### Examining Google Earth's Model of the Earth's shape and size.

OBJECTIVE: The primary objective is to examine Google Earth's model of the shape and size of the Earth to see how and whether it fits with models from your readings and lecture. Secondarily you will gain familiarity with Google Earth controls for measuring coordinates and distances.

Google Earth is an interesting resource for getting a quick image-based geographic overview, a modern, web-dependent, high-tech version of the "trusty old" map or aerial photograph. 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 particular, you will be looking to see whether GE uses a spheroid, an oblate spheroid, or a prolate one for its model. (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.)

To prime your thinking, how might GE calculate distances between points? It might...

• 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
where b and c are the co-latitudes of two points and A is their angular separtion around the North Pole.
• 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 terrain surface model and sum the length of its segments in 3D Euclidean space.
d = sum-over-segments [((X1-X2)^2 + (Y1-Y2)^2 + (Z1-Z2))^0.5]
• Store a table of all interpoint distances.
• Do something else. What?
As you do this exercise, keep an eye out for evidence for which method they use. Which do you think it is?

USING GOOGLE EARTH: 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.
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?). Select (straight) line or (connect the dots) path length. Click along the path or line. Read the result from the panel.

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").

Measure the following distances (in meters) using the "ruler" tool. Record the distances as you go.

From Place --- To Place Distance on (Google) Earth (meters)
North Pole to Equator along a meridian
Ninty (90) degrees along the Equator: 45W to 135W
One (1) degree along the Equator: 0W to 1W
One (1) degree along a meridian from the North Pole (90N) to 89N.
One (1) degree along a meridian: 0N to 1N
One (1) degree along the 60N parallel.
One (1) degree along the 45th parallel:

Examine what happens in mountainous terrain by measuring the distance between N 28.05, E 86.85 and N 27.9, E 87.0 (across Mt Everest) with the "terrain on" and "terrain off".

Examine what happens in mountainous versus flat terrain by measuring the similar distance between N 28.05, E 39.95 and N 27.9, E 40.1 in the with the terrain on and off and comapring it with the the distances above.
Mountainous       Flat
Terrain on:
Terrain off:

Finally, try it 'old school' with a pair of coordinates using the spherical trigonometry equation above. I.e., where b and c are the co-latitudes of two points, that is their angles "down from the North Pole" and A is the angle between their meridians around the North Pole. This gives an angular separation along the great circle that includes the two points. If you are working in radians, assume a spherical earth with a radius of 6,371,007 meters, and mulitply your angle in radians by that. If you are working in degrees, assume a spherical earth with a circumference of 40,030,20 m, then multiply your angle in dgrees by 111,195 meters per degree). lets you estimate the distance in linear rather than angular units.

For example, given two places...

```Point Arena, CA (Lighthouse)
latitude: 	 38.954443
co-latitude:   (90.0 - 38.954443) or 51.045557
longitude: 	-123.740678

Honolulu, HI (Aloha Tower)
latitude:	 21.306971
co-latitude:	(90.0 - 21.306971) or 68.693029
longitude: 	-157.866041

longitude separation:  (-123.740678 - -157.866041 =) 34.125363

distance = arcos( (cos(b)*cos(c) + sin(b)*sin(c)*cos(A)) * 111.195 km/degree

= acos[(cos(51.045)*cos(68.693) + (sin(51.045)*sin(68.693)*cos(34.125)]
* (111.195 km per degree)

= arcos[.6287 * .3633   +  .7776 * .9316 * .8278]  * 111.195 (km/degree)

= arcos[.22845  +  .59974]        *  111.195  (km/degree)

=    34.086 (degrees)  *  111.195  (km/degree)

=    3790.267  km

(That is pretty close to what I get with the distance measuring tool in
Google Earth...  3792.66 km.  The dfference is 2.393 km or about 0.06%.
Probably more difference due to rounding and approximatoin on the radius
than to pointing errors.)
```