Shape Metrics
Jason Parent
jason.parent@uconn.edu
Academic Assistant – GIS Analyst
Daniel Civco
Professor of Geomatics
Center for Land Use Education And Research (CLEAR)
Natural Resources and the Environment
University of Connecticut
Shlomo Angel
Adjunct Professor of Urban Planning
Robert F. Wagner School of Public Service, New York University
Woodrow Wilson School of Public and International Affairs,
Princeton University
1
Background
► Metrics
that quantify aspects of shape have
widespread applications
 Pattern analysis in landscape ecology and geography
 Identify suitability of a given area for a particular
purpose
► Hundreds
of metrics exist for measuring
characteristics of shapes.
 The relevancy and appropriate use is often not clear
2
Why measure characteristics of shape?
► Patch
shape and area influences the viability of
forest patches for certain forest species.
► Compactness of the urban footprint can be a
measure of sprawl in cities.
► Compactness of election districts may indicate
gerrymandering.
► Patch shape and area may determine the
suitability of a patch of land for a particular
purpose
3
Objectives
► Present
10 metrics for measuring various aspects
of shape that can be applied to contiguous
patches.
► Present a normalized version of each metric that is
not affected by shape area
 Measures shape compactness
 Values range between 0 and 1 with higher values
indicating greater compactness
► Present
a framework for determining the
appropriate metric(s) for a given analysis.
► Present a script tool that can calculate each of the
metrics for polygons in a feature class.
4
Defining a polygon in terms of points
►
Interior points
Points evenly
distributed
throughout shape
20,000 points
►
Vertex points
Points defining
perimeter vertices
(inflexion points)
Perimeter points
► Points equally
spaced along
perimeter
100 points
5
Normalizing shape metrics
- the Equal Area Circle
►A
circle is the most compact shape possible for a
given area (Angel et al. 2009).
► The Equal Area Circle (EAC) is a circle with an area
equal the area of the shape.
► The normalized metrics presented are normalized
using the EAC
 Creates a measurement of compactness
 Metrics normalized with the EAC are highly correlated
► Normalized
metrics are appropriate when the
influence of shape area is irrelevant or misleading
or when a measure of compactness is needed.
6
Shape characteristics
► Distribution
of the polygon around a central
point
► Distribution of points within the polygon
► Characterizing the polygon interior and
perimeter
► Characterizing the polygon as an object to
traverse or circumvent
7
Distribution of the shape around
a central point
8
Proximity index - definition
► The
average Euclidean distance from all interior
points to the centroid*
d1 + d2 +…dn
Proximity =
n
d4
d3
d1
d2
9
* The average XY coordinates for all vertices that define the shape
Normalized proximity index
ProximityEAC
normalized proximity (nPI) =
ProximityShape
ProximityEAC = 2 * radius
3
EAC
index
I = 5776
nI = 0.98
I = 4968
nI = 1.00
normalized index
I = 4944
nI = 0.69
I = 7267
nI = 0.55
I = 6177
nI = 0.44
10
Proximity index - comments
► All
points in shape are given equal weight.
► Relatively quick calculation time
► Basic index appropriate for use when
distance to the shape’s center is needed…
 i.e. Proximity calculated for an urban footprint gives an
estimate of the travel distance for residents commuting
to the urban center – used to infer travel costs (time,
fuel, pollution, etc.)
► Normalized
index appropriate for measuring
compactness
11
Spin index - definition
►
►
The average of the square of the Euclidean distances
between all interior points and the centroid.
Also known as Moment of Inertia in the literature.
spin =
d12 + d22 +…dn2
# of points
d4
d3
d1
d2
12
Normalized spin index
spinEAC
normalized spin =
spinShape
spinEAC = 0.5 * radius2EAC
index
I = 37990552
nI =0.95
I = 27768701
nI = 1.00
I = 27634615
nI = 0.48
I = 54370006 I = 50360516
nI = 0.33
nI = 0.17
normalized index
13
Spin index - comments
Similar to Proximity except more
weight is given to the polygon’s
extremities.
► Relatively quick calculation time.
► Basic metric results not very intuitive.
► Normalized metric appropriate for
measuring compactness when focus is
on shape extremities…
►
nProximity = 0.44
nSpin = 0.17
i.e. Normalized spin calculated for an urban footprint gives
a measure of compactness that is more sensitive to the
outlying parts of the footprint. This index is more capable
of identifying footprints that have “tendril-like” projections
(often perceived as an indicator of sprawl).
14
Dispersion - definition
The average distance from the centroid to all points on the
shape perimeter.
► Based on the Boyce-Clark Index (Boyce and Clarke 1964).
►
d1 + d2 +…dn
dispersion =
n
d4
d3
d1
d2
15
Normalized dispersion index
circle with dispersion
equal to shape dispersion
d3
d4
d1
normalized
=
dispersion
d2
dx is the distance between
shape perimeter and the
circle perimeter along a
radial emanating from the
centroid.
d1 + d2 +…dn
deviation =
n
dispersion – deviation
dispersion
16
Dispersion index - examples
index
I = 8664
nI = 0.90
I = 7451
nI = 1.00
I = 4539
nI = 0.56
I = 7469
nI = 0.81
I = 6802
nI = 0.53
normalized index
► Normalized
values close to 1 indicate equal
dispersion in all directions.
