Exploring Parametric Equations

By: Sydney Roberts

For this exploration, we will be investigating parametric equations of the form

LetŐs start by investigating what will happen when a and b vary at the same rate (are defined as the same constant) from 0 to 10. We can see that as this constant is around 1, we have a complete circle that is formed.

After n reaches 1, we can assume that the circle continues to loop around. Hence, allowing n to vary from 0 to 10 will allow for 10 completed revolutions to be made starting from t=0 and moving counterclockwise.

Now letŐs consider what happens if we vary a and b (again at the same rate) from -10 to 0. This time, as n approaches 0, the circle seems to ŇunwindÓ counterclockwise from t=0. Therefore, it seems that the negative values cause the parametric circle to start with 10 (because we started at -10) revolutions, and then unwind in a counterclockwise motion.

The interesting investigations come from when a and b are not the same constant. Consider some examples below and try to make your own conjectures.

At this point you should start to see a pattern form that deals with the ratio of b to a. We see that the parametric curves tend to intersect and form Ňloops.Ó It appears as if the number of these loops is . However, we have only looked at cases where a ˛ b. What happens if b ˛ a?

Again, look at the following examples.

Here, the relationship is harder to see, but there are some conclusions we can draw:

á      If  is an odd integer, then the parametric curve is non-intersecting

á      If  is an even integer, then the parametric curve greatly resembles the previous explorations of looking at when a ˛ b except this time, the parametric curve is forming on the y axis instead of the x axis.

For further exploration, look at these animations for different relationships between a and b.

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