Lecture Notes

Exercises

Great circle distance

What is a great circle?  How many meridians does it take to form a great circle?

Calculate the great circle distance between Mount Pleasant (43^35'52"N, 84^46'03"W) and Washington D.C. (38^50'N, 77W) using the formula on page 50.  If you don't have a calculator with trigonometric functions, you may use the following procedures that require access to the Internet.

1. Preparation

• Start Netscape Navigator.  Select "Communicator > navigator" to open an additional window.
• In the first navigator window, go to "http://geobli.cst.cmich.edu/bli/circular.html".  This is a program for conversions of various circular measures.  We will need it for converting decimal degrees to radian.  The calculator we use for distance calculation uses radian instead of degree for the trigonometric functions.
• In the second navigator window, go to "http://geobli.cst.cmich.edu/bli/jcintro.html".  This is an advanced calculator that allows to input the entire distance formula.

• Convert the geographical coordinates to decimal degrees.
• Use the circular conversion program to convert the decimal degrees to radian.  Write down the results.
• In the "javacalc" window, type in the mathematical expression.  Use "abs" to calculate absolute values, e.g., |-77-34.5| should be expressed as "abs(-77-34.5)".
• Press button "execute" to get the result (cosD).
• Use the function "acos" to calculate D.  The result is in radian.
• Use the circular convertor to convert D into degrees.
• Multiply D by 69.11mi to get the distance.

Now you can compare the distance you calculated by hand with that by ArcView.

1. Start ArcView.
2. Open a new view and add the two themes, "cities.shp" and "states.shp", from "L:\esri\esridata\usa".  Move theme "cities.shp" on top of "states.shp".  Make both themes visible.
3. Click on the name "cities.shp" to activate the theme.  Select "Theme > Properties".  In the properties dialog window, click the button that has a hammer icon.  Type the following statement in the window:

(( [City_name] = "Mount Pleasant") and ([State_name] = "Michigan" )) or (([City_name] = "Washington") and ([State_name] = "District of Columbia"))

Click OK to close the query window and the properties window.  If you encounter errors, check if you type in the right statement.

As a result of the above operation, you should have only two points on the map, one is Mount Pleasant, the other is Washington D.C.

4. Select "View > properties".  From the dropdown form "Map units", select "decimal degrees", from "Distance units", select "miles".  Click OK to continue.

5. Use the zoom in tool to maximize the display of Mount Pleasant and Washington DC.

6. Click the measure tool (the button with a ruler icon).  Click on Mount Pleasant, then Washington DC.  Note that the distance between the two points is displayed in the lower left corner of the window.  This is a great circle distance.  Compare the result with the one you calculated by hand.

Problems with direction

Illustrate the true azimuth between any two points on the sphere.

Between Mount Pleasant and Washington DC, would the true azimuth measurement be the same for both places?

Examine the direction diagram in the USGS quadrangle map.  What do "MN" and "GN" stand for?  How would you use them?

What do we call the line with a constant azimuth?

You now know the great circle is the shortest course between two places on the earth.  You also know that the direction along the great circle is constantly changing.  It would not be convenient to change direction constantly when you navigate.  What is the solution to this problem?

Geodetic position determination

When the sun is at its highest altitude in Mount Pleasant, the Universal Time (GMT) is approximately 17.65.  What's Mount Pleasant's longitude?

Horizontal and Vertical Control Networks

What is the North American Datum1927 (NAD27)?  How does it differ from NAD83?

What are the levels of accuracies for both horizontal and vertical controls?  Why do we have these different levels?

What's the mathematics behind the construction of these control networks?