ACM Home Page
Please provide us with feedback. Feedback
Voronoi diagrams—a survey of a fundamental geometric data structure
Full text PdfPdf (5.18 MB)
Source ACM Computing Surveys (CSUR) archive
Volume 23 ,  Issue 3  (September 1991) table of contents
Pages: 345 - 405  
Year of Publication: 1991
ISSN:0360-0300
Author
Franz Aurenhammer  Technische Univ. Graz, Schiebbstattgasse, Austria
Publisher
ACM  New York, NY, USA
Bibliometrics
Downloads (6 Weeks): 154,   Downloads (12 Months): 1110,   Citation Count: 179
Additional Information:

references   cited by   index terms   collaborative colleagues  

Tools and Actions: Request Permissions Request Permissions    Review this Article  
DOI Bookmark: Use this link to bookmark this Article: http://doi.acm.org/10.1145/116873.116880
What is a DOI?

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

1
2
3
 
4
5
 
6
AGGARWAL, A., CHAZELLE, B., GUIBAS, L. J., O'DUNLAING, C., AND YAP, C. K. 1988. Parallel computational geometry. Algorithmica 3, 293-327.
 
7
AHUJA, N. 10~2. Dot pattorn processing using Voronoi polygons as neighborhoods. IEEE Trans. Patt. Anal. Mach. Int. PAMI-4, 336-343.
 
8
AKL, S. 1983 A note on Euclidean matchings, triangulations, and spanning trees. J. Comb. Inf. Syst. Sci. 8, 169-174.
 
9
ALEXANDERSON, G. L., AND WETZEL, J. E. 1978. Simple partition of space. Math. Magazine 51, 220-225.
 
10
ALT, H., AND YAP, C. K. 1990. Algorithmic aspects of motion planning: A tutorial (part 2). Algorithms Rev. 1, 61-77.
11
 
12
ARNOLD, D. B., AND MILNE, W. J. 1984. The use of Voronoi tessellations in processing soil survey results. IEEE Comput. Graph. Appl. 4, 22-30.
 
13
ARONOV, B. 1989. On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica 4, 109-140.
14
15
 
16
ASANO, T., AND ASANO, T. 1986. Voronoi diagram for points in a simple polygon. In Perspectives in Computing 15, D. S. Johnson, T. Nishizeki, A. Nozaki, H. S. Will Eds. Academic Press, New York, pp. 51-64.
17
 
18
ASH, P. F., AND BOLKER, E. D. 1985. Recognizing Dirichlet tessellations. Geometriae Dedicata 19, 175-206.
 
19
ASH, P. F., AND BOLKER, E. D. 1986. Generalized Dirichlet tessellations. Geometrme Dedicata 20, 209-243.
 
20
ASH, P. F., BOLKER, E. D., CRAPO, H., AND WHITE- LEY, W. 1988. Convex polyhedra, Dirichlet tessellations, and spider webs. In Shaping Space: A Polyhedral Approach, M. Senechal and G. Fleck, Ed~. Birkh/iuser, Boston, pp. 231-250.
 
21
AUGENBAUM, J. M., AND PESKIN, C. S. 1985. On the construction of the Voronoi mesh on a sphere. J. Comput. Phys. 59, 177-192.
 
22
 
23
AURENHAMMER, F. 1987b. A criterion for the affine equivalence of cell complexes in R d and convex polyhedra in Rd+ 1. Discrete Comput. Geometry 2, 49-64.
 
24
 
25
 
26
AURENHAMMER, F. 1988b. Linear combinations from power domains. Geometriae Dedicata 28, 45-52.
 
27
 
28
 
29
AURENHAMMER, F., AND EDELSBRUNNER, $. 1984. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recognition 17, 251-257.
 
30
AURENHAMMER, F., AND IMAI, H. 1988. Geometric relations among Voronoi diagrams. Geometriae Dedicata 27, 65-75.
31
 
32
AURENHAMMER, F., AND STOCKL, G. 1988. On the peeper's Voronoi diagram. Rep. 264, IIG-TU Graz, Austria.
 
33
AURENHAMMER, F., ST(SCKL, G., AND WELZL, E. 1991. The post-office problem for fuzzy point sets. Rep. 91-07, FU Berlin, Germany.
 
