Investigation of Parametric Curves


Nathan Wisdom

We want to investigate the parametric equations:



0≤ t ≤50

When a and b = 1 we get the following graph


Fig 1

This is a surprising result, because I did not expect to get a straight line with these parametric curves.

Is it possible to obtain other well known curves, for instance parabola, ellipse, etc? What values of a and b would produce such curves?

We now look at the graph for a/b =1/2, 1/4, 2/3, 12/13 etc.

Fig 2. a =1, b =2

Note that you get a boa tie loop with amplitude 3. Note also the graph is bounded by a rectangle.
We see that there are two peaks at the top and two peaks at the bottom. Also there is one knot, which is located at (0,0)

Fig 3. a=1, b=4

In Fig 3. We note when b = 4 there are four peaks and three knots. The curve is still bounded by the same rectangle.

Fig 4. a=1, b =6

When b = 6 we have 6 peaks and 5 knots. The curve is closed

Fig 5. a=1, b= 10
Note the loops are not closed any more b = 10, and the right hand side of the graph is now compressed. Also there are no knots on the left hand side of the curve.

Now look at another set of graphs

Fig. 6 a=2 b=1

Compare Fig6 with Fig 2. Is this a simple transformation?

Note the graph is bounded by 3, -3 on the y-axis and 4, -4 on the x- axis.
It is evident that the number of loops is determined by a. Look at the graph when a = 6 and b =1.

Fig. 7 a=6, b=1

We now look at the curve for which a= 2 and b =3 (Fig 8). We have have 2 peaks left and right and three peaks top and bottom. The knots appear to have a simple relationship with a, and b. There are three knots on the x-axis, 2 knots above and 2 below.
Fig 8. a=2, b=3

Fig.9 a=2, b=7
Again the peaks are consistent but the knots are complex.

Here is another curve. Is this a single transformation?
Fig 10 a=3, b=2

Compare Fig10 with Fig 8.

Fig 11 a=3, b=4

Fig 12. a=3, b=8

Fig 13 a=4, b=7

Fig 14 a=12, b=13

This class of parametric curves are called Lissajous curves. Note that they are all bounded by a rectangle of dimension 2a by 2b.
These curves have significant application in Physics and Particle Mechanics.
We now compare the graphs with the equation:

We note the same set of graphs bounded by the unit square.

What values of a and b would produce this graph?

Or  This?

Here is a real Challenge. How might this graph be formed?