Assignment #10

Parametric Curves

by

Vicki Tarleton and Michelle Jones

Parametric equations are defined by a pair of functions

x = f(t) and y = g(t) OR

x = x(t) and y = y(t)

The value of t determines a point (x, y) on the coordinate plan. As t varies, the point (x, y) will trace out a curve, called a parametric curve, where t is the parameter.

In this investigation, we considered the parametric equations

 x = 4sin y = 3sin (t)

and considered the graphs for different values of .

The graph below is for the value of = . As time t changes, the curve that is formed is shaped like

a bowtie, going through the origin. There are two loops formed here, crossing the x-axis at (-4, 0) and

(4, 0). The range goes from [-3, 3].

The next graph illustrates what occurs when = . This graph also goes through the origin and its domain is [-4, 4] and the range is [-3, 3]. The graph now presents four loops.

What might you conclude from these two graphs? Click here for a quick summary.

Now for other values of . The graph shown below is for = . Notice here that there are three loops or three humps and the range for t did have to be increased to get the complete graph since more curvature is occuring.

How do you think the graph of = might look? Click here to see if your guess if corrrect.

We became curious about some other values of . The graph = was somewhat surprising.

It was not a closed or connected figure. This lead me to the conclusion that is seems as though when

a + b = an odd value, the graph is connected and when a + b = an even value, the graph is not connected.

As a final investigation, we graphed = .

It followed the same pattern and is connect since 12 + 13 = 25 is odd and has 13 humps. As the fractions get closer to 1, the curve that is traced out becomes more involved and more elaborate. The path that t follows beomes longer.