34
AVIS, D., AND BHATTACHARYA, B. K. 1983. Algorithms for computing d-dimenslonal Voronoi diagrams and their duals. Adv. Comput. Res. 1,159-180.
 
35
BABENKO, V. F. 1977. On the optimal cubature ormulae on certain classes of continuous functions. Analys~s Math. 3, 3-9.
 
36
BADDELEY, A. 1977. A fourth note on recent research in geometrical probability. Adv. Appl. Probab. 9, 824-860.
37
 
38
BENTLEY, J. L., AND MAURER, H. A. 1979. A note on Euclidean near neighboring searching in the plane. Inf. Process. Lett. 8, 133-136.
39
 
40
BERN, M. W., EPPSTEIN, D., AND GILBERT, J. 1990. Provably good mesh generation. Manuscript, XEROX Palo Alto Research Center, Palo Alto, Calif.
 
41
BESAG, J. 1974. Spatial interaction and the statistical analysis of lattice systems. J. Roy. Statist. Soc. B 36, 192-236.
 
42
BIEBERBACH, L. 1912. 'l)ber die Bewegungsgruppen der euklidischen Riiume I, II. Math. Ann. 72, 400-412.
 
43
BLUM, H. 1967. A transformation for extracting new descriptors of shape. In Proceedings of the Symposium on Models for the Perception of Speech and Visual Form. Weiant Whaten- Dunn, Ed. MIT Press, Cambridge, Mass., pp. 362-380.
 
44
BLUM, H. 1973. Biological shape and visual science (Part I). J. Theol'. Biol. 38, 205-287.
 
45
46
 
47
BOOTS, B. N. 1979. Weighting Thiessen polygons. Econ. Geography, 248-259.
 
48
BooTs, B. N. 1986. Voronoi (Thiessen) polygons. In CATMOG 45, Geo Books, Norwich, Conn.
 
49
BOWYER, A. 1981. Computing Dirichlet tessellations. Computer J. 24, 162-166.
 
50
BRASSEL, K. E., AND REIF, D. 1979. A procedure to generate Thiessen polygons. Geograph. Anal. 11,289-303.
51
 
52
BRONDSTED, A. 1983. An Introduction to Convex Polytopes. Springer, New York, Heidelberg, Berlin.
 
53
BRosTow, W., AND SICOTTE, Y. 1975. Coordination number in liquid argon. Physica A 80, 513-522.
 
54
BROSTOW, W., DUSSAULT, J. P., AND FOX, B. L. 1978. Construction of Voronoi polyhedra. J. Comput. Phys. 29, 81-92.
 
55
BROWN, K. Q. 1979. Voronoi diagrams from convex hulls. Inf. Process. Lett. 9,223-228.
 
56
 
57
BRUMBERGER, H. AND GOODISMAN. 1983. Voronoi cells: An interesting and potentially useful cell model for interpreting the small angle scattering of catalysts. J. Appl. Crystallogr. 16, 83-88.
 
58
CALABI, L., AND HARTNETT, W.E. 1968. Shape recognition, prairie fires, convex deficiencies, and skeletons. Am. Math. Monthly 75, 335-342.
 
59
 
60
 
61
 
62
 
63
 
64
65
 
66
CHrR~TON, D., AND TARJAN, R. E. 1976. Finding minimum spanning trees. SIAM J. Comput. 5, 724-742.
67
 
68
CHEW, L. P. 1989a. Constrained Delaunay triangulations. Algorithmica 4, 97-108.
 
69
HEW, L. P. 1989b. Guaranteed-quality triangular meshes. Rep. TR 89-983, Dept. Comput. Sci., Cornell Univ., Ithaca, N.Y.
70
71
 
72
 
73
CI~ISTOFIDES, N. 1976. Worst-case analysis of a new heuristic for the traveling salesman problem. Symposium on Algorithms and Complexity, Carnegie-Mellon Univ., Penn.
 
