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	Compactness Index	Urban Density Gradient Model
Campo	Urban Studies	Human Geography
Famiglia	Process / pipeline	Regression model
Anno di origine	2010	1951
Ideatore	Geographic shape-analysis tradition (Richardson, Cole; codified by Angel, Parent & Civco)	Colin Clark; Edwin Mills & Richard Muth (theory); Bruce Newling (quadratic form)
Tipo	Geometric/morphological index of how compact a settlement footprint is	Family of functional models of urban population density as a function of distance from the centre
Fonte seminale	Angel, S., Parent, J., & Civco, D. L. (2010). Ten compactness properties of circles: Measuring shape in geography. The Canadian Geographer, 54(4), 441–461.	Clark, C. (1951). Urban population densities. Journal of the Royal Statistical Society. Series A (General), 114(4), 490–496.
Alias	Shape Compactness Measure, Polsby-Popper Index, Richardson Compactness, Perimeter-Area Compactness	Urban Density Function, Population Density Gradient, Density-Distance Function, Monocentric Density Model
Correlati	4	4
Sintesi	A compactness index measures how compact the shape of a settlement, district, or built-up area is, almost always by comparing it to the circle — the most compact shape enclosing a given area. Classic indices such as the Polsby–Popper or Richardson ratio compare a polygon's area to its perimeter, while more elaborate measures compare interpoint distances or fitted circles, all returning a value of one for a perfect circle and falling toward zero as the shape becomes elongated, indented, or fragmented. Angel, Parent and Civco systematized these into a coherent family by showing that the circle is optimal on ten distinct geometric properties, clarifying which index answers which question.	The urban density gradient model is the broad family of functional relationships that describe how population density varies with distance from a city's centre. Its canonical member is Colin Clark's 1951 negative-exponential form, but the family also includes Bruce Newling's quadratic-exponential function that permits a density crater at the core, simpler linear and Smeed forms, and the economic micro-foundation supplied by the Muth-Mills monocentric city model. Together these give planners and economists a compact, comparable language for urban spatial structure.