Thursday, July 5, 2007

Google Earth and Math

This morning I shared with my co-attendees here at PCMI some information about GoogleEarth (hereafter GE) and how I have used it in the classroom. I told them how my classes had looked for circular objects in and around Salt Lake City. We used these objects as our examples when we looked at calculating the area of a sector and the length of an arc. A couple of the circles we found allowed us to see some practical applications of these calculations.

As the day wore on I was playing with the polygon tool on GE and I noticed that a path drawn between two points did not line up with the side of a triangle that shares endpoints with the previously drawn path. As our discussion continued I learned a little more about what constitutes a polygon in spherical geometry (all sides are portions of great circles). We came to the conclusion that the polygons were not true polygons in the spherical mathematics sense. It does look like a path between two points will be the shortest distance spherically, but for some reason the paths and the polygons don't seem to use the same construction when dealing with segments between the two points.

I started browsing the web to find out what I could about GE and how it constructs its paths and polygons. I still have not found the answer to this, but I did run across some interesting software. GEPath (see GE Blog article here or go to download page here) allows you to export a set of placemarks from GE as a KML file and then open this file in GEPath and create a polygon. GEPath will also calculate the area and perimeter of the polygon. This turns out to work nicely.

Using this method, I created a spherical triangle. I then zoomed in to each of the corners and copy/pasted the pictures into Geometer's Sketchpad. I calculated the angles of the triangles and they sum to greater than 180 degrees (181.69 degrees). This type of activity can be used to show students the importance of understanding the importance of the assumptions that are made in problem solving, particularly in geometry.

The other program is GEGraph, which allows you to plot data in a 3D view in GE. I have not tried it yet, but hope to try it soon. I am particularly interested in how GEGraph data can be transferred to and from Fathom.

2 comments:

Alyson said...

Well written article.

Anonymous said...

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