17
Dispersion index - comments
►
Only points in perimeter are
used to calculate compactness.
 Gaps in a shape do not affect
the metric
►
►
►
nI = 0.90
nI = 0.90
Relatively quick calculation time.
Use basic metric when average spread of a phenomena is
of interest.
Normalized metric appropriate for measuring shape
compactness when gaps in the shape should be ignored…
 i.e. Normalized dispersion can indicate whether a
phenomena (i.e. invasive species spread) is propagating
from an epicenter equally in all directions. This can give
an idea of the effectiveness of containment efforts.
18
Distribution of points within
the shape
19
Cohesion index - definition
►
The average distance between all pairs of interior points.
cohesion =
d1 + d2 +…dn
# of point pairs
d3
d4
d5
d1
d6
d2
20
Normalized cohesion index
cohesionEAC
normalized cohesion =
cohesionShape
cohesionEAC = 0.9054 * radiusEAC
index
I = 7881
nI = 0.98
I = 6739
nI = 1.00
I = 6719
nI = 0.69
I = 9386
nI = 0.58
I = 8282
nI = 0.45
normalized index
21
Cohesion index - comments
All points in shape given equal weight
► Computationally intensive for large numbers of points
►
 Only calculated for a sample of points to improve calculation
time.
►
Appropriate when the average distance between
points in a shape is needed or when distribution of
the shape about the center is not relevant.
 i.e. Cohesion calculated for an urban footprint gives an
estimate of the travel distance for residents commuting
within the city – does not assume residents
predominantly travel to the urban center. Can be used
to infer travel costs (time, fuel, pollution, etc.).
22
Characterizing the shape
interior and perimeter
23
Depth index - definition
► The
average distance from the shape’s interior
points to the nearest point on the perimeter.
24
Normalized depth index
depthShape
normalized depth =
depthEAC
depthEAC = 1 * radiusEAC
3
index
I = 2530
nI = 0.89
I = 2487
nI = 1.00
I = 563
nI = 0.33
I = 711
nI = 0.35
I = 552
nI = 0.41
normalized index
25
Depth index - comments
Measures average distance from the interior of a
polygon to the edge of the polygon.
► Indicates how susceptible the patch may be to
disturbances outside the shape perimeter.
►
 Larger distances indicate greater insulation of the shape’s
interior to external events.
►
Appropriate when the insulation of a patch’s interior
from the surrounding environment is important.
 i.e. The depth of a forest patch can indicate the
suitability of the patch for species that do not tolerate
close proximity to development, open fields, or other
land cover types.
26
Viable interior index - definition
► The
area of the shape that is beyond the depth of
the edge-effect.
Edge-width
27
The edge-effect
Occurs when a patch can be influenced or degraded by the
surrounding environment.
 i.e. The edge of a forest patch can be affected by
increased exposure to wind, light, invasive species, etc.
► The distance over which the “edge-effect” can occur
depends on the issue or species or study
 In ecological literature, distances ranges from 25 meters
to several hundred meters depending on species and
land cover type.
 Timber harvest may not be practical within some
distance from developed areas.
► We typically assume a 100 meter edge-width for general
purposes studies.
►
28
Normalized viable interior index
interiorEAC
normalized interior =
interiorShape
interiorEAC = Π * (radiusEAC – edge)2
Index (ha)
I = 17,127
nI = 0.97
I = 13,068
nI = 1.00
I = 353
nI = 0.07
I = 2,176
nI = 0.27
I = 339
nI = 0.11
normalized index
29
Viable interior index - comments
Measures area of shape that is not susceptible to
influence from the surrounding environment.
► An appropriate edge-width distance must be used for
meaningful results.
► Easy and quick to calculate.
► Appropriate when the insulation of a patch’s interior
from the surrounding environment is important and
an appropriate edge-width is known.
►
 i.e. The interior index of a forest patch can indicate
whether the patch contains enough suitable area to
support the desired diversity of interior forest species
30
Girth index - definition
► The
radius of the largest circle that can be
inscribed in the shape
d
31
Normalized girth index
girthShape
normalized girth =
radiusEAC
index
I = 7417
nI = 0.87
I = 7344
nI = 1.00
I = 1461
nI = 0.28
I = 1700
nI = 0.28
I = 1200
nI = 0.29
normalized index
32
Girth index - comments
Measures the largest circular area that can be fully
contained within a shape.
► Can be used to determine if a polygon can
accommodate the footprint of a feature such as a
proposed development.
►
 i.e. A suitability analysis indicates patches of area that
are suitable for development. The girth index can be
used to determine which patches contains a contiguous
area large enough to contain the footprint of the
proposed development.
