The picture above was obtained from the Astronomy Picture of the Day Archive. This archive contains a number of striking and informative images, many of which come from NASA. To go to this archive click on the image below.
The earth is round -- as we read these words, we have the benefit of pictures of the earth like the one above, taken from satellites like Apollo 17. Without pictures like these this idea -- the idea that the earth is round -- was historically a very difficult idea for humans to accept.
The best models or maps of the earth are globes since they have the same shape as the earth.
The diameter of the earth measured from pole to pole is 7,899.834 miles or 12,713.550 kilometers. Find a map of your state. Determine the scale of the map -- that is, how many miles on the earth are represented by each inch on the map or how many kilometers on the earth are represented by each centimeter on the map. Sometimes this scale is represented as a ratio -- for example, 50,000 : 1. This means that each centimeter on the map represents 50,000 centimeters or 50 kilometers on the earth. How large would a globe need to be if it had the same scale as your map of your state. Find a map that you might use for planning a hike or a map that you might use to find your way around your city or town. How large would a globe need to be if it had the same scale as this map.
The link below leads to the Tiger Map Service maintained by the United States Census Bureau. Using the Tiger Map Sevice you can create your own maps for a variety of purposes.
The Gazetteer at the United States Census Bureau Tiger Map Service
Working in the Tiger Map Service window, follow the steps below to create a map.
If you're interested in finding out more about the Tiger Map Service, click on the link below to open a new window and go to its home page.
If you're interested in finding about more about the services and data available over the World Wide Web from the United States Census Bureau, click on the link below to open a new window and go to its home page.
The best way to work through the remainder of this module is with a globe. We need a system to describe the location of various places on the earth or on a globe. There are two very natural points on the earth -- the north pole and the south pole. These points are especially easy to locate on most globes because most globes revolve about an axle and that axle goes through the north pole and the south pole. Because we live in the northern hemisphere -- closer to the north pole than the south pole -- the north pole on our globes is generally on the "top" of the globe. If you lived in the southern hemisphere a globe would be more convenient if the south pole were on top. Look at your globe. How easy is it to look at the United States or at England or another European country? How easy is it to look at Australia or Zimbabwe? If you have a contact with email in the southern hemisphere, send him or her email and ask about the globes available locally. Is the north pole on the "top" or is the south pole on the "top""
Locating the north pole and the south pole on the earth is more difficult. Although the earth revolves about an axis and the axis runs through the north pole and the south pole, there is no axle running through the earth. We do have several clues. In the northern hemisphere we can see the north star or Polaris. The north pole of the earth points almost directly at Polaris. You can see this for yourself. Leave a camera out on a clear night with the lens open, taking a picture of the part of the sky near the north pole. As the earth turns around its axis, the stars appear to move in circles. Your picture will show circular "star tracks" for every star except one -- Polaris. Because the earth's axis is pointing toward Polaris, Polaris does not appear to move. The picture below a view of the night sky near Polaris. It will look different at different times of the night and of the year because of the earth's rotation and its motion about the sun.
Because a compass can be used during the day or on a cloudy night, compasses are especailly valuable for navigation. It is important to know the correction that must be made to determine the direction of the north pole from the direction of the north magnetic pole. If you have contacts with email in different places, contact them by email and ask them to repeat the experiment you did above -- comparing the direction of the north magnetic pole to the direction of the north pole. Then compare your results.
The next feature we want to notice is the equator. This is the circle that runs around the earth's "waist" -- exactly halfway between the north pole and the south pole. The equator should be clearly marked on your globe. The picture below shows a portion of the equator passing through South America.
Points along the equator are labeled in degrees. There are 360 degrees measured along the equator. Starting at 0 we say a point on the equator is 0 to 180 degrees of longitude west if it is west of the point 0 and 0 to 180 degrees of longitude east if it is east of 0.
A line starting at the north pole, passing through a point on the equator, and ending at the south pole is called a meridian and is labeled according to where it hits the equator. The prime meridian passes through Greenwich, England and the point 0 on the equator.
