Compactness measure of a shape

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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.

Properties[edit]

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.

Examples[edit]

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 . Another measure in use is ,[1] which is proportional to .

Applications[edit]

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.[2] Another use is in zoning, to regulate the manner in which land can be subdivided into building lots.[3] Another use is in pattern classification projects so that you can classify the circle from other shapes.

See also[edit]

References[edit]

  1. ^ U.S. Patent 6,169,817
  2. ^ Rick Gillman "Geometry and Gerrymandering", Math Horizons, Vol. 10, #1 (Sep, 2002) 10-13.
  3. ^ MacGillis, Alec (2006-11-15). "Proposed Rule Aims to Tame Irregular Housing Lots". The Washington Post. p. B5. Retrieved 2006-11-15.