437:
433:
425:
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375:
364:
359:
348:
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206:
204:
200:
196:
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185:
181:
177:
164:
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155:
154:
153:
151:
150:triangulating
147:
143:
133:
131:
121:
119:
115:
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99:
95:
91:
87:
83:
78:
76:
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68:
64:
54:
50:
40:
31:
18:
602:
593:
575:
559:
530:
517:
429:
252:
248:
244:
240:
236:
229:
225:
221:
217:
213:
207:
175:
174:
171:In polyhedra
141:
139:
129:
127:
107:
97:
93:
89:
85:
79:
66:
60:
52:
490:2-connected
470:2-connected
420:2-connected
404:2-connected
388:3-connected
372:3-connected
356:3-connected
340:2-connected
324:7-connected
308:2-connected
292:2-connected
276:3-connected
251:faces, and
634:Categories
551:Dimensions
509:References
243:vertices,
203:1-skeleton
163:dissection
146:dissecting
109:Degenerate
432:monogonal
186:, like a
184:polyhedra
98:connected
75:dissected
640:Polygons
567:(1987),
525:(1929),
496:See also
448:dihedron
440:cylinder
259:Examples
114:monogons
63:geometry
464:monogon
247:edges,
71:polygon
49:annulus
617:
582:
450:hole.
239:, for
224:= 2(1-
192:square
118:digons
572:(PDF)
484:digon
444:digon
234:genus
195:faces
180:faces
88:. An
615:ISBN
580:ISBN
228:) +
208:The
188:cube
130:left
65:, a
607:doi
182:in
148:or
120:).
61:In
47:An
636::
613:,
574:,
529:,
232:,
220:+
216:-
140:A
100:.
609::
539:N
253:H
249:F
245:E
241:V
237:g
230:H
226:g
222:F
218:E
214:V
112:(
96:-
94:H
90:H
53:n
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