Find your location on the globe. Look at the meridian containing your location and notice where it crosses the equator. For example, if you live in Chicago, Illinois, your meridian goes north along Lake Michigan, crosses Lake Superior, Ontario, and Hudson Bay, before reaching the north pole. It goes south through Illinois, and crosses the Gulf of Mexica, a bit of Mexico, and Central America before reaching the equator and then going on to the south pole. It crosses the equator approximately 87 degrees west. This is a very rough measurement from the globe. We say that the longitude of Chicago is roughly 87 degress west. Compare the rough reading you obtained from your globe with the more exact reading you get from your Global Positioning System. We got a rough reading of 113 degrees west for Helena, Montana from our globe and a reading of 112 degrees, 3.008 minutes from our GPS. Our rough reading was surprizingly good.
The earth's radius measured from the center of the earth to a point on the equator is 3963.205 miles or 6378.160 kilometers. This is called the earth's equatorial radius. Its equatorial circumference is 24,901.55 miles or 40,075.16 kilometers. Thus, at the equator, each degree of longitude corresponds to 69.1710 miles or 111.3199 kilometers .
Each meridian is divided into 180 degrees -- 90 degrees north of the equator and 90 degrees south of the equator. These degrees are called degrees of latitude. The equator is located at latitude 0; the north pole at latitude 90 north; and the south pole at latitude 90 south.
Determine the latitude of your location roughly using your globe and compare your rough determination with a more precise measurement using your Global Positioning System.
The radius of the earth measured from the center of the earth to either pole is 3949.917 miles or 6356.775 kilometers, so its polar circumference is 24,818.06 miles or 39,940.80 kilometers.
For the remainder of this module we work with a simplified model of the earth. We work with a sphere whose radius is 3957 miles or 6367 kilometers -- the average of the equatorial radius and the polar radius. By making this simplification we will introduce errors of up to 12 miles in the distance between two points on opposite sides of the earth. The error for closer points will be smaller. These errors are small enough for our purposes here and the resulting simplification is sufficient payoff for tolerating these relatively small errors. Of course, if you were piloting an airplane you would need a more accurate model.
This idea -- the idea of working with a simplified model first and then when necessary working with more realistic and more complex models -- is extremely important. Our world is complicated and often we cannot understand it in one giant step, so we take smaller steps, building up progressively better and better models or pictures of the real world.
In our simplified model each degree of latitude or longitude near the equator represents 69.06 miles or 111.1 kilometers. Angles of less than one degree are measured in minutes. There are 60 minutes in each degree and, thus, each minute represents 1.15 miles or 1.85 kilometers.
Now look at your globe. If you were to travel around the globe starting at a point above or below the equator and traveling due west staying at the same latitude then your trip would be shorter than if you traveled along the equator. The picture below shows several possible routes like this at different latitudes. Notice the routes closer to the poles are shorter than the routes closer to the equator. Each of these routes -- that circumnavigate the globe at a fixed latitude -- is called a parallel.
where D is the diameter of the equator -- 7914 miles or 12,734 kilometers.
and the location of my favorite coffeee house, Patterns, is
This is close to but not exactly the same as the way angles are measured. When we measure angles we usually divide each minute into 60 seconds, so that an angle of 4 and one half minutes would be 4 minutes and 30 seconds. The GPS would report 4.5 minutes.
Patterns is 0.1 minute south of the fountain and 1.92 minutes east. Thus, Patterns is 0.12 miles south of the fountain and 1.52 miles east of the fountain. By the Pythagorean Theorem the distance from the Carroll College fountain to Patterns is 1.53 miles.
Explain how the distances above were determined from latitude and longitude. Use your GPS to measure distances between various landmarks in your area. For each landmark have several different people use the GPS to determine the latitude and longitude. Notice that there will be differences because of the errors inherent in the GPS and because of errors introduced by the Department of Defense. How large is the variation in the measuremnts of latitude and longitude? How large is the variation in your distance measurements?
The module Spherical Coordinates and the GPS describes how we can work with the information provided by the GPS. For now, you should answer the following questions. Remember sometimes it is best to work with a simplified model and get rough answers but it is also important to be aware of the limitations of your model.
Click on the appropriate name(s) below to send email to any of the authors.
Department of Mathematics, Engineering, Computer Science, and Physics Carroll College, 1601 N. Benton Avenue, Helena, MT 59625