Reference

Honda, H. 1978. "Description of cellular patterns by Dirichlet domains: the two-dimensional case." Journal of Theoretical

Midtb¯, T., 1996. URL: http://www.iko.unit.no/tmp/term/node7.html

Okabe A, Boots B. and K. Sugihara. 1992. Spatial Tessellations. Concepts and applications of Voronoi diagrams. New

Pielou, E. 1977. Mathematical Ecology. New York: John Wiley and sons

Kapraff, J. 1991. Connections. The geometric bridge between art and science. New York: McGraw-Hill, Inc.

Wiens, J. A. 1969. "An approach to the study of ecological relationships among grassland birds." Am. Ornith. Union

G. Albinus. (1983). "Estimates of Agmon-Miranda Type for Solutions of the Dirichlet Problem in Plane Domains with Corners." (1983). (17 ref.) Math. Nachr. 112: 187-208. Akad. Wissensch. DDR, Inst. Math., Mohrenstr. 39, DDR-1086 Berlin, German Democratic Republic.

B. N. Boots and D. J. Murdoch. (1983). "The Spatial Arrangement of Random Voronoi Polygons." (43 ref.) Comput. Geos. 9 (3): 351-365. Wilfrid Laurier University, Department of Geography, Waterloo, Ontario, N2L 3C5, Canada.

H. Brumberger and J. Goodisma (?). (1983). "Voronoi Cells: An Interesting and Potentially Useful Cell Model for Interpreting the Small-Angle Scattering of Catalysts." (1983). (20 ref.) J. Appl. Crys. 16 (Feb): 83-88. Syracuse University, Department of Chemistry, Syracuse, NY 13210.

J. Cohen and J. Gosselin. (1983). "The Dirichlet Problem for the Biharmonic Equation in a C1 Domain in the Plane." (1983). (5 ref.) Indi. Math. J. 32 (5): 635-685. University of Tennessee, Knoxville, TN 37916.

C. W. David. (note) (1984). "Voronoi Polyhedra and Cleft Recognition in Aquated Macromolecules." (5 ref.) Comput. Chem. 8 (3): 225-226. University of Connecticut, Department of Chemistry, Storrs, CT 06268.

E. E. David and C. W. David. (1983). "Voronoi Polyhedra for Studying Solvation Structure (IV)." J. Chem. Phys. 78 (3): 1459-1464. University of Connecticut, Department of Chemistry, Storrs, CT 06268.

J. Fairfield. (1983). "Segmenting Dot Patterns by Voronoi Diagram Concavity". (Letter). (19 ref.) IEEE Patt. A 5 (1): 104-110. James Madison University, Department of Mathematics & Computer Science, Harrisonburg, VA 22807.

B. J. Gellatly and J. L. Finney. 1982. "Calculation of protein volumes: An alternative to the Voronoi procedure." J. Molecular Biology 161(2): 305-322 (October 25).

I. G. Gowda and D. G. Kirkpatr., D. T. Lee and A. Naamad. (1983). "Dynamic Voronoi Diagrams." (30 ref.) IEEE Info T 29 (5): 724-731. Bell No. Res., P.O. Box 3511, STNC, Ottawa, Ontario K1Y 4H7, Canada.

H. G. Hanson. 1983. "Voronoi cell properties from simulated and real random spheres and points." (37 ref.) J. Stat. Phys. 30 (3): 591-605. University of Minnesota, Department of Physics, Duluth, MN 55812.

Hisao Honda. 1978. "Description of cellular patterns by Dirichlet Domains: The two-dimensional case." J. Theoretical Biology 72: 523-543.

Hisao Honda and Goro Eguchi. 1980. "How much does the cell boundary contract in a monolayered cell sheet." J. Theoretical Biology 84: 575-588.

V. G. Mazja, S. A. Nazarov, and B. A. Plamenev. (1981). "Asymptotics of Solutions of the Dirichlet Problem in a Domain with a Truncated Thin Tube". (18 ref.) Math. U.SSR S. 116 (2): 167-194.

F. M. Richards. 1974. "The interpretation of protein structures: total volume, group volume distributions, and packing density." J. Molecular Biology 82: 1-14.

* M. Tanemura, T. Ogawa and N. Ogita. 1983. "A New Algorithm for 3-D Voronoi Tessellation". (18 ref.) J. Comput. Ph. 51 (2): 191-207. Institute of Stat. Math, 4-6-7 Minami Azabu, Minato Ku, Tokyo 106, Japan.