Shape area
Circle area
A = 1.19
a = 0.21
A = 1.01
a = 0.74
33
Perimeter index - definition
► The
perimeter of the shape
34
Normalized perimeter index
normalized perimeter =
perimeterEAC
perimeterShape
index
I = 60,357
nI = 0.89
I = 46,918
nI = 1.00
I = 134,879
nI = 0.24
I = 97,017
nI = 0.39
I = 53,447
nI = 0.48
normalized index
35
Normalized perimeter index - comments
Measures the length of the perimeter of a given
shape.
► Very quick and easy to calculate
► Metric gives an indication of the shape’s exposure to
external conditions.
►
 i.e. The normalized perimeter index for a forest patch
will indicate the exposure of the patch to the
surrounding environment. Patches for which the
normalized index is maximized will have less exposure.
36
Characterizing the shape as
an object to traverse or
circumvent
37
Detour Index
► The
perimeter of the shape’s convex hull
*
Convex
hull
* The convex hull is the convex polygon with the shortest
possible perimeter that fully encompasses it.
38
Normalized detour index
normalized detour
=
perimeterEAC
perimeterConvex Hull
► Larger
normalized values indicate that a relatively
shorter path is needed to circumvent the shape
index
I = 66,373
nI = 0.81
I = 48,349
nI = 1.00
normalized index
I = 48,118
nI = 0.67
I = 65,447
nI = 0.58
I = 75,747
nI = 0.34
39
Detour index - comments
The convex hull is the shortest path needed to
circumvent a shape.
► The normalized index indicates how large an obstacle
a shape presents relative to its area.
► Relatively quick to calculate.
► Appropriate to use when the shape is an obstacle to
passage and cannot be traversed.
►
 i.e. The normalized detour index can be used quantify
the degree to which a highway obstructs wildlife
movement. The effectiveness of wildlife crossings in
reducing the obstruction can be analyzed.
40
Traversal Index
►
The average distance of the shortest paths connecting any
two points on the shape perimeter.
 The paths must remain inside the shape.
traversal =
d1 + d2 +…dn
# of point pairs
d1
d2
d3
41
Normalized traversal index
normalized traversal =
traversalEAC =
traversalEAC
traversalShape
4 * radiusEAC
Π
index
I = 11,019
nI = 0.98
I = 9397
nI = 1.00
normalized index
I = 7679
nI = 0.85
I = 13,357
nI = 0.57
I = 9039
nI = 0.58
42
Traversal index - comments
The traversal index is the shortest average distance
between two points on the perimeter – the paths
between points cannot intersect the shape boundary.
► Very long calculation time
►
 10-15 minutes per feature
►
Appropriate to use when the shortest interior distance
between any two points on the patch’s perimeter is
relevant.
 i.e. The traversal index can estimate the distance
required to cross a lake from any direction.
43
Choosing the appropriate metric
►
The basic metrics each provide different information
about shape characteristics.
 The metric used should make logical sense for the
analysis.
►
The normalized metrics tend to be highly correlated with
each other.
 One metric may be a good proxy for another to measure
compactness.
44
Summary: shape aspect
and suggested metrics
►
Distribution of the shape
around a central point…
►
 Proximity
 Spin
 Dispersion
►




The shape as an object to
traverse or circumvent
 Traversal
 Detour
Characterizing the shape
interior and exposure to
external conditions
►
Perimeter
Girth
Depth
Viable interior
Distribution of points
within the shape…
 Cohesion
45
The Shape Metrics Tool
►
►
►
A Python script has been
developed to calculate the
metrics presented
Script will run out of
ArcToolbox for ArcGIS 9.3
Will be available through
Center for Land use
Education And Research
(CLEAR) website
 http://clear.uconn.edu/tools/Sh
ape_Metrics/index.html
46
Conclusions
► The
basic form of the metrics are influenced by
shape area
 The appropriate metric to use depends on the
application – the metric should make logical sense.
► The
normalized version of the metrics provides a
measure of compactness.
 Normalized metrics tend to be highly correlated with
each other and with people’s perception of
compactness.
► The
script tool will facilitate calculation of shape
metrics for polygon feature class data.
47
References
► Angel,
S., J. Parent, and D.L. Civco. 2009. Ten
Compactness Properties of Circles: A Unified
Theoretical Foundation for the Practical
Measurement of Compactness. Canadian
Geographer. (in press)
► Angel, S and GM Hyman (2009). Ten Theorems
Concerning the Compactness of Circles.
(forthcoming).
48
QUESTIONS?
Jason Parent
jason.parent@uconn.edu
Academic Assistant – GIS Analyst
Daniel Civco
Professor of Geomatics
Center for Land Use Education And Research (CLEAR)
Natural Resources and the Environment
University of Connecticut
Shlomo Angel
Adjunct Professor of Urban Planning
Robert F. Wagner School of Public Service, New York University
Woodrow Wilson School of Public and International Affairs,
Princeton University
Download script at:
http://clear.uconn.edu/tools/Shape_Metrics/index.html
49