Brenda King

Parametric Curves

Introduction:

Parametric curves use a function to define the coordinates (x,y) for any point on a curve.  Usually the function is based on t for time. As t changes the path of the graph changes and can be mapped out.  This investigation will vary the parameters a and b in the equations

x=cos (at) and y = sin (bt) for 0 < t < 2p.

Change parameter a holding b=1

The boundaries, or extent, of these graphs appears to be a square area the size of a 1 by 1.  Is there a consistent pattern in the curves?

In diagram 1, the first two graphs look like the familiar unit circle and parabola.  Doing the conversion from parametric to rectangular form should verify this to be the case.

The other graphs however are more interesting.  It would appear the number of times the graph reaches the outer boundary, a loop, depends on the values of a or b. The number of side loops depending on a and the number of top loops depending on b.

With b=1, there is one loop along the top of each graph.  The number of loops along the side appears in the graphs below agree with the value of a with 1, 2, 3, 4, and 8 loops.

Diagram 1

Change parameter b holding a=1

In a likewise manner, with a=1, there is one loop along the side of each graph and the number of loops along the top are 1, 2, 3, 4, and 8 loops.

Diagram 2

Parameter a = b

When the two parameters are equal, then the graph stays a circle. For example,

NO common factors in parameters

I found some consistent graphs supporting the idea that loop counts are linked to parameters a and b .

Diagram 3

Parameter with common factors of two

However, when the parameters have something in common, then the number of loops is reduced by the common factor.

Diagram 4

Parameter with common factors of three

In the case of odd factors, the exchange of a and b values orients the graph in a 90 degrees rotation.

Diagram 5