Compactness Measure of A Shape

The compactness measure of a shape, sometimes called the shape factor, is a numerical quantity representing the degree to which a shape is compact. The meaning of "compact" here is not related to the topological notion of compact space. 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 in the shape.
  • They agree with intuitive notions of what makes a shape compact.

A common compactness measure is the Isoperimetric quotient, the ratio of the area of the shape to the area of a circle (the most compact shape) having the same perimeter.

Compactness measures can be defined for three-dimensional shapes as well, typically as functions of volume and surface area. One example of a compactness measure is sphericity Ψ. Another measure in use is .

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 gerrymandering. Another use is in zoning, 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.

Famous quotes containing the words shape and/or measure:

    The “universal moments” of child rearing are in fact nothing less than a confrontation with the most basic problems of living in society: a facing through one’s children of all the conflicts inherent in human relationships, a clarification of issues that were unresolved in one’s own growing up. The experience of child rearing not only can strengthen one as an individual but also presents the opportunity to shape human relationships of the future.
    Elaine Heffner (20th century)

    The measure discriminates definitely against products which make up what has been universally considered a program of safe farming. The bill upholds as ideals of American farming the men who grow cotton, corn, rice, swine, tobacco, or wheat and nothing else. These are to be given special favors at the expense of the farmer who has toiled for years to build up a constructive farming enterprise to include a variety of crops and livestock.
    Calvin Coolidge (1872–1933)