74
CLARKSON, K. L. 1987. New applications of random sampling to computational geometry. Discrete Comput. Geom. 2, 195-222.
 
75
CLARKSON, K. L. 1989. An algorithm for geometric minimum spanning trees requiring nearly linear expected time. Algorithmica4, 461-468.
76
 
77
 
78
 
79
CONWAY, J. M., AND SLOAN~, N. J. A. 1982. Voronoi regions of lattices, second moments of polytopes, and quantization. IEEE Trans. Inf. Theory IT-28, 211-226.
 
80
CRAIg, L. K. 1978. The Monte Carlo generation of random polygons. Comput. Geosci. 4, 131-141.
 
81
CRAPO, H. 1979. Structural rigidity. Struct. Topol. 1, 13-45.
 
82
CRuz ORIVE, L.-M. 1979. Distortion of certain Voronoi tessellations when one particle moves. J. Appl. Probab. 16, 95-103.
 
83
DAVID, E. E., A~D DAVID, C. W. 1982. Voronoi polyhedra as a tool for studying solvation structure. J. Chem. Phys. 76, 4611-4614.
 
84
DE FLORIANI, L., FALCIDIENO, B., PIENOVI, C., AND NAG~, G. 1988. On sorting triangles in a Delaunay tessellation. Rep. Inst. Mat. Appl., Consiglio Nazionale della Richerche, Genova, Italy.
 
85
DEHNE, F., AND KLEIN, R. 1987. An optimal algorithm for computing the Voronoi diagram on a cone. Rep. SCS-TR-122, School Comput. Sci., Carleton Univ., Ottawa, Canada.
86
 
87
DELAUNAY, B. N. 1932. Neue Darstellung der geometrischen Kristallographie. Z. Kristallograph. 84,109-149.
 
88
DELAUNAY, B. N. 1934. Sur la sphere vide. Bull. Acad. Science USSR VII: Class. Sci. Math., 793-800.
 
89
DELAUNAY, B. N. 1963. Theorie der regul~iren Dirichletschen Zerlegungen des n-dimensionalen euklidischen Raumes. Schrifienre~he Int. Math. Dtsch. Akad. Wiss. Berlin 13, 27-31.
 
90
DELAUNAY, B. N., DOLBILIN, N. P., AND STOGRIN, M. I. 1978. Combinatorial and metric theory of planigons. Trudy Mat. Inst. Steklov 148, 109-140.
 
91
DEWDNEY, A. K. 1977. Complexity of nearest neighbour searching in three and higher dimensions. Rep. 28, Univ. of Western Ontario, London, Ontario.
 
92
DEWDNEY, A. K., AND VRANCH, J. K. 1977. A convex partition of R3 with applications to Crum's problem and Knuth's post-office problem. Util. Math. 12, 193-199.
 
93
 
94
95
 
96
DIRICHLET, G. L. 1850. Uber die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Re~ne u. Angew. Math. 40, 209-227.
 
97
DOBKIN, D. P., AND LASZLO, M. J. 1989. PrimL tives for the manipulation of three-dimensional subdivisions. Algorithmica 4, 3-32.
 
98
DOBKIN, D. P., AND LmTON, R. J. 1976. Multidimensional searching problems. SIAM J. Comput. 5, 181-186.
 
99
 
100
 
101
DWYER, R. A. 1987. A faster divide-and-conquer algorithm for constructing Delaunay triangulations. Algorithmica 2, 137-151.
102
103
 
104
 
105
106
 
107
EDELSBRUNNER, H., AND MAURER, H. A. 1985. Finding extreme points in three dimensions and solving the post-office problem in the plane. Inf. Process. Lett. 21, 39-47.
 
108
 
109
EDELSBRUNNER, H., AND WELZL, E. 1985. On the number of line separations of a finite set in the plane. J. Combin. Theory Ser. A, 15-29.
 
110
EDELSBRUNNER, H., GUIBAS, L. J., AND S~A~m, M. 1989. The upper envelope of piecewise linear functions: algorithms and applications. Discrete Comput. Geom. 4, 311-336.
 
