Weaving Graphs from Parametric Equations

Consider the following parametric equations:

x = cos (3t)

y = sin (7t)  where t ranges from 0 to 2 pi .

We see that the graph weaves around the origin and the pattern has an overall square shape to it. We can take different ranges of t to show more clearly how this weave develops.

So we are interested in looking at the graphs of the parametric equations:

x = cos (a*t)

y = sin (b*t) where t ranges from 0 to 2 pi.

In particular we are interested in the positive integer values of a and b that produce a weave or mesh-like graph.

This generally occurs when a and b are both greater than 1, and neither a nor b is a multiple of the other.

Let's first consider the case where a > b.  We will fix a equal to 7 and show the graphs where b is equal to 3, 4, 5, and 6.

So we can see that as b increases the graph becomes a more tightly would weave.

Now we will look at the case where b > a.  We will fix b constant at 7 and let a take on the values 3, 4, 5, 6.

We see a slightly different phenomenon here. The odd values of a behave much the same way as in the first case. As a (odd values) increases the graph becomes a more tightly wound weave. But when a is even the weave does not appear closed. In fact, the graph traces the half of the curve to a point and then retraces its steps back to the starting position. It then traces the second half of the graph to a second point before retracing its steps again back to the starting point. This is demonstrated below.