J. Toriwaki, K. Mase, Y. Yashima, and T. Fukumura. 1982. "Modified Voronoi Diagram and Relative Neighbors on a Digitized Picture and their applications to tissue image-analysis." (5 ref.) P. Soc. Photo. 375: 362-367. Toyohashi Univ. Technol, Deptartment of Information and Computer Science, Toyohashi, Aichi 440 Japan.

? "Segmenting Blobs into Subregions". (1983). (26 ref.) IEEE Syst. M. 13 (3): 363-384.

John R. Jungck and Wendy Black Grady

Algorithms:

Brown, K. Q.; "Voronoi Diagrams from Convex Hulls", Information Processing Letters, vol. 9, no. 5: pp. 223 - 228, (1979).
• Construction of a planar Voronoi diagram. Convex Hull calculated via algorithm by Preparata & Hong. Uses a geometric transform (inversion). Easily generalized to higher dimensions, but not always easily applied to generalized metrics.

Guibas, L. & Stolfi, J.; Ruler, Compass and Computer: The Design and Analysis of Geometric Algorithms, DEC Systems Research Center, Dept 37, CA; (1989).
• Location of points within a given Voronoi Diagram. Use of the "Locus Method". Geometric transformation discussion touches on the "Delauney Diagram via convex hull". (Projection to a paraboloid or by stereographic projection).

Gowda, I. G., Kirkpatrick, D. G., Lee, D. T. and Naamad, A.; "Dynamic Voronoi Diagrams", IEEE Transactions on Information Theory, vol. 29, no 5: pp. 724 - 731 (September, 1983).
• Studies the "dynamization" of Voronoi diagrams - insertion and deletion of points - and discusses the applications to both nearest- and farthest-neighbor problems.

Kirkpatrick, D. G.; "Optimal Search in Planar Subdivisions", SIAM Journal of Computing, 12(1):28 - 35; (1983).
• Algorithm for searching triangular subdivisions of the plane for a given point. Based on an edge-ordered representation of a finite set. Proofs on storage and time requirements.

Klein, R.; "Concrete and Abstract Voronoi Diagrams", Lecture Notes in Computer Science #400, Springer-Verlag; (1989).
• References Three O (n log n) algorithms for construction of Voronoi Diagrams:

Lee & Schachter; " Two Algorithms for Constructiong the Delaunay Triangulation", International Journal of Computer and Information Sciences, vol. 9 no. 3; pp. 219 - 242 (1980).
• Review of properties of Delaunay Triangulations and its applications. Algorithms based first on a divide-and-conquer approach and secondly on an iterative approach.

Preparata & Hong; "Convex Hulls of Finite Sets of Points in Two and Three Dimensions", Communications of the ACM, vol. 20 no. 2; (February, 1977).
• Specific discussion of algorighms as well as number of operations and storage requirements. Uses a recursive approach and "giftwrapping"

Preparata, F. P. & Shamos, M. I. ; Computational Geometry, Texts and Monographs in Computer Science , Springer - Verlag (85).
• Discussion of a variety of topics including Voronoi Diagrams, Convex Hulls, Minimum Spanning Trees. Time and Storage requirements for calculations are discussed, as well as some applications. Shows the relationships between these structures.

Applications:

Billings, M. P.; Structural Geology: 2nd Edition; Prentice‚Hall, NY; (1954).
• Description and Classification of Geological formations. Types of materials in which specific fracturing patterns occur. Discussion of causes.

Cannings, C. & Cruz Orive, L.-M.; "On the adjustment of the sex ratio and tyhe gregarious behavior of animal populations", Journal of Theoretical Biology 55 pp. 115 - 136 (1975).
• Consideration on the probability of detection of females of a population by males in relation to location.

Coxeter, H. S. M.; Introduction to Geometry, John Wiley & Sons, Inc., NY; (1969).
• Good introduction to lattices and tessellations of the plane. Some extension to three dimensions. Limited discussion on Phyllotaxis.

Fischer, R.A. and Miles, R.E.; "The Role of Spatial Pattern in the Competition between Crop Plants and Weeds. A Theoretical Analysis", Mathematical Biosciences, 18 pp. 335 - 350 (1973).
• Consideration of maximizing yields of crop plants by means of spacing. Effects of yields when growth and germination times of crop plants vs. weeds are equal and as they vary.

Gray, N.H., Anderson, J.B., Devine, J.D. and Kwasnik, J.M.; "Topological Properties of Random Crack Networks"; Mathematical Geology; 8:617 - 626; (1976).
• Discussion of various types of fractures - properties of the material in which they occur.