111
 
112
EDELSBRUNNER, H., KIRKPATRICK, D. G., AND SEIDEL, R. 1983. On the shape of a set of points in the plane. IEEE Trans. Inf. Theory IT-29,551-559.
 
113
 
114
 
115
EHRLICH, P. E., AND IM HOF, H. C. 1979. Dirichlet regions in manifolds without conjugate points. Comment. Math. Helvet. 54,642-658.
 
116
EISELT, H. A., AND PEDERZOLI, G. 1986. Voronoi diagrams and their use: A survey. Part I: Theory; Part II: Applications. In S. Goyal, Ed. Proceedings of the ASAC 7, 2. pp. 98-115.
 
117
ENGEL, P. 1981. 'lJber Wirkungsbereichsteilungen mit kubischer Symmetrie. Z. Kristallograph. 154, 199-215.
 
118
 
119
FAIRFIELD, J. 1979. Contoured shape generation: Forms that people see in dot patterns. In Proceedings of the IEEE Conference on Systems Man Cybernetics. pp. 60-64.
 
120
FAIRFIELD, J. 1983. Segmenting dot patterns by Voronoi diagram concavity. IEEE Trans. Part. Anal. Mach. Int. PAMI-5, 104-110.
 
121
FEDEROFF, E. S. 1885. Elemente der Lehre yon den Figuren. Verh. Russ Min. Ges. St. Petersburg 21, 1-279.
122
 
123
FINNEY, J. L. 1979. Procedure for the construction of Voronoi polyhedra. Note. J. Comput. Phys. 32, 137-143.
 
124
 
125
FORTUNE, S. 1987. A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 153-174.
 
126
FORTUNE, S. 1988. Sorting helps for Voronoi diagrams. Manuscript, ATT Bell Lab., Murray Hill, N.J.
 
127
FRANK, F. C., AND KASPER, J. S. 1958. Complex alloy structures regarded as sphere packings. Acta Crystallogr. 11,184-190.
 
128
GABRIEL, K. R, AND SOKAL, R. R. 1969. A new statistical approach to geographic variation analysis. Syst. Zoology 18, 259-278.
 
129
GAMBINI, R. 1966. A computer program for calculating lines of equilibrium between multiple centers of attraction. Manuscript, Center of Regional Studies, Univ. of Kansas, Lawrence, Kan.
130
 
131
GAUSS, C. F. 1840. Recursion der 'Untersuchungen fiber die Eigenschaften der positiven tern~iren quadratischen Formen von Ludwig August Seeber. J. Reine Angew. Math. 20, 312-320.
 
132
GILBERT, E. N. 1962. Random subdivisions of space into crystals. Ann. Math. Star. 33, 958-972.
 
133
 
134
GOWDA, I. G., KIRKPATRICK, D. G., LEE, D. T., AND NAAMAD, A. 1983. Dynamic Voronoi diagrams. IEEE Trans. Inf Theory IT-29, 724-731.
 
135
GREEN, P. J., AND SIBSON, R. 1977. Computing Dirichlet tesselations in the plane. Comput. J. 21, 168-173.
 
136
GRUBER, P. M., AND LEKKERKERKER, C. G. 1988. Geometry of Numbers. North-Holland Publishing Co., Amsterdam.
 
137
GRUBER, P. M., AND RYSKOV, S. S. 1987. Facet-tofacet implies face-tofface. Manuscript.
 
138
GRfiNBAUM, B. 1967. Convex Polytopes. Interscience, New York.
 
139
GRfJNBAUM, B., AND SHErHARD, G. C. 1980. Tilings with congruent tiles. Bull. Am. Math. Soc. 3, 951-973.
 
140
141
 
142
 
143
GUIBAS, L. J., MITCHELL, J. S. B., AND ROOS, T. 1991. Voronoi diagrams of moving points in the plane. Springer LNCS. In press.
 
144
 
145
 
146
 
147
 
148
HORTON, R. E. 1917. Rational study of rainfall data make~ possible bettor estimates of water yield. Eng. News-Record, 211-213.
 
