View source for Compactness measure

{{Short description|Measure of the degree to which a geometric shape is compact}}
'''Compactness measure''' is a numerical quantity representing the degree to which a [[Shape|shape]] is compact. The circle and the sphere are the most compact planar and solid shapes, respectively.

==Properties==
Various compactness measures are used.  However, these measures have the following in common:

*They are applicable to all geometric shapes.
*They are independent of scale and orientation.
*They are dimensionless numbers.
*They are not overly dependent on one or two extreme [[Point (geometry)|points]] in the shape.
*They agree with intuitive notions of what makes a shape compact.

==Examples==
A common compactness measure is the isoperimetric quotient, the ratio of the area of the shape to the area of a [[Circle|circle]] (the most compact shape) having the same perimeter.  In the [[Plane (geometry)|plane]], this is equivalent to the Polsby–Popper test.  Alternatively, the shape's area could be compared to that of its bounding circle,<ref name="r1"/><ref name="r2"/><ref name="r4"/> its [[Convex hull|convex hull]],<ref name="r1"/><ref name="r3"/> or its [[Minimum bounding box|minimum bounding box]].<ref name="r3"/>

Similarly, a comparison can be made between the perimeter of the shape and that of its convex hull,<ref name="r3"/> its bounding circle,<ref name="r1"/> or a circle having the same area.<ref name="r4"/>

Other tests involve determining how much area overlaps with a circle of the same area<ref name="r2"/> or a reflection of the shape itself.<ref name="r1"/>

Compactness measures can be defined for three-dimensional shapes as well, typically as functions of [[Physics:Volume|volume]] and surface [[Area|area]].  One example of a compactness measure is [[Sphericity|sphericity]] <math>\Psi</math>.  Another measure in use is <math>(\text{surface area})^{1.5}/(\text{volume})</math>,<ref>{{US patent|6169817}}</ref> which is proportional to <math>\Psi^{-3/2}</math>.

For [[Raster graphics|raster]] shapes, ''i.e.'' shapes composed of pixels or cells, some tests involve distinguishing between exterior and interior edges (or faces).<ref name="r2"/><ref name="r4"/><ref name="r5"/>

More sophisticated measures of compactness include calculating the shape's [[Physics:Moment of inertia|moment of inertia]]<ref name="r2"/><ref name="r3"/> or boundary [[Curvature|curvature]].<ref name="r3"/><ref name="r4"/>

==Applications==
A common use of compactness measures is in redistricting.  The goal is to maximize the compactness of electoral districts, subject to other constraints, and thereby to avoid [[Social:Gerrymandering|gerrymandering]].<ref>Rick Gillman "Geometry and Gerrymandering", Math Horizons, Vol. 10, #1 (Sep, 2002) 10-13.</ref>  Another use is in [[Engineering:Zoning|zoning]], to regulate the manner in which land can be subdivided into building lots.<ref>{{cite news |first=Alec |last=MacGillis |title=Proposed Rule Aims to Tame Irregular Housing Lots
 |url=https://www.washingtonpost.com/wp-dyn/content/article/2006/11/14/AR2006111401154.html |newspaper={{wipe|The Washington Post}} |page=B5 | date=2006-11-15 | access-date=2006-11-15 }}</ref>
Another use is in pattern classification projects so that you can classify the circle from other shapes.{{citation needed|date=November 2017}}

==Human perception==
There is evidence that ''compactness'' is one of the basic dimensions of shape features extracted by the human visual system.<ref>{{Cite journal|last=Huang|first=Liqiang|title=Space of preattentive shape features|journal=Journal of Vision|year=2020|language=en|volume=20|issue=4|pages=10|doi=10.1167/jov.20.4.10|pmid=32315405|pmc=7405702 |doi-access=free}}</ref>

==See also==
* Reock degree of compactness
*Surface area to volume ratio
*''How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension''

==References==
{{reflist|refs=
<ref name="r1">{{cite web|url=https://fisherzachary.github.io/public/r-output.html|title=Measuring Compactness|access-date=22 Jan 2020}}</ref>
<ref name="r2">{{cite web|url=https://keep.lib.asu.edu/items/129674|title=An Efficient Measure of Compactness for 2D Shapes and its Application in Regionalization Problems|first1=Wenwen|last1=Li|first2=Michael F|last2=Goodchild|first3=Richard L|last3=Church|access-date=1 Feb 2022}}</ref>
<ref name="r3">{{cite web|url=http://www.cyto.purdue.edu/cdroms/micro2/content/education/wirth10.pdf|title=Shape Analysis & Measurement|first=Michael A|last=Wirth|access-date=22 Jan 2020}}</ref>
<ref name="r4">{{cite web|title=State of the Art of Compactness and Circularity Measures|first1=Raul S|last1=Montero|first2=Ernesto|last2=Bribiesca|s2cid=14813591 }}</ref>
<ref name="r5">{{cite web|url=https://www.sciencedirect.com/science/article/pii/S0898122197000825|title=Measuring 2-D Shape Compactness Using the Contact Perimeter|first=E|last=Bribiesca|access-date=22 Jan 2020}}</ref>
}}

[[Category:Geometric measurement]]

{{Sourceattribution|Compactness measure}}