The compactness measure of a shape is a numerical quantity representing the degree to which a
shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie ...
is compact. The meaning of "compact" here is not related to the topological notion of
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
.
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
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* 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
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
(the most compact shape) having the same perimeter. In the
plane, this is equivalent to the
Polsby–Popper test The Polsby–Popper test is a mathematical compactness measure of a shape developed to quantify the degree of gerrymandering of political districts. The method was developed by lawyers Daniel D. Polsby and Robert Popper, though it had earlier been ...
. Alternatively, the shape's area could be compared to that of its
bounding circle,
its
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
,
or its
minimum bounding box
In geometry, the minimum or smallest bounding or enclosing box for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure ...
.
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.
Other tests involve determining how much area overlaps with a circle of the same area
or a reflection of the shape itself.
Compactness measures can be defined for three-dimensional shapes as well, typically as functions of
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
and surface
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
. One example of a compactness measure is
sphericity . Another measure in use is
, which is proportional to
.
For
raster shapes, ''i.e.'' shapes composed of pixels or cells, some tests involve distinguishing between exterior and interior edges (or faces).
More sophisticated measures of compactness include calculating the shape's
moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
or boundary
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
.
Applications
A common use of compactness measures is in
redistricting
Redistribution (re-districting in the United States and in the Philippines) is the process by which electoral districts are added, removed, or otherwise changed. Redistribution is a form of boundary delimitation that changes electoral distri ...
. The goal is to maximize the compactness of
electoral districts, subject to other constraints, and thereby to avoid
gerrymandering. Another use is in
zoning
Zoning is a method of urban planning in which a municipality or other tier of government divides land into areas called zones, each of which has a set of regulations for new development that differs from other zones. Zones may be defined for a si ...
, to regulate the manner in which land can be subdivided into
building lots.
Another use is in pattern classification projects so that you can classify the circle from other shapes.
Human perception
There is evidence that ''compactness'' is one of the basic dimensions of shape features extracted by the human visual system.
See also
*
Reock degree of compactness The Reock degree of compactness, or Reock compactness score, is a ratio that quantifies the compactness of the geographic area of a voting district. The score is sometimes used as an indication of the extent to which a voting district may be consi ...
*
Surface area to volume ratio
*''
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension''
References
{{reflist, refs=
[{{cite web, url=https://fisherzachary.github.io/public/r-output.html, title=Measuring Compactness, access-date=22 Jan 2020]
[{{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]
[{{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]
[{{cite web, url=https://pdfs.semanticscholar.org/8364/d027fcaea36786472f1a71e9e0a9ea2b5298.pdf?_ga=2.24601394.920866423.1581298705-411922795.1576410314, title=State of the Art of Compactness and Circularity Measures, first1=Raul S, last1=Montero, first2=Ernesto, last2=Bribiesca, access-date=22 Jan 2020]
[{{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]
Geometric measurement