149
HOWE, S. E. 1978. Estimating regions and clustering spatial data: Analysis and implementation of methods using the Voronoi diagram. Ph.D. dissertation, Brown Univ., Providence, R.I.
150
151
 
152
IMAI, H., IRI, M., AND MUROTO, K. 1985. Voronoi diagram in the Laguerre geometry and its applications. SIAM J. Comput. 14, 93-105.
153
 
154
JOHNSON, W. A., AND MEHL, R. F. 1939. Reaction kinetics in processes of nucleation and growth. Trans. Am. Instit. Mining Metall. A.LM.M.E. 135, 416-458.
 
155
KANTABUTRA, V. 1983. Traveling salesman cycles are not always subgraphs of Voronoi duals. Inf. Process. Lett. 16, 11-12.
 
156
 
157
KL~NG, T. 1966. Random fragmentation in two and three dimensions. Z. Astrophysik 64, 433-439.
 
158
KmKPATRICK, D. G. 1979. Efficient computation of continuous skeletons. In Proceedings of the 20th Annual IEEE Symposium on FOCS. pp. 18-27.
 
159
KmKPATRICK, D. G. 1980. A note on Delaunay and optimal triangulations. Inf Process. Lett. 10, 127-128.
 
160
KmKPATRICK, D. G. 1983. Optimal search in planar subdivisions. SIAM J. Comput. 12, 28-35.
 
161
KLEE, V. 1980. On the complexity of d-dimensional Voronoi diagrams. Archiv Math. 34, 75-80.
 
162
KLEIN, R. 1989. Concrete and Abstract Voronoi D~agrams. Springer LNCS 400.
 
163
 
164
 
165
KocH, E. 1973. Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. Z. Kristallograph. 138, 196-215.
 
166
KOrEC, R. J. 1963. An alternative method for the construction of Thiessen polygons. Professional Geographer 15, 24-26.
 
167
KUUSKAL, J. B. 1956. On the shortest spanning subtree of a graph and the traveling salesman problem. Problem. Am. Math. Soc. 7, 48-50.
 
168
LANTUEJOUL, C., AND MAISONNEUVE, F. 1984. Geodesic methods in quantitative image analysis. Pattern Recogn. 17, 177-187.
 
169
LAVES, P. 1930. Ebeneneinteilung in Wirkungsbereiche. Z. Kristallograph. 76, 277-284.
 
170
LAWSON, C. L. 1972. Generation of a triangular grid with applications to contour plotting. Tech. Memorandum 299, California Inst. Tech. Jet Propulsion Lab.
 
171
LAWSON, C. L. 1977. Software for C1 surface interpolation. In Mathematical Software III, J. Rice Ed., Academic Press, New York.
172
 
173
LEE, D. T. 1982a. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput. C-31,478-487.
 
174
LEE, D. T. 1982b. Medial axis transform of a planar shape. IEEE Trans. Patt. Anal. Mach. Intell. PAMI-4, 363-369.
 
175
LEE, D. T., AND DRYSDALE, R. L. 1981. Generalization of Voronoi diagrams in the plane. SIAM J. Comput. 10, 73-87.
 
176
LEE, D. T., AND LIN, A. K. 1986. Generalized Delaunay triangulation for planar graphs. Discrete Comput. Geom. 1,201-217.
 
177
LEE, D. T., AND PREPARATA, F. P. 1984. Computational geometry--A survey. IEEE Trans. Cornput. C.33, 12, 1072-1101.
 
178
LEE, D. T., AND SCHACHTER, B. J. 1980. Two algorithms for constructing a Delaunay triangulation. Int. J. Comput. Inf. Sci. 9,219-242.
 
179
LEE, D. T., AND WONG, C. K. 1980. Voronoi diagrams in the Ll(L~) metrics with 2- dimensional storage applications. SIAM J. Comput. 9,200-211.
 
180
LEE, D. T., AND YANG, C. C. 1979. Location of multiple points in a planar subdivision. Inf. Process. Lett. 9, 190-193.
 
