Ten compactness properties of circles: Measuring shape in geography

Canadian Geographer

Volumn 54, Issue 4, 2010, Pages 441-461

  Angel, Shlomo ^a   Parent, Jason ^b   Civco, Daniel L ^b  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

circle; compactness; landscape metrics; morphology

Indexed keywords

GIS; LANDSCAPE;

EID: 78649851956     PISSN: 00083658     EISSN: 15410064     Source Type: Trade Journal    
DOI: 10.1111/j.1541-0064.2009.00304.x     Document Type: Article

References (31)

1. ANGEL, S., SHEPPARD, S. C., CIVCO, D. L., BUCKLEY, R., CHABAEVA, A., GITLIN, L., KRALEY, A., PARENT, J., and, PERLIN, M., 2005 The Dynamics of Global Urban Expansion, Transport and Urban Development Department, The World Bank, (Washington, DC: The World Bank), September (Available at:, Accessed 16 October 2009)
2. BLAIR, D. J., and, BISS, T. H., 1967 The Measurement of Shape in Geography: An Appraisal of Methods and Techniques Bulletin of Quantitative Data for Geographers 11
3. BOYCE, R. R., and, CLARK, W. A. V., 1964 The concept of shape in Geography The Geographical Review 54, 561-572
4. BURCHFIELD, M., OVERMAN, H. G., PUGA, D., and, TURNER, M. E., 2006 Causes of sprawl: a portrait from space Quarterly Journal of Economics 121 (2), 587-633
5. DE SMITH, M. J., 2004 Distance and Path: The Development, interpretation and Application of Distance Measurement in Mapping and Spatial Modelling. Unpublished PhD dissertation, Department of Geography, University College, University of London (Available at:, Accessed 19 October 2009)
6. DE SMITH, M. J., GOODCHILD, M. F., and, LONGLEY, P. A., 2007 Geospatial analysis: A Comprehensive Guide to Principles, Techniques and Software Tools (Leicester: Matador)
7. EHRENBERG, K., 1892 Studies on the measurement of horizontal shapes of areas Vehandl. der. phys.-mediz Gesellschaft zu Würzburg, Neue Folge 25, n.p., quoted in Y. S. Frolov 'Measuring of Shape of Geographical Phenomena: a History of the Issue'Soviet Geography: Review and Translation 16, 1975, 676-687
8. FROLOV, Y. S., 1975 Measuring of shape of geographical phenomena: a history of the issue ', Soviet Geography: Review and Translation 16, 676-687
9. GUSTAFSON, E. J., 1998 Quantifying landscape spatial pattern: what is the state of the art? Ecosystems 1, 143-156
10. HILBERT, D., and, COHN-VóSSEN, S., 1952 Eleven properties of the sphere ', in Geometry and the Imagination, trans. P. Nemenyi (New York: Chelsea), 215-232
11. HORN, D. L., HAMPTON, C. R., and, VANDENBERG, A. J., 1993 Practical application of district compactness Political Geography 12, 103-120
12. KENDALL, M. G., and, MORAN, P. A. P., 1963 Geometrical Probability (New York: Hafner Publishing Company)
13. MACEACHREN, A. M., 1985 Compactness of geographic shape: comparison and evaluation of measures Geografisca Annaler, Series B, 67 (1), 53-67
14. McGARIGAL, K., and, MARKS, B. J., 1995 Fragstats: Spatial Pattern Analysis Program for Quantifying Landscape Structure (Available at:, Accessed 16 October 2009)
15. MILNE, B. T., 1988 Measuring the fractal geometry of landscapes Applied Mathematics and Computation 27, 67-79
16. NAGEL, DR,. 1835 On Coastal Development of Continents Annal von Berghaus 12, n.p., quoted in Y. S. Frolov 'Measuring of Shape of Geographical Phenomena: a History of the Issue' Soviet Geography: Review and Translation 16, 1975, 676-687
17. POLSBY, D. D., and, POPPER, R. D., 1991 The third criterion: Compactness as a procedural safeguard against gerrymandering Yale Law and Policy Review 9, 301-353
18. RITTER, C., 1822 Die Erdkunde im Verhältniss zur Natur und Geschichte des Menschen, oder allgemeine vergleichende Geographie, 2nd edition, Part I, Book 1, Berlin, quoted in Y. S. Frolov 'Measuring of Shape of Geographical Phenomena: a History of the Issue'Soviet Geography: Review and Translation 16, 1975, 676-687
19. ROHRBACH, C., 1890 On Mean Frontier Distances ', Petermanns Geogr. Mitteil 36, n.p., quoted in Y. S. Frolov 'Measuring of Shape of Geographical Phenomena: a History of the Issue' Soviet Geography: Review and Translation 16, 1975, 676- 687
20. SULLIVAN, L., 1896 The tall office building artistically considered Lippincott's Monthly Magazine March, 403-409.
21. THOMPSON, D. W., 1952 On Growth and Form, 2nd edition, vol. 1, (Cambridge: Cambridge University Press)
22. THüNEN, J. H. VON, 1966 Isolated state; an English edition of Der isolierte Staat, ed., P. G. Hall, trans. Carla Wartenburg (Oxford: Pergamon Press), quoted in Y. S. Frolov 'Measuring of Shape of Geographical Phenomena: a History of the Issue'Soviet Geography: Review and Translation 16, 1975, 676-687
23. TIKHOMIROV, V. M., 1990 Stories about Maxima and Minima, transl. by Abe Shenitzer (Providence, RI: American Mathematical Society)
24. US SUPREME COURT 1996 Bush v. Vera, 517 US 952 (O'Connor, J., opinion of the Court)
25. US SUPREME COURT. 2004 Vieth v. Jubelirer, 541 US 267, (Souter, J., dissenting)
26. US SUPREME COURT. 2006a League v. Perry, 548 US 399 (Kennedy, J., opinion of the Court)
27. US SUPREME COURT. 2006b League v. Perry, 548 US 399 (Roberts, J., dissenting)
28. WILDGREN, J. K., 1995 Fractal Geometry and the Boundaries of Voting Districts ' in Spatial and Contextual Models in Political Research, ed., M. Eagles, (London: Taylor & Francis), 107-126
29. WHITTINGTON, G., BEAVON, K. S. O., and, MABIN, A. S., 1972 Compactness of Shape: Review, Theory, and Application (Johannesburg: Department of Geography and Environmental Studies, University of Witwatersand)
30. WURM, S. A., and, HATTORI, S., eds. 1981 Language Atlas of the Pacific Area Pacific Linguistics Series C No. 66 (Canberra: Department of Linguistics, Research School of Pacific Studies, Australian National University)
31. YOUNG, H. P., 1988 Measuring the compactness of legislative districts Legislative Studies Quarterly 13 (1), 105-115