### Examining Google Earth's Model of the Earth's shape and size and your mental map's geometry.

OBJECTIVE: In Part A, the primary objective is to apply your critical thinking skills and Google Earth's distance measuring tool to see whether its model of the shape and size of the Earth fits with models from our readings and lectures. In Part B, the primary objective is to see how your mental maps' representations of distances and angles fit with the local geography/geometry in GE.

BACKGROUND: Google Earth is an interesting resource for getting a quick image-based geographic overview. It is a modern, web-dependent, high-tech version of the "trusty old" globe, a "mash-up" satelite imagery, aerial photographs and vector map data. You will use Google Earth to determine coordinates for several places, the distances between them (measured in meters), and compare those measures to see what they might reveal about the model of the Earth's shape used in this software. In part A, you will be looking to see whether GE uses a spheroid, an oblate spheroid, or a prolate one for its model, and whether it uses that model or its terrain model in calculating distances between points. In part B, we'll see how well an affine transformation can bring your sketch map into correspondence with GE's "reality".

(If you do not have access to GE on a computer of your own, we can arrange access via one of the university's computer labs.)

Using Google Earth, it becomes easy to determine and record geographic coordinates, and to measure distances and the lengths of the routes between points. (I hope that you'll enjoy using this tool and will find it helpful this term. Take a little time to become familiar with it.)

Set up and Use Hints.
Tools : options : 3D View tab... Show Lat/Lon as decimal degrees. Show elevation in meters, and distances in meters,.
View : check "Status Bar".

Location of the pointer (hand) is displayed in the Status Bar as decimal degrees of latitude and longitude and meters of elevation. (You could transcribe these from there.)

Push-pin (place-marker), Polygon, and Path Tools. Click the tool. Enter a mnemonic name. Set style and color. Digitize. (finally) click "OK" to close the panel. (You can reset the "properties" of a place-marker, polygon or path; right-click its name in the side-bar.)

Ruler (measurement) Tool. Set the units (meters). (How many different units of distance do they offer? What is the oddest one?) Select (straight) line or path length (connect the dots). Click the end points (or along the path). Read the result from the panel.

Image Overlay Tool. This tool lets you overlay an image file (.jpg etc.) on the GE globe. The "handles" let you slide the image around, and tweak its orientation.

You can save the image (including your digitized paths) as a 'jpg' file. File : Save : Save Image...

You can save the coordinates of your push-pins and paths to a .kml file: In the side-bar, under the "Places" tab, Right-click the "My Places" (or "Temporary Places") line and select "Save as..." from the drop-down menu. Enter a filename (like, your initials + "-366.kml").

### Part A: Is the Google Earth Model Spherical, Oblate or Prolate?

What distances could you compare to answer that question? (How about 1 degree on both a Meridian and the Equator? Or how about 90 degrees on each?) What would you expect in each case? What reasoning would you use to conclude from those measurements?

Measure them and use your observations to see whether you find evidence for the earth shape used in GE. Are there any issues that might reduce your confidence in the conclusion?

### Part B: Compare your mental map's local geometry to Google Earth's

You should have a .jpg file of a mental map showing PSB 310 and several other points.

GE allows you to overlay photos on its model. Place the jpg of your mental map so that PSB 310 is lined up. Keeping that point in place, adjust the overlay as best you can to get the other points to line up. At least one other point should also be able to coincide exactly; but probably no more given that this is an affine transformation rather than a rubber-sheet fit. When you get it as good as you can...

For each of your mapped points, what are the displacements (distances between "true" and "mental map" points and angles between them around that PSB central point? Use the Measurment tool to measure and plot the displacements. Find the difference between azimuths from PSB to the "GE" and "mental" positions. Record them in a table like:

```

Place...          Distance displacement (m)    Angular
Displacement (deg)
Az(GE)  Az(mental)
Difference

1. __________

2. __________

3. __________

4. __________

5. __________

sum:

avg:

```

### THE REPORT:

Adress both Parts A and B, each with a brief description of your observations of distances (and directions) in Google Earth and what they suggest about the model of the shape of the earth that GE uses (part A) and what kinds of distortion you see in the sketch map (part B). In part A: Is the GE model spherical, oblate, or prolate? Why do you think so? For part B: Do you see any patterns in the distortions? Use a table (like above) and a map (export and print a .jpg image showing your overlaid sketch and the displacement vectors) to facilitate your answer.

All told, including the map and table, this should be less than 3 pages. Submit in hardcopy.

### Something to think about: How does GE compute distance between points?

Among the ways that GE might calculate distances between points seem to be at least these...

• Calculate the arc between the points with the Law of Cosines from spherical trigonometry and multiply by a scale factor to get a linear distance in km.
d = arcos( (cos(b)*cos(c) + sin(b)*sin(c)*cos(A)) * 111 km/degree
• Project the coordinates to a flat Cartesian (2D) space and calculate the distance with the Pythagorean Theorem:
d = ((X1-X2)^2 + (Y1-Y2)^2 )^0.5
• Work-out a geodesic on the 3D terrain surface model and sum the length of its segments in that 3D Euclidean space.
d = sum-over-segments [((X1-X2)^2 + (Y1-Y2)^2 + (Z1-Z2)^2)^0.5]
• Store a table of all possible inter-point distances from which to "look-up" the distances as needed?
• Some other way?
As you do this exercise, keep an eye out for evidence for which method GE uses. Which do you think it is?