Slide1: Great Circles & Map Projections Teacher Guide This activity explores the Earth’s spherical geometry and the concept of the Great Circle to help students comprehend and appreciate the importance of map projections and to gain a better understanding of relative distances between places on the Earth’s surface. Great Circles & Map Projections Teacher Guide Overview High School. Beginning to intermediate. Should be completed in one to two 50-minute classes. Process Standards 1-1, 2-2, 2-3, 4-1, 4-2, 4-3, 4-4, Geometry Standards 3-2, 4-4 PASS Objectives Grade Level GIS Skill Level Time Further enrichment would be beneficial to the advanced students of mathematics who have some background in trigonometry, as they can learn and apply the great circle distance formulas. Other Skills GIS Skills Vector and Raster Data, Measuring Distances, Map Projections, Spatial and Aspatial Queries, Spatial Analyst Extension Developed by William R. Flynn
william@musicgeographer.com Teachers can personalize this activity to the locations of their choice. Please contact William Flynn if you would like to have a copy of the data for this exercise. Lesson Preparation A few beach balls are recommended for illustrating Part One of this activity. Materials Geometry, Trigonometry, Geography Subjects
Slide2: Great Circles & Map Projections Today we will learn about the concept of the map projection, which enables us to represent our three-dimensional world in two dimensions. It turns out that mathematics plays a significant part in determining how appropriately and accurately we can model the Earth. Depending on the projection style, we might employ a cylindrical shape to project one version of the world while at other times it may be preferable to choose a conic or planar projection. Since the planet is not a two-dimensional flat object, we will always need to compromise some aspect in order to accommodate others. The give and take comes between things like trying to preserve true shape, equal area, relative distances, and accurate directions. One of the projections we will study in depth today is the Gnomonic projection, which maintains true direction and allows us to introduce the wonderful Great Circle, which represents the shortest surface distance on a spheroid.
Slide3: 2 As summer vacation is just around the corner, it seems that it is time to get in the spirit of the warm season by using a familiar prop to help illustrate why we have map projections. An inflated beach ball will play the role of the Earth and we are going to draw two points on the surface. Next we will draw the shortest route that connects the two points and measure this route with string. Then, we will cut the beach ball and attempt to lay it flat like a two-dimensional map. On the lines below, describe what we learned in this step, commenting on why the cut-up beach ball Earth would not lay flat and if anything changed in regards to measurement, shape, area, etc. PART ONE Having a Ball with the Earth Great Circles & Map Projections
Slide4: 3 PART TWO Real Life Example: Flight Paths Suppose your fondness for things like Sudoku puzzles, Godzilla, sushi, and anime have made you want to travel to Tokyo. Well, in order to fly to Japan, you need to get to a nearby international hub airport, one of which might be Atlanta, Georgia’s Hartsfield-Jackson International Airport.
On the Mercator projection world map below, draw what you think is the most efficient direct path between:
Atlanta (32° 53' 49" N / 97° 02' 17" W) Tokyo (35° 45' 53" N / 140° 23' 11" E) Great Circles & Map Projections Now let’s check how close you were…
Slide5: 4 Part 2 – Step 1 Open the ArcMap File Place a checkmark beside the layer Atlanta to Tokyo and you will see the optimal path between the two cities. This path represents an arc portion of a Great Circle, which identifies the shortest surface distance between two points on a sphere. If you have studied trigonometry, you will realize how the Great Circle is a nice applied example of trigonometric functions.
