Assignment #10

Parametric Equations

By

Michelle Nichols

A parametric curve in the plane is a pair of functions:

x = f(t)

y = g(t)

where the two continuous functions define ordered pairs (x,y).  The two equations are usually called the parametric equations of a curve.  The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention to the range of t.  In many applications, we think of x and y “varying with time t” or the angle of rotation that some line makes from an initial location.

A cycloid is the locus of a point on a circle that rolls along a line.  Let’s look at the parametric equations for the cycloid and graph it.

Parametric Equation for the Cycloid:

And it’s graph…

Now let’s look at the GSP sketch.

·        Start off by constructing a line and a point on our line.

·        Next, we’ll construct a line through the point and perpendicular to the original line.  Then construct a circle (NOTE: this will be the size of our rolling circle), using our point on the line as the center.  Choose your point on the circle and the center, and construct a segment (this will be used below to create our rolling circle).

·        Next step:  Select the point on the circle, and the original line, and construct the line parallel.

·        Place a point on the parallel line just created.  Choose the segment created above in the circle and our point just created to construct a second circle.  Construct a segment from center to point on circle to be the spoke on the wheel.

·        All of the construction lines can now be hidden.  We have no need for them.  Next, we’ll animate our sketch.  Choose the center of the circle and the point on the circle.  Go to “Edit” and “Action Buttons” and “Animation”.  We also need to trace the outer point on the circle.  (Highlight the point, “display”, “trace point”)  Click on our Animate button, and we have our cycloid.

Click here to see an animation of this sketch.  Note:  You must have GSP.