Parametric Curves of Sin and Cosine

I will be investigating the parametric curves of the form x = cos(at) and y = sin(bt) for various values of a and b. For all of the following curves, 0< t<2p

a = 1, b =1

This figure shows that the graph travels through one exactly; not only here but at +/-1 and on both axes.

The next two figures have a and b interchanged with the values of 10 and 1:

a = 10, b = 1

a = 1, b = 10 (above)

When both are equal to ten, the 'circle' has a definite thickness that is well illustrated in the figure after this one:

a = b = 10

Close up of the above graph:

Here is the 'thickness' of the circle illustrated as one of its characteristics.

For a = 100, b = 1 (below):

The spear-like projections are evident here and in the next picture:

Notice the symmetry of the middle lines of the top and bottom 'spikes'. Both are pointing toward the x axis. The middle line is vertical only on the axis.

Below, a = 1, b = 100:

When the values for a and b are 'switched' the pointed sides rotate 90 degrees.

The points ascend to 1 eventually, and only toward the left side; The spikes on the bottom ascend to -1 on the left side as well:

a = 100, b = 100:

Note the symmetrical property of the 'mesh' or net:

a = 1000, b = 100:

This is a very unique design; the boundaries are very interesting.

Now, as the values become smaller the design is less ornate:

a = .01, b = 1:

a = .05, b = 1:

Below, a = 0.1, b = 1:

a = 0.5, b = 1, below:

a = .9, b = 1, below:

a = .98, b = 1:

a = 1, b = .01:

a = 1, b = .05:

a = 1, b = .09:

a = 1, b = -.09; Note, with b = -.01 and -.05 the graphs are like those above with the exception that they are located below the horizontal axis as below:

When a = b = -.01, -.05 respectively, the graph is of a much shorter curve falling below the horizontal axis at 1 and moving down. Here is where a = b = -.09:

There is a great deal going on that needs much more attention in future work.