Honda, H.; "Description of Cellular Patterns by Dirichlet Domains: The Two-Dimensional Case", Journal of Theoretical Biology, vol. 72: pp. 523 - 543 (1978).
• Given a cellular pattern, how closely can it be approximated by Dirichlet domains. Discussion includes identifying "good" and "fair" approximations, and use of Dirichlet domains to predict the removal or addition of cells from tissue samples.

Jasny; "Exploiting the Insights from Protien Structure"; Science; 240:722 - 3; May 6; (1988).
• Discussion of modelling protiens to study compressability and spaces in which water molecules may deter bonding to other protiens.

McPherson; "Macromolecular Crystals..."; Scientific American; 260:62 - 9; March (1989); (see May '89 also).
• Aspects of crystallization of molecules for study by x-ray crystallography. Does not directly address Voronoi polyhedra.

Nelson, R. A.; "Natural Fracture Systems: Descriptions & Classification"; Bulletin of the American Association of Petrol Geol; 63 pp. 2214 - 2219 (1979)
• Introduction to geological classification of fracture patterns and their causes.

O' Shea, T.; "Dirichlet Polygons - An Example of Geometry in Geography", Mathematics Teacher, pp. 170 - 173 (March 1986).
• Brute force method and use of Delaunay Triangulation to determine Voronoi regions. Several Applications introduced, as a topic to be introduced to high school students.

Pielou, E. C.; Mathematical Ecology, John Wiley & Sons; (1977).
• More of an introduction to sampling techniques. Although there is some discussion of S‚Mosaics, it is very limited, and not the main focus of the book.

Pieri, D. C. ;"Lineament and Polygon Patterns on Europa",
• Classification of polygonal fracture patterns based on average number of sides. Poisson distributions compared to Voronoi distributions.

Richardson, J.S.; "Anatomy and Taxonomy of Protein Structures", Advances in Protein Chemistry, 34 pp. 167 - 339. (1981).
• Description of the physical structure of proteins as a result of the dihedral angles between molecules.

Basic Properties:

Cruz-Orive, L..; "Distortion of Certain Voronoi Tessellations when One Particle Moves"; Journal of Applied Probability, 16, pp. 95 - 103 (1979).
• Consideration of a regular set of points. Identifies properties necessary for a point to contribute to the boundary of a Voronoi Region and the effect of arbitrarily adding a point to a square tessellation. Results are related to the concept of the circumcircles of the Delaunay triangulation of the set.

Earnshaw, R. A., ed.; Theoretical Foundations of Computer Graphics and CAD, NATO ASI series, Series F: Computer and Systems Sciences, Vol. 40, Springer-Verlag, Great Britain (1988)
• Collection of articles on various fields. Articles include:

• Dobkin, D.; "Computational Geometry - Then and Now", pp. 71- 109.
  •• Presentation of data structures, development and open questions.
• Overmars, M.; "Computational Geometry on a Grid - An Overview", pp. 168 - 184.
  •• Example problems, which include the convex hull, and solutions with discussion of time requirements.
• Akman, V.; "Geometry and Graphics Applied to Robotics", pp. 619 - 638.
  •• Short discussion on path planning using Voronoi diagrams.

Hargittai, I. (ed); Symmetry: Unifying Human Understanding,Pergamon Press; (1986).
• Collection of several articles on symmetry. Quite a variety of subjects: Loeb, A. L.; "Symmetry and Modularity": Specifically discusses Voronoi regions in three dimensional space.
MacKay; "Generalized Crystallography": Symmetry as related to natural vs. theoretical crystals.
Gray; "Symmetry in a Natural Fracture Pattern: The Origin of Columnar Joint Networks": Properties of contractional cracking formations such as mud cracks and lava flows.
Senechal, M. "Geometry and Crystal Symmetry": Relationship of crystal structure to lattice structures.
MacKay, A. L. & Klinowski, J.; "Towards a grammar of Inorganic Structure": discusses the unit cells used for periodic structures.

Graham, R., and Yao, F.; "A Whirlwind Tour of Computational Geometry", pp. 687 - 701 (October, 1990).
• Introduces concepts of the convex hull, minimum spanning trees and Voronoi diagrams. Application to space partitions and range search queries such as the post-office problem. Also discuss common techniques used such as Divide-and-Conquer, geometric transformations, etc.

Loeb, A. L.; Space Structures, Addison-Wesley; (1976).
• Properties of Dirichlet domains on a random, finite set of points and on a lattice. Use of such properties to determine the Dirichlet center when the regions are known.

