• 1. Translation
• 2. Scaling
• 3. Rotation
• 4. Reflection

• Simple Example
• City Fire Study Example

A. INTRODUCTION

• coordinate transformations are required when you need to register different sets of coordinates for objects in the same area that may have come from maps of different (and sometimes unknown) projections
• will need to transform one or more sets of coordinates so that they are represented in the same coordinate system as other sets
• the term conflation is often used to talk about the process of making two geographic data sets fit, or of combining the contents of both
• conflation has to overcome lack of fit by adjusting positions
• may also have to replace, combine, or average attributes
 
• (x,y) is the location of the object before transformation (in the old coordinate system)
• (u,v) is the location of the object after transformation (in the new coordinate system)
• are major groups of transformations:
• affine transformations are those which keep parallel lines parallel
• as we will see they are a class of transformations which have 6 coefficients - 6 parameters have to be specified
• curvilinear transformations are higher order transformations that do not necessarily keep lines straight and parallel
• these transformations may require more than 6 coefficients
• piecewise transformations break the map into regions, apply different transformations in each region
• e.g., photogrammetry uses mosaics of air photos
• need to make sure the map doesn't rip, fold where regions meet - regions must edgematch
• transformations can be simple or complex
• complex may produce a better fit
• but it may make no sense given what we know about the production process
• transformations are defined by using control points
• a small number of points whose locations are known in both coordinate systems
• by knowing locations of a small number of points in both systems, we can build transformations that let us convert all of the locations on a map

• affine transformations keep parallel lines parallel
• are four different types (primitives):

• origin is moved, axes do not rotate

u = x - a;  v = y - b

• origin is moved a units parallel to x and b units parallel to y

• both origin and axes are fixed, scale changes

u = cx;  v = dy

• scaling of x and y may be different
• e.g. if x and y are latitude and longitude
• if the scaling is different, the shape of the object will change

• origin fixed, axes move (rotate about origin)

u = x cos(a) + y sin(a);  v = -x sin(a) + y cos(a) (note: a is measured counterclockwise)

• coordinate system is reversed, objects appear in mirror image

 
• 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

• 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
• sometimes these equations are simplified
• e.g., assume uniform scale change in all directions, simple rotation - requires only 4 parameters
• u = a + bx + cy;  v = d - cx + by

• 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 or conflate 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 of affine transformations
• control points must not be on a straight line (not collinear)
• best located to cover the edge, but also include some in the middle if possible
• tics should be located much more carefully and accurately than the regular data
• what kinds of points to use as tics?
• grid intersections, e.g. lat/long degrees
• corner points, if they are defined by lat/long
• prominent features, e.g. major intersections, depending on scale
• marks made on the ground for the project, e.g. white crosses for air photos
• the ICBM solution
• sometimes there aren't any

• control points are:     x         y         u         v

                                0         0         1         10
                                1         0         1           9
                                0         1         3         10
                                1         1         3           9

• solution:
• hint: note that y and u are related, a change in y always produces the same change in u; x and v are similarly related
• therefore:

v = 10 - x; u = 1 + 2y

• complete equations are:

u = 1 + 0x + 2y;  v = 10 - 1x + 0y

• note that bf = 0, ce = -2
• therefore, bf > ce, there is no reflection involved

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?
• Problem: two sets of data, different coordinate systems (UTM and the digitized data)

Oxford and Sanatorium             2.90    9.80        4757500        472700
Wonderland and Southdale       4.86    5.96        4753800        476500
Wharncliffe and Stanley             7.32    8.56        4758200        478600
Oxford and Wharncliffe             7.58    9.67        4759800        478400
Wellington and Southdale           8.24    4.90        4754300        481500
Highbury and Hamilton             11.32    6.67        4758200        484100
Trafalgar and Clarke                 13.17    7.56        4759700        486200
Adelaide and Dundas                 9.49      8.59        4759400        481100
Clarke and Huron                     11.48    10.62        4763700       484700
Highbury and Fanshawe            11.50    12.68        4765400        481500
Richmond and Fanshawe            9.67    12.78        4763600        476900
Richmond and Huron                  8.28    10.73        4761500        478800

Best fit equations:

UTM North = 4744301 +  520.8 x + 1182.9 y            R² = 0.988   SEE = 438m
UTM East   =   474085 + 1258.0 x -   563.3 y            R² = 0.938    SEE = 1060m

Clarke and Huron, Richmond and Fanshawe have high residuals. After deleting:

UTM North = 4744135 +  501.0 x + 1222.2 y            R² = 0.998    SEE = 151m
UTM East   =   473668 + 1207.9 x -   464.6 y            R² = 0.999    SEE = 167m
 
 

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

• simple linear affine transformation equations can be extended to higher powers:

u = a + bx + cy + gxy  or  u = a + bx + cy + gx²  or  u = a + bx + cy + gx² + hy² + ixy

• equations of this form create curved surfaces
• provides rubbersheeting in which points are not transformed evenly over the sheet, transformations are not affine (parallel lines become non-parallel, possibly curved)
• rubber-sheet transformations may also be piecewise
• map divided into regions, each with its own transformation equations
• equations must satisfy continuity conditions at the edges of regions
• curvilinear transformations usually give greater accuracy
• accuracy in the sense that when used to transform the control points or tics, the equations faithfully reproduce the known coordinates in the other system
• however if error in measurement is present, and it always is to some degree, then greater accuracy may not be desirable
• a curvilinear transformation may be more accurate for the control points, but less accurate on average