181
LEVEN, D., AND SnARra, M. 1986. Intersection and proximity problems and Voronoi diagrams. Adv. Robotics 1, 187-228.
 
182
LEwd, D., AND SHAem, M. 1987. Planning a purely translational motion of a convex robot in two-dimensional space using Voronoi diagrams. Discrete Comput. Geom. 2, 9 31.
 
183
 
184
LINGAS, A. 1986b. A fast algorithm for the generalized Delauuay triangulation. Manuscript, Univ. of LinkSping, Sweden.
 
185
 
186
LINHART, J. 1981. Die Beleuchtung yon Kugeln. Geom. Dedicata 10, 145-154.
 
187
LITTLE, D. V. 1974. A third note on recent research in geometric probability. Adv. Appl. Probab. 6, 103-130.
 
188
LOEB, A. L. 1970. A systematic survey of cubic crystal structures. J. Solid State Chem. I, 237-267.
 
189
MANACHER, G. K., AND ZOBRIST, A. L. 1979. Neither the greedy nor the Delaunay triangulation of a planar point set approximates the optimal triangulation. Inf. Process. Lett. 9, 31-34.
 
190
MATULA, D. W., AND SOKAL, R. R. 1980. Properties of Gabriel graphs relevant to geographic variation research and the clustering of points in the plane. Geograph. Anal. 12, 205-221.
 
191
MATZKE, E. B., AND NESTLER, J. 1946. Volumeshape relationships in variant foams. Am. J. Botanics 33, 130-144.
 
192
MAUS, A. 1884. Delaunay triangulation and the convex hull of n points in expected linear time. BIT 24, 151-163.
 
193
MAXWELL, J. C. 1864. On reciprocal diagrams and diagrams of forces. Phil. Mag. 4, 27, 250-261.
 
194
McLAIN, D. H. 1976. Two dimensional interpolation from random data. Comput. J. 19, 178-181.
 
195
MEGIDDO, N. 1983. Linear time algorithms for linear programming in R3 and related problems. SlAM J. Comput. 12, 759-776.
 
196
 
197
MEIJERING, J. L. 1953. Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rept. 8, 270-290.
 
198
MILES, R. E. 1970. On the homogenous planar Poisson process. Math. Biosci. 6. 85-127.
 
199
 
200
MOLLISON, D. 1977. Spatial contact models for ecological and epidemic spread. J. Roy. Stat. Soc. B 39,283-326.
201
 
202
MORAN, P. A. P. 1966. A note on recent research in geometric probability. J. Appl. Probab. 3, 53-463.
 
203
MORAN, P. A. P. 1969. A second note on recent research in geometrical probability. Adv. Appl. Probab. 1, 73-89.
 
204
MOUNT, D. M. 1985. Voronoi diagrams on the surface of a polyhedron. Rep. 1496, Univ. of Maryland.
205
206
 
207
MUDER, D. J. 1988a. How big is an n-sided Voronoi polygon? Manuscript, MITRE Corp., Bedford, Mass.
 
208
MUDER, D. J. 1988b. Putting the best face on a Voronoi polyhedron. Proc. London Math. Soc. 56, 329-348.
 
209
 
210
MURTADH, F. 1983. A survey of recent advances in hierarchical clustering algorithms. Computer J. 26, 354-359.
 
211
NEWMAN, C. M., RINOTT, Y., AND TVE~SKY, A. 1983. Nearest neighbor and Voronoi regions in certain point processes. Adv. Appl. Probab. 15, 726-751.
 
212
NIGGLI, R. 1927. Die topologische Strukturanalyse. Z. KristaUograph. 65, 391-415.
 
213
NOWACKI, W. 1933. Der Begriff 'Voronoischer Bereich'. Z. Kristallograph. 85, 331-332.
 
214
NOWACKI, W. 1976. Uber allgemeine Eigenschaften von Wirkungsbereichen. Z. KristaUograph. 143, 360-368.
 
215
O'DUNLAING, C., AND YAP, C. K. 1985. A "retraction'' method for planning the motion of a disc. J. Algorithms 6, 104-111.
 
