Compiled with assistance from David H. Douglas, University of
Ottawa and David M. Mark, State University of New York at
Buffalo
NOTES
UNIT 32 - SIMPLE ALGORITHMS I - INTERSECTION OF LINES
Compiled with assistance from David H. Douglas, University of
Ottawa and David M. Mark, State University of New York at
Buffalo
A. INTRODUCTION
- the intersection of two lines is a critical operation in
GIS
- is used in polygon overlay operations, merging and
dissolving polygons and lines
- is the basis for point in polygon operations and
critical for sliver removal
- therefore, an efficient algorithm to determine the
intersection of two lines is very important in any GIS.
- GIS algorithms for complex processes are often built up
from simple ones
- this section will review a few simple algorithms,
later sections will show how they can be built up
into complex operations
- the first operation here determines if two lines cross
- begin by examining the algorithm for two straight
lines, then go on to two complex lines or polygons.
- eventually will see that this algorithm forms the
core of numerous GIS operations, including the point
in polygon and polygon overlay processes.
- the algorithm illustrates one of the principles of this
type of programming, that there are numerous special
cases which have to be dealt with.
B. DEFINITIONS
Algorithms
- an algorithm is a procedure consisting of a set of
unambiguous rules which specify a finite sequence of
operations that provides the solution to a problem, or to
a specific class of problems
- each step of an algorithm must be unambiguous and
precisely defined
- the actions to be carried out must be rigorously
specified for each case
- an algorithm must always arrive at a problem solution
after a finite number of steps
- this must also be a reasonable number of steps
- every meaningful algorithm provides one or more outputs
- it is preferable that the algorithm be applicable to any
member of a class of problems rather than only to a
single problem
- in general, the cost of obtaining a solution increases
with the problem size
- if the size of the problem is sufficiently small,
even an inefficient algorithm will not cost much to
run
- consequently, the choice of an algorithm for a small
problem is not critical unless the problem has to be
solved many times.
Heuristics
- an heuristic is a rule of thumb, strategy, trick,
simplification, or any other kind of device which
drastically limits the search for solutions in large
problem spaces
- they do not guarantee optimal solutions, in fact
they do not guarantee any solution at all
- (definition from Feigenbaum, E.A. and J.
Feldman, eds., 1963, Computers and Thought,
McGraw-Hill, p.6)
C. SIMPLEST CASE
Question
Does the line from (4,2) to (2,0) cross the line from (0,4)
to (4,0)? If so, where?
Procedure
Solution
- for line 1 above
b = (2 - 0)/(4 - 2) = 1
- using point 1
2 = a + 4
thus a = -2
- the equation is y = -2 + x
- for line 2 similarly, y = 4 - x
- solving simultaneously, the two lines intersect at (3,1)
General form
Simple program
overhead/handout - Simple program to compute the
intersection of two lines
- the handout lists a rudimentary program for determining
if two lines cross
- x and y are used for line 1, u and v for line 2
input x1,y1
input x2,y2
input u1,v1
input u2,v2
b1 = (y2-y1)/(x2-x1) (A)
b2 = (v2-v1)/(u2-u1) (B)
a1 = y1-b1*x1
a2 = v1-b2*u1
xi = - (a1-a2)/(b1-b2) (C)
yi = a1+b1*xi
if (x1-xi)*(xi-x2)>=0 AND
(u1-xi)*(xi-u2)>=0 AND
(y1-yi)*(yi-y2)>=0 AND
(v1-yi)*(yi-v2)>=0
then print "lines cross at",xi,yi
else print "lines do not cross"
end if
D. SPECIAL CASES
- unfortunately this program will get into trouble in
certain special cases:
Vertical lines
- if line 1 is vertical, the instruction labeled (A) will
cause an error because of an attempt to divide by zero,
as numerical processors cannot deal with infinity
- similarly if line 2 is vertical, line (B) will cause an
error
Parallel lines
- if the two lines are parallel, line (C) will cause an
error
Solution
E. COMPLEX LINES
- consider two complex lines of n1 and n2 straight line
segments respectively:
diagram
- these can be processed for all intersections by
looping the simple algorithm, testing every segment
in one line against every segment in the other
- the amount of work to be done is proportional to the
product (n1 x n2)
- can reduce the amount of work by using a heuristic that
will save future computation
- although this requires an additional processing
step, overall processing time should be reduced
- examples of such methods are:
Minimum enclosing rectangle
- a minimum enclosing rectangle (MER) of a line is defined
by the minimum and maximum x and y coordinates of the
line
diagram
- a very rough check for intersection can be made by seeing
if the enclosing rectangles of two lines overlap
overhead - Minimum enclosing rectangles
- if they do not intersect, the lines cannot intersect
- if they do intersect, then find the MERs for each
straight segment of each line to see which, if any
of these intersect
Monotonic sections
- can divide each line into sections which are
monotonically increasing or decreasing in x and y
- monotonic segments are such that:
- a straight line parallel to either x or y axis cuts
the section at most once
- there is a break where ever x or y hits a local
maximum or minimum
overhead - Monotonic sections
- this sets up conditions which can be used to reduce the
amount of work done in looking for intersections.
- as the segment continues to increase in one
direction, it cannot turn and intersect the other
line again
overhead - Intersection of monotonic sections
- this is the approach used by ARC/INFO and other vendor
GISs
- the number of calculations required to determine the
intersection of two complex lines drops from a value
proportional to (n1 x n2) to a value proportional to
(n1 + n2)
Sorting lines
- when there are many lines with many intersections to be
processed, as in the overlay of two complex coverages
- can sort the lines by their ranges so that only
lines in similar ranges will be considered together
REFERENCES
Douglas, David H., 1974. "It makes me so cross," a paper
distributed by the Laboratory for Computer Graphics and
Spatial Analysis, Graduate School of Design, Harvard
University, September 1974. (This note has been
reprinted numerous times in computing magazines, and in
Marble, Calkins and Peuquet, 1984.)
Lee, D.T. and F.P. Preparata, 1984. "Computational geometry:
a survey," IEEE Transactions on Computers C-33(12):1072-
1101. A good introduction to basic algorithms for
geometrical problems.
Little, J.J. and T.K. Peucker, 1979. "A recursive procedure
for finding the intersection of two digital curves,"
Computer Graphics and Image Processing 10:159-71.
Marble, Duane F., Calkins, Hugh W. and Peuquet, Donna J.,
eds., 1984. Basic Readings in Geographic Information
Systems, Williamsville N.Y., SPAD Systems Ltd.,
Saalfeld, Alan, 1987. "It doesn't make me nearly as CROSS,"
International Journal of Geographical Information Systems
1(4), pp 379-386
Taylor, George, 1989. "Letters," International Journal of
Geographical Information Systems 3(20):192-3. Another
look at the line intersection problem.
DISCUSSION AND EXAM QUESTIONS
1. Why is the "crossing segment" or line intersection
problem so important in GIS?
2. Identify the rules which can be used to limit searching
for intersections between two lines which are both monotonic
in x and y. Each line will be one of four combinations -
either increasing or decreasing in x, and either increasing
or decreasing in y. You will need to deal with 16
combinations when discussing the options for two lines.
3. Review and discuss the technique for line intersection
described in Little and Peucker, 1979.
4. Compare raster and vector approaches to the determination
of the intersection between two lines. Are there
circumstances under which a raster approach might be
preferable?
5. Since geographical data is never perfectly accurate, the
special cases identified in the algorithm should never occur
precisely. Modify the algorithm to deal with special cases
by treating data as imprecise. What are the advantages and
potential problems with such an approach?
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