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Similarly, a comparison can be made between the perimeter of the shape and that of its convex hull, its bounding circle, or a circle having the same area.
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Other tests involve determining how much area overlaps with a circle of the same area or a reflection of the shape itself.
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shapes composed of pixels or cells, some tests involve distinguishing between exterior and interior edges (or faces).
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Compactness measures can be defined for three-dimensional shapes as well, typically as functions of
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377:"An Efficient Measure of Compactness for 2D Shapes and its Application in Regionalization Problems"
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Various compactness measures are used. However, these measures have the following in common:
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Rick
Gillman "Geometry and Gerrymandering", Math Horizons, Vol. 10, #1 (Sep, 2002) 10-13.
37:. The circle and the sphere are the most compact planar and solid shapes, respectively.
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is one of the basic dimensions of shape features extracted by the human visual system.
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How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension
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More sophisticated measures of compactness include calculating the shape's
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94:. Alternatively, the shape's area could be compared to that of its
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They agree with intuitive notions of what makes a shape compact.
190:{\displaystyle ({\text{surface area}})^{1.5}/({\text{volume}})}
281:, to regulate the manner in which land can be subdivided into
446:"Measuring 2-D Shape Compactness Using the Contact Perimeter"
86:(the most compact shape) having the same perimeter. In the
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is a numerical quantity representing the degree to which a
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Measure of the degree to which a geometric shape is compact
82:, the ratio of the area of the shape to the area of a
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Li, Wenwen; Goodchild, Michael F; Church, Richard L.
273:, subject to other constraints, and thereby to avoid
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They are not overly dependent on one or two extreme
477:"Proposed Rule Aims to Tame Irregular Housing Lots"
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19:Not to be confused with the topological notion of
269:. The goal is to maximize the compactness of
52:They are independent of scale and orientation.
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49:They are applicable to all geometric shapes.
265:A common use of compactness measures is in
123:. One example of a compactness measure is
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475:MacGillis, Alec (2006-11-15).
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146:. Another measure in use is
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228:{\displaystyle \Psi ^{-3/2}}
90:, this is equivalent to the
307:Reock degree of compactness
197:, which is proportional to
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503:Huang, Liqiang (2020).
348:"Measuring Compactness"
293:There is evidence that
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80:isoperimetric quotient
561:Geometric measurement
432:U.S. patent 6,169,817
277:. Another use is in
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139:{\displaystyle \Psi }
57:dimensionless numbers
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104:minimum bounding box
522:10.1167/jov.20.4.10
482:The Washington Post
271:electoral districts
27:Compactness measure
406:Wirth, Michael A.
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92:Polsby–Popper test
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251:moment of inertia
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380:. Retrieved
351:. Retrieved
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261:Applications
253:or boundary
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158:surface area
119:and surface
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295:compactness
100:convex hull
489:2006-11-15
325:References
125:sphericity
41:Properties
515:(4): 10.
255:curvature
210:−
206:Ψ
134:Ψ
102:, or its
55:They are
555:Category
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301:See also
242:shapes,
74:Examples
532:7405702
35:compact
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451:22 Jan
416:22 Jan
353:22 Jan
279:zoning
240:raster
181:volume
117:volume
98:, its
84:circle
64:points
411:(PDF)
382:1 Feb
88:plane
31:shape
537:PMID
453:2020
418:2020
384:2022
355:2020
244:i.e.
238:For
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527:PMC
517:doi
167:1.5
33:is
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