UNIT 34 - THE POLYGON OVERLAY OPERATION
UNIT 34 - THE POLYGON OVERLAY OPERATION
Compiled with assistance from Denis White, Environmental
Protection Agency, Corvallis, OR
A. INTRODUCTION
- the simple algorithms discussed previously form the basis
for one of the most complex operations of vector GIS
systems - polygon overlay
Traditions of polygon overlay use
B. GENERAL CONCEPTS OF POLYGON OVERLAY OPERATIONS
- in GIS, the normal case of polygon overlay takes two map
layers and overlays them
- each map layer is covered with non-overlapping
polygons
- if we think of one layer as "red" and the other as
"blue", the task is to find all of the polygons on the
combined "purple" layer
diagram
- attributes of a "purple" polygon will contain the
attributes of the "red" and "blue" polygons which formed
it
- number of polygons formed in an overlay is difficult to
predict
- there may be many polygons formed from a pair of
"red" and "blue" polygons, with the same "purple"
attributes
- when two maps are overlaid, will result in a map with a
mixture of 3 and 4 arc intersections
- four arc intersections do not generally occur on
simple polygon maps
Operations requiring polygon overlay
1. Windowing
- the windowing operation, in which a window is
superimposed on a map and everything outside the
window is discarded, is a special case of polygon
overlay
2. Buffering
- buffering around points, lines and polygons is
another case
- buffers are generated around each point or
straight line segment
- the combined buffer is found by polygon overlay
diagram
3. Planar Enforcement
- the process of building points, lines and areas from
digitized "spaghetti" (see Unit 12)
- wherever intersections occur between lines, the
lines are broken and a point is inserted
- the result is a set of points, lines and areas
which obey specific rules:
C. OVERLAY ALGORITHMS
Objective
- overlay two maps of different themes and determine the
combined attributes of the new polygons
- e.g. overlay a soils map on a vegetation map and
create a new set of polygons with a new set of
attributes
Given
- two overlapping polygons as follows:
- red map: a polygon with attribute A
- blue map: a polygon with attribute 1
- the outside world labelled 0 on both maps
- two intersecting arcs defining the boundaries of the
polygons:
1. Red Map (light lines): (0,1) (0,3) (2,3) (2,1)
(0,1)
Polygons - Right: A Left: 0
2. Blue Map (heavy lines): (1,0) (3,0) (3,2) (1,2)
(1,0)
Polygons - Right: 0 Left: 1
Procedure
- after intersections have been found, six new arcs are
formed, three from arc 1 and three from arc 2:
1. (0,1) (0,3) (2,3) (2,2)
2. (2,2) (2,1) (1,1)
3. (1,1) (0,1)
4. (1,0) (3,0) (3,2) (2,2)
5. (2,2) (1,2) (1,1)
6. (1,1) (1,0)
- because the right and left polygon labels are known for
each input arc, we know the labels of the new polygons as
soon as the intersections have been found
- there are four new polygons
- their attributes combine red and blue attributes:
00, A0, A1 and 01
- the arc right and left labels, deduced from the geometry
of the intersections, are:
Arc Right Left
1 A0 00
2 A1 01
3 A0 00
4 00 01
5 A0 A1
6 00 01
- a more complex example:
overhead - Complex overlay example
- in this case the right and left polygon labels for arcs
1, 2, 4, 5 and 7 would be known from the geometry of the
intersections:
1R: A0 1L: 00
2R: A1 2L: 01
4R: A1 4L: 01
5R: 01 5L: 00
7R: A1 7L: A0
- the labels of the remaining arcs must be determined
- labels can be passed from one arc to another around a
polygon:
3R: must be the same as 2R and 4R
6L: from 2L, 4L
- arc 3 was part of the red network, so its soils labels
are known, the remaining (blue) part of its left label
must be the same as the blue part of its right label
- 3R is A1 - thus 3L is B1
- thus 6R is B1
- how to get the blue labels of arc 8?
- use a point in polygon routine to find the enclosing
blue polygon
- use a data structure in which arcs on the inside of
the polygon boundary "point" to arcs on the outside
of any enclosed islands
- e.g. 5R -> 4L -> 6L -> 8L -> 2L -> 5R
- thus the labels of arc 8 are 8L: 01, 8R: C1
- the final step in the algorithm is to identify all new
polygons by following around each polygon from one arc to
the next until every right and left side of every arc has
been identified with a uniquely numbered polygon
D. COMPUTATIONAL COMPLEXITY
- polygon overlay is numerically intensive and time
consuming, therefore it is the most complex operation of
most vector-based GIS programs
- notation: if computing time to process n objects is
proportional to n, the computational complexity is "order
n", or O(n)
- if it is proportional to n2, we way it is "order n
squared" or O(n2)
- it is important to know:
- how long does it take to overlay a given number of
polygons?
