Jernita Randolph

Exploration of Parametric Curves

Let’s start with an observation of and for , when a=b.

Notice that regardless of the integer value as long as a=b, we have a circle.

Next, let’s observe values where a ≤ b,

Here we have half a curve.

Here we have 2 loops.

Notice a pattern?

When a ≤ b the number of loops is with symmetry along the x-axis.

Now, let’s observe values where a ≥ b,

We have 5 loops with symmetry to the y-axis.

Here we have 3 loops with symmetry to the y-axis. The same pattern follows as with a ≤ b, except now the number of loops is equal to with symmetry to the y-axis.

What happens when a is not evenly divisible by b? Here we have 3 loops with symmetry to the y-axis superimposed over 2 loops with symmetry to the x-axis.

And here we have 5 loops with symmetry to the y-axis superimposed over 2 loops with symmetry to the x-axis.

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