Aggarwal, Alok, Guibas, Saxe, Shor; "A linear-time algorithm for computing the Voronoi Diagram of a convex polygon" Discrete Computational Geometry 4 no. 6 pp. 591 - 604 (1989)

Aronov, B. ; "On the Geodesic Voronoi Diagram of Point Sites in a Simple Polygon", Algorithmica, 4 pp. 109 - 140 (1989).

Ash, P. F. & Bolker, E. D.; "Generalized Dirichlet Tesselations", Geometriae Dedicata, 20:209 - 243; (1986).

"   "; "Recognizing Dirichlet Tesselations", Geometriae Dedicata, 19:175 - 206; (1985).

Aurenhammer; "A New Duality Result Concerning Voronoi Diagrams", ICALP, (1986).

Aurenhammer & Imai; "Geometric Relations among Voronoi Diagrams", STACS; (1987).

Bramble, J. H.; "A Survey of Some Finite Element Methods Proposed for Treating the Dirichlet Problem", Advances in Mathematics, 16pp. 157 - 165 (1975).

Brostow, W. et all; "Construction of Voronoi Polyhedra", Journal of Computational Physics, 29 pp 81 - 92; (1978).

"   " , "Geometric Transforms for Fast Geometric Algorithms", Technical Report, Carnegie-Mellon; (1980).

Canny, J.; "Simplified Voronoi diagrams", Third ACM Symposium on Computational Geometry, Discrete Computational Geometry3, 3 pp. 219 - 236 (1988).

Chazelle & Edelsbrunner; "An Improved Algorithm for Construction k-th Order Voronoi Diagrams", Computational Geometry, (1985).

Chang, R. C. & ; "Average Performance Analysis of Closest pair ...", International J. of Computational Mathematics, 16 (84) no. 2 pp. 125 - 130.

Chew, P.L.; "Constrained Delaunay Triangulations", Algorithmica, 4 , pp. 97 - 108 (1989).

Cover, T. M. & Hart, P. E.; "Nearest Neighbor Pattern Classification", IEEE Transactions on Information Theory, IT - 13 pp 21 - 27; (1967).

Dershem, H.L.; "Dirichlet Regions", Hope College.

Finney, J. L.; "A Procedure for the Construction of Voronoi Polyhedra", Journal of Computational Physics, 32 pp. 137 - 143; (1979).

"   " "Random Packing and the Structure of Simple Liquids". Proceedings of the Royal Society, A319:479 - 493, (1970).

Fortune, S.; "A Sweepline Algorithm for Voronoi Diagrams", Algorithmica, 2(2):153 - 174 (1987).

"   " ; "   "   " Computational Geometry, (1986).

"   " ; "   "   " ACM on Computational Geometry, (June,1986).

Garey, et al (Preparata); "Triangulating a simple polygon", Information Processing Letters, vol. 7 no. 4; (June 1978).

Gilbert, E. N.; "Random subdivision of space into crystals", Ann. Math. Stat. 33 pp. 958 - 972 (1962).

Green, J. J. & Sibson, R.; "Computing Dirichlet Tesselations in the Plane", The Computer Journal, 21 pp. 168 - 173; London ; (1978). or (1977).

Guibas & Stolfi; "Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams", ACM on Graphics, vol. 4 no. 2; (April, 1985).

Hertel, Stefan; "Fast Triangulations ...", Foundations of Comp. Theory Lecture Notes in Computer Science, No. 158 (83).

Hinrichs, K., Nievergelt,J., and Schom,P.; "A Sweep Algorithm for the All-Nearest-Neighbors Problem", Lecture Notes in Computer Science #333, ed. Noltemeier "Computational Geometry and it's Applications", CG '88 International Workshop on Computational Geometry, Wurzburg, FRG. March 1988 proceedings. Springer-Verlag (1988)

Kendall, D.G.; "A Survey of the Statistical Theory of Shape", Statistical Science, 4 no. 2, pp. 87 - 99.

Kiang, T.; "Random fragmentation in two and three dimensions", Z. Astrophysik 64 pp. 433 - 439 (1966).

Lawson; "Generation of a Triangular Grid with Applications to Contour Plotting", Technical Memo 229 Jet Propulsion Laboratory; (1972).

Lee; "Proximity and Reachability in the Plane", Technical Report R- 831 University of Illinois, Urbana; (1978).

Lee, D. T. "Two Dimensional Voronoi diagrams in the Lp metric", J. Ass. Comput. Mach., pp. 604 - 618, Oct. 1980.

"   " ; "Farthest neighbor Voronoi diagrams and applications", Tech Report #80-11-FC-04, Dept. Elec. Eng/Comput. Sci., Northwestern Univ., Evanston, IL, Nov. 1980.