- what affects the amount of time taken?
- obviously, the number of arcs and polygons affects
the number of computations required
- it is usually possible to determine the number
of polygons being overlaid
- the number of arcs is roughly 3 times the
number of polygons
- other factors, such as the wiggliness of boundaries,
affect the time, but it is difficult to measure
these
- if n1 polygons are overlaid on n2, how many polygons
result? (assuming maps are different)
- minimum is n1+n2, polygons on the two maps do not
intersect at all
- maximum is infinity, lines have infinite wiggliness
- typical is 3 or 4 times (n1+n2), discounting spurious
polygons
- if every one of n1 "red" polygons has to be compared to
every one of n2 "blue" polygons, the algorithm will be
O(n1n2)
- if we could immediately find all "blue" arcs likely to
intersect with a given "red" arc, we could build an
algorithm which would be O(n1), which means it would be
much more efficient for a given size of problem
- to find arcs in this way we would need an efficient
method of spatial indexing
- one of the most successful methods uses the moving
band concept:
- sort both "red" and "blue" arcs in ascending
order of x
- process the arcs beginning at the left, moving
to the right, sweeping a "band" across the map
- only those arcs falling in the band are
examined
- since the arcs are sorted, we can find those in
the band on either red or blue maps quickly
- some of the best polygon overlay routines now available
in the GIS market operate in close to O(n1+n2) time
- a map with tens of thousands of polygons can be
overlaid on another map with a similar number in an
hour of computing time on a moderate-sized machine
E. INTERSECTION PROBLEMS
- because of computer precision, lines will be represented
in the computer with great precision even though the
accuracy of the representation is low
1. Adjacent lines
- the following diagram represents a case where two lines
cross
diagram
- the following diagram represents a similar case where the
two lines do not actually cross
diagram
- it is necessary to provide algorithms which distinguish
between these very different conditions
2. Sliver polygons
- overlay algorithms compute the exact intersections
between lines (see Unit 14)
- in any overlay operation, it is likely that there will be
pairs of lines which should coincide, but do not because
of differences in digitizing
- these are called slivers, spurious polygons or
"coastline weave"
overhead - Overlay of two images
- if an arc or polygon of n1 points is overlaid on one of
n2 points, up to:
( 2n1n2/(n1+n2) - 3 ) slivers may be generated
- two possible approaches:
a. delete during the overlay operation, or
b. delete afterwards
Removing slivers during overlay
- this is the most common approach used in commercial GIS
programs
- this approach deals with the line as if it were fuzzy
diagram
- define a tolerance limit for each line which indicates
the amount of uncertainty that exists regarding the true
position of the digitized line
- provides a band of width "epsilon" around every line
- for digitized lines this width may be 1 mm
- using epsilon, can then conclude that lines which have a
difference in position less than epsilon are the same
line
- these two lines can then be collapsed to represent a
single line
Problem
- it is easy to get into difficulty:
- lines A and B within epsilon, therefore the same:
- lines B and C within epsilon, therefore the same:
- but lines A and C are not necessarily within epsilon
of each other
- polygon overlay routines which do sliver removal "on the
fly" must deal with this problem
Removing slivers after overlay
- need intelligent criteria to distinguish between slivers
and real polygons
- criteria for removal include:
- area: slivers are small
- shape: slivers are long and thin
- number of arcs: slivers generally have only 2
bounding arcs while real polygons rarely have only 2
- alternating attributes: if a "red" arc between
polygons A and B is overlaid on a "blue" arc between
polygons 1 and 2, the slivers will alternate between
A2 and B1
diagram
- junctions: slivers terminate in 4 arc junctions,
but 3 arc junctions are more common in real polygons
- chaining: slivers tend to occur in chains
- it is likely that the confidence with which it
can be concluded that two arcs are forming
slivers will increase steadily as we work along
the arc
- i.e. the attribute "sliver" is strongly
correlated
- if a sliver is detected, it can be replaced by an arc
along its center line
REFERENCES
McHarg, I.L., 1969. Design with Nature, Doubleday, New York
Goodchild, M.F., and N. S. Lam, 1980. "Areal Interpolation,"
Geoprocessing 1:297-312.
DISCUSSION AND EXAM QUESTIONS
1. McHarg described the overlay technique well before the
advent of GIS and polygon overlay. Discuss the advantages
and possible disadvantages of a computer implementation of
the technique.
2. Write out and illustrate the 16 Boolean combinations of
two polygons A and B.
3. Review and discuss the alternative forms of areal
interpolation described by Goodchild and Lam, 1980.
4. Discuss the relative advantages of raster and vector
approaches to polygon overlay. Identify the application
areas likely to adopt each method given their advantages.
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Last Updated: August 30, 1997.