An Investigation using Parametric Equations

James W. Wilson

We begin with the following problem:

Find the locus of the third vertex of an equilateral triangle when two of its vertices are moved along the x-axis and y-axis respectively.

Let us explore this problems and some of its extensions.


OBSERVATION:  The Vacuum Grinder

A mechanical device that physically rotates in this way is a cutter for oval openings in picture matting. There is also an adult toy, sometimes called "the vacuum grinder," that uses the same principle. I have one on my desk.

The vacuum grinder is a toy for keeping executives busy and therefore not interfere with the work of those they supervise. Moving the handle, the attached points slide along the two slots at right angles. Most of the time if you ask someone operating the vacuum grinder "What path is traced out by the handle?" they will say "It moves in a circle."

Being more aware, as mathematics teachers, we know the path is an ellipse. Further, fixed points on the handle other than the pivot points (these move along the axes) and the midpoint between the axes (it moves in a circle) will have an ellipse for its locus.

The matting cutter for cutting ovals at a frame shop works on the same principle.


Let the base of the triangle be positioned with vertices at (0,0) and (1,0) for the initial position of the triangle. Let the third vertex be at (a,h). Therefore, the altitude of the triangle is h and the projection of the vertex onto the x-axis is a.

As the triangle rotates, let t be the angle the base makes with the x-axis. Then parametric equations for x(t) and y(t) are: (proof left as an exercise for the reader)

x(t) = a cos(t) + h sin(t)
y(t) = (1-a) sin(t) + h cos(t)

Note: The above graph was generated using Mathematics Teacher Workstation from Sterling Swift Software, Austin, Texas. The author is Ray Carry.

Please continue the INVESTIGATION using a function grapher.

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