Jernita Randolph

Exploration of Parametric
Curves

LetÕs
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, letÕs observe values
where a ² b,

Here we have half a curve.

Here we have 2 loops.

Notice a pattern?

When a ² b the number of
loops is with symmetry along the x-axis.

Now, letÕs observe values
where a ³ 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 ² 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.