Planar GPS Unit

Index

What is a planar GPS unit?

Using the planar GPS Unit

Mathematics applications of the planar GPS

Exercises

What is a planar GPS unit?

Suppose we have a large plane surface, say about the size of Texas, and GPS broadcasting units place somewhere in the area. A planar GPS unit can read the location and distance from each unit. Then we have a plane triangle to solve. See the figure below
MATH
There is a lot of math in this figure. For example the fundamental GPS problem is to find the position of the GPS unit (I.e. the red dot above). To this end we want to solve the system.
MATH
This nonlinear system can be solved on a calculator or with a computer algebra system. However, when the student does that, the only result is a number, and that's all. It is far better for the student to do more than input a number. By squaring both equations and substracting we obtain a single linear equation
MATH
Now this is nice because linear systems are much simpler to solve. The only problem is that there is only one such equation. Here is where we can use a second reading from our GPS unit. Just move one of the satellites to obtain a new distance and new set of satellite coordinates. The result is a second linear equation, and thus a set of two linear equations in two unknowns.
MATH
This is one example where we have put over-sampling to great use. In the GPS unit above all of the coordinates are visible. In the "student version" below, we show just the distances and the satellite locations.
MATH




Using the Planar GPS unit

Both planar GPS units exist and are available for your use. Click on the links below to obtain the particular links. To use it you will need the Flash 6 or greater player plug-in to your browser. The Flash player is available for free from Macromedia. (URL: http://www.macromedia.com/shockwave/download/triggerpages_mmcom/flash.html)
MATH

Directions for use:

  1. Load the application by selecting the link.

  2. Place the mouse over any of the three objects, locator, or Satellite $S_{1}$, or Satellite $S_{2}$ and drag the object where you wish it to be.

  3. Read the results in the boxes above. The only difference between the teacher version and the student version is that the teacher version gives the coordinates at all three points.

Remark

The grid is a $300\times 300$ unit area. In miles, then each subdivision represents 30 miles. You can place the satellites at any fixed coordinates and read the answer, that is the GPS coordinates. The student version does not give the GPS unit coordinates.




Mathematics of the planar GPS

The GOPS unit can be used for a variety of applications in many subjects.

  1. College algebra

    1. Finding distances: to the GPS, between Satellites

    2. Linear systems

    3. Solving quadratics

    4. Finding areas and perimeters of the covering circles.

    5. Analyze the results various errors in the values given upon the values calculated.

  2. Trigonometry

    1. Finding the angles between objects

    2. Solving triangles.

    3. Analyze the results various errors in the values given upon the values calculated.

  3. Precalculus/calculus

    1. Finding the area of the GPS-Satellite-Satellite system. Can be done using cross product, Heron's formula, or trig.

    2. Find GPS locations to maximize/minimize the sum of the areas of the covering circles. (Also perimeters.)

    3. Using multiple over sampling and equation differences, apply least squares to determine the position of the system. Remember, from the differences approach, the minimum number of distances (or readings) need is three. With four readings and up, least squares can be applies.

    4. Analyze the results various errors in the values given upon the values calculated. How does the least squares reduce the error?

Exercises

  1. Make up a lesson plan for a trigonometry class.

  2. Make up a lesson plan for a college algebra class.

  3. Make up a lesson plan for a basic algebra class.

  4. Make up a lesson plan for a precalculus class.