216
O'DUNLAING, C., SHARIR, M., AND YAP, C. K. 1986. Generalized Voronoi diagrams for moving a ladder: I. Topological analysis. Comm. Pure Appl. Math. 39,423-483.
 
217
O'DUNLAING, C., SHARm, M., AND YAP, C. K. 1987. Generalized Voronoi diagrams for moving a ladder: II. Efficient construction of the diagram. Algorithmica 2, 27-59.
 
218
 
219
OHYA, T., IRI, M., AND MUROTA, K. 1984b. Improvements of the incremental methods for the Voronoi diagram with computational comparison of various algorithms. J. Operations Res. Soc. Japan 27, 306-337.
 
220
OKABE, A., YOSHIKAWA, T., FUJII, A., AND OIKAWA, K. 1988. The statistical analysis of a distribution of active points in relation to surface-like elements. Environ. Plan. A 20, 609-620.
 
221
OVERMARS, M. 1981. Dynamization of order decomposable set problems. J. Algorithms 2, 245-260.
 
222
PACH, J., AND SHARIR, M. 1989. The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates. Discrete Comput. Geom. 4,291-310.
 
223
PAPADTMITRIOU, C. H. 1977. The Euclidean traveling salesman problem is NP-complete. Theor. Comp. Sci. 4,237-244.
 
224
PASCHINGER, I. 1982. Konvexe Polytope und Dirichletsche Zellenkomplexe. Ph.D. Dissertation, Inst. f, Math., Univ. Salzburg, Austria.
 
225
PmLBRICK, 0. 1968. Shape recognition with the medial axis transform. In P~ctorial Pattern Reeognitwn, G. C. Cheng, R. S. Ledley, D. K Pollock, and A. Rosenfeld, Eds. Washington, D.C.
 
226
PRErARATA, F. P. 1977. Steps into computational geometry. Rep. R-760, Coordinated Science Lab., Univ. of Illinois, Urbana, Ill., pp, 23-24.
227
 
228
 
229
PRIM, R. C. 1957. Shortest connection networks and some generalizations Bell Syst. Tech. J. 36, 1389-1401.
230
 
231
RHYNSBURGER, D. 1973. Analytical delineation of Thiessen polygons. Geograph. Anal. 5, 133-144.
 
232
 
233
ROGERS, C. A. 1964. Packing and Covering. Cambridge University Press, London, New York.
 
234
 
235
ROSENBERCER, H. 1988. Order-k Voronoi diagrams of sites with additive weights in the plane. Rep. UIUCDCS-R-88-1431, Dept. Cornput. Sci., Univ. Illinois, Urbana, Ill.
 
236
ROSENKRANTZ, D. J., STEARNS, R. E., AND LEWm, P. M. 1977. An analysis of several heuristics for the traveling salesman problem. SIAM J. Comput. 6, 563-581.
 
237
 
238
SAI<AMOTO, M., AND TAKAGI, M. 1988. Patterns of weighted Voronoi tessellations. Sct. Form 3, 103-111.
 
239
SCH()NFLIES, A. 1891. Kristallsysteme und Kirstallstruktur. Teubner, Leipzig.
 
240
SCHWARTZ, J., AND YAP, C. K. 1986. Advances in Robotics. Lawrence Erlbaum Associates, Hillside, N.J.
 
241
 
242
 
243
SEIDEL, R. 1982 The complexity of Voronoi diagrams in higher dimensions. In Proceedings of the 20th Annual Allerton Conference on CCC. pp. 94-95.
 
244
SEIDEL, R. 1985 A method for proving lower bounds for certain geometric problems. In Computational Geometry, G. T. Toussaint Ed. pp. 319-334.
245
246
 
247
SEIDEL, R. 1988. Constrained Delaunay triangulations and Voronoi diagrams with obstacles. In Rep. 260, IIG-TU Graz, Austria, pp. 178-191.
248
 
249
 
250
SR^MOS, M. I., AND HOEY, D. 1975. Closest-point problems. In Proceedings of the 16th Annual IEEE Symposium on FOCS pp. 151-162.
 
