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| Compactness Index× | Urban Density Gradient Model× | |
|---|---|---|
| Field≠ | Urban Studies | Human Geography |
| Family≠ | Process / pipeline | Regression model |
| Year of origin≠ | 2010 | 1951 |
| Originator≠ | Geographic shape-analysis tradition (Richardson, Cole; codified by Angel, Parent & Civco) | Colin Clark; Edwin Mills & Richard Muth (theory); Bruce Newling (quadratic form) |
| Type≠ | 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 |
| Seminal source≠ | Angel, S., Parent, J., & Civco, D. L. (2010). Ten compactness properties of circles: Measuring shape in geography. The Canadian Geographer, 54(4), 441–461. DOI ↗ | Clark, C. (1951). Urban population densities. Journal of the Royal Statistical Society. Series A (General), 114(4), 490–496. DOI ↗ |
| Aliases | Shape Compactness Measure, Polsby-Popper Index, Richardson Compactness, Perimeter-Area Compactness | Urban Density Function, Population Density Gradient, Density-Distance Function, Monocentric Density Model |
| Related | 4 | 4 |
| Summary≠ | 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. |
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