NOTES
UNIT 28 - AFFINE AND CURVILINEAR TRANSFORMATIONS
A. INTRODUCTION
B. AFFINE TRANSFORMATION PRIMITIVES
- affine transformations keep parallel lines parallel
- are four different types (primitives):
handout - Affine transformation primitives
1. Translation
- origin is moved, axes do not rotate
diagram
u = x - a
v = y - b
- origin is moved a units parallel to x and b units
parallel to y
2. Scaling
- both origin and axes are fixed, scale changes
diagram
u = cx
v = dy
- scaling of x and y may be different
- if the scaling is different, the shape of the object
will change
3. Rotation
4. Reflection
- coordinate system is reversed, objects appear in mirror
image
diagram
- to reverse y, but not x:
u = x
v = c - y
- this transformation is important for displaying images on
video monitors as the default coordinate system has the
origin in the upper left corner and coordinates which run
across and down
C. COMPLEX AFFINE TRANSFORMATIONS
- usually a combination of these transformations will be
needed
- the combined equations are:
u = a + bx + cy
v = d + ex + fy
- often cannot actually separate the needed transformations
into one or more of the primitives defined above as one
transformation will cause changes that appear to be
caused by another transformation, and order is important
- e.g. translation followed by scale change is not the
same as scale change followed by translation, has
different effect
- exception:
- reflection has occurred if bf < ce
D. AFFINE TRANSFORMATIONS IN GIS
- frequently, when developing spatial databases for use in
GIS, data will be provided on map sheets which use
unknown or inaccurate projections
- in order to register two data sets, a set of control
points or tics must be identified that can be located on
both maps
- must have at least 3 control points since 3 points
provide 6 values which can be used to solve for the
6 unknowns
- control points must not be on a straight line
(not collinear)
Simple Example
City Fire Study Example
handout - City fire study example (2 pages)
Problem:
- given: two sets of spatial data:
1. Census tract boundaries in the city with
coordinates given in UTM
2. Fire locations in the city plotted on a crude
road map
- UTM is to be the destination system
- count the number of fires in each census tract and
analyze the numbers of fires in relation to
characteristics described by census data
- e.g. is number of fires related to number of houses
constructed of wood?
Solution:
- use major street intersections as control points
- for destination system, determine UTM coordinates of
the intersections
- for the fire map system, use any arbitrary
rectangular coordinate system (i.e. digitizer table)
with fixed origin and axes
- using the two sets of coordinates for the control points
and linear regression techniques, solve for the 6
coefficients in the two affine transformation equations
- examine the residuals to evaluate the accuracy of the
analysis
- the spatial distribution of residuals may indicate
weaknesses of the model
- may show that map has been distorted unevenly
- magnitude of the residuals gives an estimate of the
accuracy of the transformation
- in this example, the magnitude of residuals
indicates an accuracy of 150 m
- i.e. UTM coordinates of fire locations are +/- 150 m
from their locations as indicated on the road map
- for this analysis, this is sufficient accuracy
E. CURVILINEAR TRANSFORMATIONS
REFERENCES
Goodchild, M.F., 1984. "Geocoding and Geosampling," Spatial
Statistics and Models, G.L. Gaile and C.J. Willmott,
eds., Reidel Publishing Company, Dordrecht, Holland,
pp. 33-53.
DISCUSSION AND EXAM QUESTIONS
1. A map has been digitized using an origin in the lower
left corner. The x axis is the lower edge of the map, and
the y axis is the left edge. The study area ranges from 0
to 100 in x, and 0 to 50 in y. Calculate the
transformations necessary to show this area on a screen
containing 640 columns and 480 rows, with the columns and
rows numbered from the top left corner, assuming that
positions on the screen are referred to by row and column
numbers.
2. Describe the transformations between x and y and UTM
Northing and UTM Easting given in the text of the unit for
the fire study example in terms of (a) translation of the
origin, (b) scaling, (c) rotation and (d) reflection.
3. Discuss the factors which contribute to lack of fit
between maps of the same area, and failure of affine
transformations to register maps perfectly using control
points. Discuss the relative importance of each of these
factors.
4. Discuss the criteria you would use to select control
points, including the number and distribution of such
points, in setting up a coordinate transformation.
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Last Updated: August 30, 1997.