Core Lab 5 - Affine and Projection Transformation

Geog 378
CCSU

To understand the principles and techniques of affine transformation and projection transformation in  ArcInfo.

1. TRANSFORMATION OF PROJECTED UNITS -  Arcinfo

In the following  lab you will do some digitizing.  Some people like to digitize using the coordinates returned from the tablet which are either in inches or centimeters and then transform the data to real-world coordinates later.  Others prefer to locate the control point or tic marks in real world coordinate and let the digitizer transform the X,Y pairs as they digitize. If you digitize a paper map using inches as your coordinate system, eventually you will have to transform that data into a real world plane (flat) coordinate system. If your digitized data is plane (and it is since it would have been done on a flat digitizing tablet) and the coordinate system you plan to use is plane there is no need to use complicated spherical trigonometry to transform the data, a simple affine transformation will do it. The export file LANDCL.E00 is in N:\LABS378. Copy the LANDCL.E00 file into the new workspace. Next, import the coverage. [ArcToolbox/Conversion Tools/Import to Coverage/Interchange File to Coverage]. This coverage is in the digitizer coordinate system where the lower left hand corner is 0,0 and the upper right is 11,11 - assuming you had a tablet that was 11 x 11 inches.

1) Create a new, blank coverage from this digitized coverage.  This is where you will place the transformed coverage in real work units. [ArcCatalog/File/New/Coverage].  Specify landcl as the coverage to copy the tics and bnd values from (control points and geographic extent).  We will change these later since they are in the tablet's coordinate system.  The feature type is poly and precision is double.

Open up ArcMap and add landcl as a layer.  As you move around the screen notice the changing coordinates at the bottom.  ArcMap thinks the data is in decimal degrees but it is actually inches on the tablet.

1. This tablet was at least ________________ inches across (x) and
________________ inches high (y).

This data is in inches (off the digitizing tablet) and you need to transform it into Universal Transverse Mercator meters.  This is a plane (the tablet is flat) to plane (the coordinate system is presumed flat as well) transformation known as an affine transformation.  It does as may of the following as the data requires:

• scale change: If you got data in kilometers but wanted it in meters this is only a change in scale
• translation: this is a shift in x and/or y parallel to the coordinate system.  This adds (or subtracts) the same values (in x and or y) to the existing data.
• rotation: This rotates around either the x or y axis or around both. [Still working on the picture, sorry].

To do this transformation first you need an empty coverage with the new tic marks in it but nothing else.  These tic marks must correspond exactly to the tics in the orignal coverage.  From the File menu in ArcCatalog choose New/Coverage. Give the new coverage the name landrw (rw = real world).  From the Define projection window click on Next - we will skip that step.  From the next window choose polygon as the base feature type, double as the precision and click finish. This makes a completely blank coverage called landrw.

Now we need to enter the tic marks (in real world coordinates) into this coverage that will match with the tic marks in the original coverage (in digitizer inches.)  Right click on the landrw coverage and click on the "tics and extent" tab in the next window.  Use the "Add" button and put the following tics into the table.

These are the four corners of the coverage in the new coordinate system.  Tics do not have to be the four corners but sometimes are.  In the Xmin, Xmax, Ymin and Ymax fields place the correct numbers (432000,438000,5019000,5024000)

You notice that our area extends 6,000 units (meters) from east to west (438000-432000) and 5,000 meters from north to south (5024000-5019000).

Check the values that are in the file and make sure they are correct. It is much easier to fix mistakes early. You have just entered the real world coordinates in projected units for the 4 tics. Now you can use these 4 tics, which are in real-world coordinates,  to transform the whole coverage from previous digitizer inches to real world coordinates.

3) Transformation

Use ArcTools/Data Management/Projections/Tranform and specify landcl as the input coverage and landrw as the output coverage.  Specify Affine transformation.  This process creates a transformation equation for new x and y values.  The new x/y  values are a function of the old x/y values and a set of parameters.  These parameters perform shifting, rotation and scale changes of the coordinates but the changes are from one flat coordinate system (the digitizer) to another (a plane coordinate system). The parameters are estimated using least squares analysis.  Ask to see the output log. It should look like this: [notes in brackets are from me and do not appear in the log]