Lee & Wong; "Voronoi Diagrams in L1 and LÉ Metrics with 2-Dimensional Storage Applications" Rep RC 6848 (#29359) IBM (1977).

Lengal, T.; "On the Computational Complexity of Some Geometrical and Clustering Problems"

Lingas, A; "Voronoi Diagrams with barriers and the shortest diagonal problem", Information Processing Letters 32 no. 4 pp. 191 - 198 (1989).

MacKay, A. L.; "Stereological Characteristics of Atomic Arrangements in Crystals" ,Journal of Microscopy, 95:217 - 227 , (1972).

Melter, R.A. and Rudeanu, S.; "Voronoi Diagrams for Boolean Algebras".

Namba, T.; "Competition for space in a heterogeneous environment", Journal of Mathematical Biology, v.27 no.1, pp. 1 - 16 (1989).

Ohya, Iri & Murota; "A Fast Voronoi Diagram Algorithm with Quaternary Tree Bucketing",

Olivier, Martin " ... Cellular Automata", Comm Math Phys. 93 (84) no. 2 pp. 219 - 258.

Post, M. J.; "A Minimum Spanning Ellipse Pattern", IEEE, (October, 1981).

Overmars, Mark H. "Computational Geometry on a grid: An Overview", Theoretical Foundations of Computer Graphics and CAD: Proceedings of the NATO Advanced Stydy Institute held in Il Ciorro, July 4 - 17, 1987 ed. R. A. Earnshaw. Nato Advanced Science Institute Series F: Computer & Systems Sciences 40 Springer-Verlag, New York, (1988)

Richards, F. J.; "The Geometry of Phyllotaxis and its Origin"; Symp Soc Exp Biol. 2 (1948) pp. 217 - 245.

"   " , "Phyllotaxis: Its Quantitive expression and relation to growth in the Apex"; Philosophical Transactions of the Royal Society B; 235 (1951) pp. 509 - 564.

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Shamos, M. I. & Hoey, D.; "Closest Point Problems", Proceedings of the 16th IEEE symposium of Foundations of Computer Science, pp. 152 - 162. (1975).

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1. Geometry in Action
   http://www.ics.uci.edu/~eppstein/gina/voronoi.html
   A very comprehensive site on the subject. Here you can find explanations on the definition, but most of the site is dedicated to providing links to other sites on the subject.very good site - a very long list of sites where you are almost guaranteed to find something of interest
2. Voronoi Diagrams
   http://www.iko.unit.no/tmp/term/node6.html
   A page with detailed explanation of Voronoi Diagrams. With a little history introduction and two graphs this page explains what Voronoi Diagrams are all about. a very nice page - good explanation supported with graphs
3. Delaunay Triangulation
   http://www.iko.unit.no/tmp/term/node7.html
   A page with detailed explanation of the Delaunay Triangulation - the formal definition as well as three diagrams are present to better the understanding of the reader. a very good page - an explanation that will make this mathematical term as easy to understand as 2+2
4. Project Orion
   http://ndsun.cs.ndsu.nodak.edu/www/ORION/report1.html
   Deptartment of Defense's project for Path Computation for Remote Autonomous Vehicles. They use Voronoi diagrams and Delaunay Triangulations to help plan the path of next-generation RAVs. interesting report - if you are at all interested in the subject you should check this site out
5. The Geometry Junkyard
   http://www.ics.uci.edu/~eppstein/junkyard/topic.html
   A FEAST for anyone with love for math. A great number of links.amazing!
6. The Geometry Junkyard - Nearest Neighbors
   http://www.ics.uci.edu/~eppstein/junkyard/nn.html
   A site with a small but very carefully selected number of links to other pages of interest in the subject of Voronoi Diagrams and nearest neighbors. definetely worth checking out
7. Delaunay Triangulation and Voronoi Tessalation
   http://www.erc.msstate.edu/~pt1/voronoi/voronoi.html
   A site with a great number of pictures that speak for themselves. a very GOOD resource in the quest of understanding the subject better
8. Proximity
   http://www.cs.brown.edu/courses/cs252/proximity/home.html
   A page that contains 3 PS lectures and a paper on the subject.looks very good although I didn't have a chance to actually read them
9. LEDA - LEP - Abstract Voronoi Diagrams
   http://www.mpi-sb.mpg.de/LEDA/friends/avd.html
   A site that offers a program for calculations of Abstract Voronoi diagrams.both the documentation and the software are provided - I didn't have a chance to test it personally so I cannot testify to usefulness or quality

Contact: Zdravko Jeremic [jeremicz@alum.beloit.edu]

posted: 07/25/97

updated: 04/22/07