Compactness measure is a numerical quantity representing the degree to which a
shape
A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
. 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 number
Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...
s.
*They are not overly dependent on one or two extreme
points
A point is a small dot or the sharp tip of something. Point or points may refer to:
Mathematics
* Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
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 point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
(the most compact shape) having the same perimeter. In the
plane, this is equivalent to the
Polsby–Popper test. Alternatively, the shape's area could be compared to that of its
bounding circle,
its
convex hull
In geometry, the convex hull, 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 bounding box or smallest bounding box (also known as the minimum enclosing box or smallest enclosing box) for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dime ...
.
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 regions in 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 measure of a region's size on a 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 open surface or the boundary of a three-di ...
. One example of a compactness measure is
sphericity
Sphericity is a measure of how closely the shape of a physical object resembles that of a perfect sphere. For example, the sphericity of the ball (bearing), balls inside a ball bearing determines the quality (business), quality of the bearing, ...
. 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/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...
or boundary
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
.
Applications
A common use of compactness measures is in
redistricting
Redistricting in the United States is the process of drawing electoral district boundaries. For the United States House of Representatives, and state legislatures, redistricting occurs after each ten-year census.
The U.S. Constitution in Art ...
. The goal is to maximize the compactness of
electoral districts, subject to other constraints, and thereby to avoid
gerrymandering
Gerrymandering, ( , originally ) defined in the contexts of Representative democracy, representative electoral systems, is the political manipulation of Boundary delimitation, electoral district boundaries to advantage a Political party, pa ...
. Another use is in
zoning
In urban planning, zoning is a method in which a municipality or other tier of government divides land into land-use "zones", each of which has a set of regulations for new development that differs from other zones. Zones may be defined for ...
, to regulate the manner in which land can be subdivided into
building lots.
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
*
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://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