Transforming coordinates for coverage N:\harmon\temp\landcl
Scale (X,Y) = (636.054,634.184) Skew (degrees) = (0.017)
Rotation (degrees) = (0.345) Translation = (423736.547,5015391.741)
RMS Error (input,output) = (0.000,0.000)  this is a measure of the "goodness of fit" between the two coordinate systems.  If this number is not close to 0 in both cases it almost always means a mis-entered tic coordinate in the output coverage.  The values correspond to the measurement units - in our case the first was inches and the second meters.  This is a good fit.  
It also  performed a scale change in x and y, almost the same.  Skew means that it was not quite perfectly straight up and down.  It rotated .345 degrees - this is very little out of a possible 360 degree rotation.  The most important thing it did was translate - adds value to x and y.  In the case of all x values 423736.547 was added and for the y values in the polygons 50152391.741 was added

Affine this identifies the type of transformation that was made; the simplest linear one
X = Ax + By + C
Y = Dx + Ey + F
A = 636.043 B = -3.630 C = 423736.547
D = 3.832 E = 634.174 F = 5015391.741

X = transformed x coordinate values - meters
x = original x coordinates - inches
Y = transformed Y values - meters
y = original y coordinate - inches

The coefficients were the ones plugged into the equation.  For those with some statistics in their backgrounds, these were calculated using least squares regression analysis. If your RMS error was high, you should be able to find the offending tic mark(s) in this table.

Agains, the RMS (Root Mean Square)  error describes the deviation between the tic locations in the input coverage (in inches in this case) and those in the output coverage (in meters here). If your digitizer unit is inch, your input RMS should usually be less than 0.005. High RMS errors indicate that the input and output coverage tics do not match well, and that the output coverage features will not register well with the original source map. This can happen for a number of reasons, such as:

• original coverage tics were not digitized accurately in the input layer,
• there is map sheet distortion such as stretching or shrinking; this usually occurs more in one direction (x or y) than in another,
• the real-world coordinate values for tics were not recorded correctly. What is the accuracy that you got? If you discover a high RMS error, check your record of real-world coordinates, correct it, and redo the transformation.  

Copy the export files SLOPECL and WELL from N:\labs378   and transform them from digitizer units to real world units in the same way. Do you need completely repeat what you did for landrw?  When you create the new coverage, in the first dialog box, specify a template coverage to be landrw.  This will copy over the tic and bnd files from landrw into your new coverages sloperw and wellrw.  It saves having to enter them manually which is more prone to errors.  I will check your landrw, sloperw and wellrw coverages to make sure you did this correctly.

2. Projection transformation from Lat/Long to CT State Plane - Arcinfo

The next part of the exercise takes data in latitude and longitude coordinates and projects them to a plane coordinate system, a UTM projection. This is not a simple translation/rotation/scale change operation because we are going from a spherical coordinate system to a plane coordinate system. Copy the  QULL.E00 export file from n:\ctdata24\state\ to your workspace for this exercise and import it. [ArcToolbox/Conversion Tools/Import to Coverage/Interchange File to Coverage].

This coverage is the outlines of the quads that cover Connecticut in latitude and longitude - 7.5 minutes of longitude wide and 7.5 minutes of latitude high. So, unprojected, they would appear square but at this latitude 7.5 minutes of latitude covers substantially more distance than 7.5 minutes of latitude.

Run ArcToolbox/Data Management/Projection/Project Wizard.  Think about the questions and let the wizard walk you through the process. Your input dataset (qull) is in decimal lat/long, North American Datum of 1927 (NAD27) and you want the output dataset to be in Connecticut State Plane feet in NAD27.  Call the output coverage qu_sp27. You do not need to shift the x or y values any, other than what the coordinate transformation automatically does.

The projection transformation does not rebuild planar topology as it finishes you use ArcToolbox/Data Management/Topology/Build qu_sp27.

Bring this coverage into   ArcMap and call me over to look at it.

Congratulations! Your coverages now have topology, and real world coordinates that would match with any other CT Stateplane dataset.

Things to be handed in for this part: 1) Answers to the question 1, 3, and a printed copy from question 2. 2) Summarize the procedures of transforming projected unit and reference unit to real world coordinates. What is(are) the major difference(s)?

Extra Credit:  Go on the web, find some geographic data from somewhere in the world that is in decimal latitude and longitude and project it into an appropriate coordinate system.  Some hints:

• you may find a lot of shapefile data which will need to be converted to coverages before you can do a projection change.[ArcToolbox/Data Conversion]
• If you find an import file, .e00, you will have to import that to a coverage.
• Unless projection information is included in the data you may have to run the Define Projection Wizard where you tell it that the data is geographic. You may have to guess at the datum for the original data; not every web provider is so careful in providing metadata.