251
SHARm, M. 1985. Intersection and closest-pair problems for a set of planar discs. SIAM J. Comput. 14,448-468.
 
252
SmsoN, R. 1977. Locally equiangular triangulations. Comput. J. 21,243-245.
 
253
SmsoN, R. 1979. The Dirichlet tessellation as an aid in data analysis. Scandinavian J. Stat. 7, 14-20.
 
254
SmsoN, R. 1980. A vector identity for the Dlrichlet tessellation. Math. Proc. Camb. Phil. Soc. 87, 151-155.
255
 
256
SMITH, C. S. 1954. The shape of things. Sci. Am. 58-64.
 
257
SNYDEe, D. E. 1962. Urban places in Uruguay and the concept of a hierarchy. Festschrift: C. F. Jones. Northwestern University Studies in Geography 6, 29-46.
 
258
STIFTER, S. 1989. An axiomatic approach to Voronoi diagrams in 3D. Manuscript, RISC, Univ. of Linz, Austria.
 
259
SUGISARA, K., AND IRI, M. 1988. Geometric algorithms in finite-precision arithmetic. Research Memorandum RMI 88-10, Faculty of Engineering, Univ. of Tokyo, Japan.
260
 
261
SUZUKI, A., AND IRL M. 1986. Approximation of a tessellation of the plane by a Voronoi diagram. J. Operations Res. Soc. Japan 29, 69-96.
 
262
TANEMURA, M., OGAWA, T., AND OGITA, N. 1983. A new algorithm for three-dimensional Voronoi tessellation. J. Comput. Phys. 51,191-207.
 
263
T~SKI, A. 1951. A decision method for elementary algebra and geometry. Univ. of California Press, Berkeley, Calif.
 
264
THmSSEN, A. H. 1911. Precipitation average for large area. Monthly Weather Rev. 39, 1082-1084.
 
265
TOKUYAMA, T. 1988. Deformation of merged Voronoi diagrams with translation. Rep. TR- 88-0049, IBM Tokyo Research Lab.
 
266
TOUSSA~NT, G. T. 1980. The relative neighborhood graph of a finite planar set. Pattern Recogn. 12, 261-268.
 
267
TOUSSAINT, G. T., ANn BHATTACHARYA, B. K. 1981. On geometric algorithms that use the furthestpoint Voronoi diagram. Rep. 81-3, McGill Univ., Montreal, Quebec, Canada.
 
268
TUOMIN~N, O. 1949. Das EinfiuBgebeit der Stadt Turku ins System der Einfiuflgebiete SW--Finnlands. Fennia 71-5, 1-138.
 
269
URQUHART, R. 1980. A note on the computation of subgraphs of the Delaunay triangulation. Rep., Univ. of Glasgow, U.K.
 
270
271
 
272
VORONOI, M. G. 1908. Nouvelles applications des parametres continus a la theorie des formes quadratiques. J. Reine Angew. Math. 134, 198-287.
273
 
274
WATSON, D. F. 1981. Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Comput. J. 24, 167-172.
 
275
WEAmE, D., ANn RWmR, N. 1984. Soap, cells, and statistics--random patterns in two dimensions. Contemp. Phys. 25, 59-99.
 
276
WmTELEY, W. 1979. Realizability of polyhedra. Structural Topol. 1, 46-58.
 
277
 
278
WIGNER, E., AND SEITZ, F. 1933. On the constitution of metallic sodium. Phys. Rev. 43, 804-810.
 
279
WmLIAMS, R. E. 1968. Space-filling polyhedron: Its relation to aggregates of soap bubbles, plant cells, and metal crystalites. Science 161, 276-277.
 
280
YAO, A. C. 1975. An O(E log log V) algorithm for finding minimum spanning trees. Inf. Process. Lett. 4, 21-23.
 
281
YAO, A. C. 1982. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput. 11,721-736.
 
282
YAP, C. K. 1987. An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete Comput. Geom. 2, 365-393.

CITED BY  180