You will need to print the following script or view the script in a split screen with the GSP presentation linked below:
To understand the mathematics that the Global Positioning System (GPS) uses when calculating it's position in three-dimensional space, we can explore a simpler two-dimensional version using GSP. The basic mathematical idea behind Trilateration goes like this.
Let the aqua colored circle represent Earth (in 2-dimensions).
We know that we are somewhere on Earth, but we're not sure where. Let's say we're at point A (show Point A).
It's a good thing we have a GPS receiver. We turn on the GPS receiver and it begins to pick up signals from the satellites above.
It detects satellite B (show satellite B) and begins to calculate it's distance from satellite B using the distance formula (distance = rate x time).
Our GPS receiver now knows that it is located somewhere along the circumference of the circle centered at satellite B and passing through our location, point A (show circle BA).
Our receiver detects satellite C (show satellite C) and calculates it's distance.
Now our receiver knows that it is also located somewhere along the circumference of circle CA (show circle CA).
Since it is located on the circumference of both circles and the two circles only intersect in two places (show intersection E), our receiver must be located on either point A or on point E.
Finally, our GPS receiver locates a third satellite (show satellite D) and calculates its distance from D (show circle DA).
Since circle DA passes through point A and does not pass through point E, we must be located at point A.
How does this work algebraically?
(Show Axes and Show Circle Equations)
The equations for these circles are based on the Cartesian Coordinate System. The Global Positioning System uses our longitude/latitude coordinate system to formulate three-dimensional spherical equations.
Have students explore the following:
Expand any two of the equations, set them equal to each other and solve for "y" (if "y" cancels, solve for "x" to produce the vertical line equation). Have students graph their equations. They should get a linear equation that passes through the intersection point(s) of the two circles.
Example Solution for Linear Equation:
Homework:
After expanding the three equations and finding the linear equations that pass throught the points of intersection, students should be prepaired to solve this system of equations using substitution and/or elimination methods.