Parametric Curve Exploration

by

Chad Crumley

 

A parametric curve in the plane is a pair of functions

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 the range of t .  In many applications, x and y "vary with time t "or an angle of rotation that some line makes from an initial location.  This definition is from Dr. Wilson at UGA.

For this exploration, we are going to vary values of a and b in the following parametric equation (where t is between 0 and 2 * pi):

 

Here is a graph when a and b are both 1:

The orientation is important.  Remember that cos(0) = 1 and sin(0) =0 and that any point in the plane can be written (cos x, sin x), then the circle starts at (1,0) and moves in a counterclockwise direction. In fact as long as a=b, then the resulting figure will be an arc of a circle or a circle.  As long as a and b are larger than 1 (and still equal), then a circle will result since t is between o and 2 * pi.  If a and bare values between 0 and 1 (and still equal), then an arc will result since the path would be some portion of 360 degrees.  Again, if a and b are still equal and larger than 1, then the path would make more than 1 rotation. If the value for a and b are negative, then the path would travel clockwise. 

Here is a movie to try to convince you:  Click here.  (Here a and b values are equal and run between 0 and 10).

Look at the graphs below:

The above examples are ellipses.  The pink graph has foci on the x-axis.  The blue graph has foci on they-axis because the larger number in front of cos t or sin t.  Also notice that the ellipse is stretched to the number in front of cos t along the x-axis and the number in front of sin t along they-axis.  Interesting?

Back to the assignment, lets look at  for various values of a and b, were they are not equal. 

Here is a movie of the parametric equations above with b =1 and a varies between -10 and 10 in 100 steps.  Click here.

Let us look at the graph when a=0. 

Since sine alternates between -1 and 1 and cos (0) = 1, the graph results in a vertical line segment at x=1.  (Sine is the y-coordinate.)

Here is a movie of the parametric equations with a=1 and b varies between -10 and 10 in 100 steps.  Click here.

Similar to above, here is the graph when b=0.

Look closely and you can see another line segment from (-1, 0) to (1, 0) since cosine alternates between -1 and 1 and sin (0) = 1.  (Cosine is the x-coordinate.) 

In conclusion, any equation can be separated into parametric equations by letting x equal a variable, usually t, and solving for y in terms of that variable.  For example, the parabola y = x2 can be written as follows:

Parametric equations can results in many “neat” graphs.  For example here is the butterfly curve discovered by Temple H. Fay.

 



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