How are the Great Circle path and your estimate different? Does this make sense? In ArcMap, open the file that we will be using to explore map projections and great circles. The location of the file is:
C:\GreatCircles.mxd Great Circles & Map Projections Part 2 – Step 2 Display the Atlanta-Tokyo Flight Path
Slide6: You can check to see how close your estimate was to the actual Great Circle surface distance. Using the measure tool on the Toolbar palette, click on one end of your predicted flight path (Atlanta or Tokyo) and measure to the other end, clicking wherever needed. In the lower-left corner of your window, you will see the distance under the Miller Cylindrical projection displayed. Notate it below: 5 Part 2 - Step 3 Measure Distances Great Circles & Map Projections MILES To check the Great Circle distance, hold down the SHIFT key. While keeping SHIFT depressed, click on one end of the flight path and then click on the destination city on the other side. With the SHIFT key held down, ArcGIS automatically calculates the Great Circle arc distance, which should be lower than your guessed flight path. How far apart are Tokyo and Atlanta based on Great Circle distance? MILES
Slide7: 6 PART THREE Perkins, OK as the Center of the World GIS software allows you to customize your map projections and one of the nice features is that you can set your map center as any location you would like. For example, let’s place Perkins, Oklahoma as the center of the world. By doing this, we can calculate the shortest surface distance to anywhere else on the planet. A raster layer based on distances at all one-degree latitude / longitude confluences in the world has been provided for you.
Great Circles & Map Projections Place a checkmark next to Distance from Perkins, OK to turn on the raster layer containing worldwide distances in miles from your high school. Study the distance patterns and answer the following questions:
What part of the world (general region) is farthest from Perkins? Part 3 – Step 1 Turn on the Distance Grid By querying the distance raster layer (you can use the identify tool), give approximate distances from Perkins, Oklahoma for the following features: HAWAII IRAQ MADAGASCAR
Slide8: 7 Great Circles & Map Projections Part 3 – Step 2 Turn on Spatial Analyst Extension Activate the Spatial Analyst extension by going into the TOOLS menu and selecting Extensions. Place a check next to Spatial Analyst if there is not already one.
Then turn on the Spatial Analyst toolbar under VIEW menu – Toolbars – Spatial Analyst.
On the Spatial Analyst toolbar is a button titled Create Contours. If you click anywhere on the map, locations of equal distances from Perkins, Oklahoma will be connected with “isolines.” Try placing a few isodistance contours on the map and discuss the kinds of shapes that you see both near and far from Perkins, Oklahoma. Part 3 – Step 3 Isodistance Contours
Slide9: 8 Great Circles & Map Projections PART FOUR Great Circles and the Gnomonic Projection American Airlines uses Miami’s international airport to connect the United States with Central America, South America, and Europe. Some of AA’s international flights out of Miami have been mapped for you to see. American Airlines flies from Miami to London, Paris, and Madrid in Europe as well as many stops in South America including Buenos Aires, Lima, and Sao Paulo. A flight between Miami and Mexico City has also been included for your reference. In this part of the activity, you will learn about the unique Gnomonic projection, which has the advantage of keeping direction uniform across the map. The Gnomonic projection is believed to be the oldest map projection in the history of cartography.
Place a checkmark next to International Flights – MIA to see a sampling of American Airlines flights originating from Miami, Florida. Try to describe the way the flight trajectories arc differently for flights to the west, south, or east. Part 4 – Step 1 Turn on the Miami International Flights
Slide10: 9 Great Circles & Map Projections Part 4 – Step 2 Switch to a Gnomonic Projection for Miami Right-click on the Data Frame “Great Circles” and choose Properties.
Click on the Coordinate System tab. Click on New… on the right side of the dialog box and select Projected Coordinate System.
Under Name, type “Miami Gnomonic”. For Projection Name, drop down and select Gnomonic. Change the Longitude_Of_Center to -80 and the Latitude_Of_Center to 25. Next to Geographic Coordinate System, choose Select and then browse to the World folder and select WGS_1984.prj datum. Click OK. Accept any coordinate system warnings and return to the map. What happened to the Great Circle path lines for the flights originating from Miami, Florida?
Slide11: 10 Great Circles & Map Projections Bonus Question: If you are standing at the North Pole facing south and then turn to your left, which direction are you facing? Further Enrichment: Trigonometry students can calculate the great circle distances between any two point locations on the Earth’s surface using the following formulas (Source: Wikipedia): Great Circle Mapper http://gc.kls2.com/
Great Circle Calculator http://216.147.18.102/dist/ Also worth checking out…