Using Graphing Calculator let us see how the graphs of these parametric equations appear for the given
For this choice of a and b we get a curve that looks like a bowtie. The domain of the curve is from -4 to 4 and the range -3 to 3 which are our coefficients for the x and y respectively. The point of intersection (or where the curve overlaps) is (0,0).
When we increase b to 4 we see that now we have four loops in the same domain and range. When we increase b to any positive even integer we get that many loops, so for instance a=1 and b=6 will give us six loops. The curve overlaps at (0,0) and at around x=-2.75 and 2.75.
Here we see that a=2 and b=3 gives us a curve which looks like two boomerangs facing each other. The curve overlaps at (0,0), around (-2,2), (-2,-2), (2,2), (2,-2), and around x=-3.5 and 3.5.
Increasing a and b to 12 and 13 respectively we get a more busy curve. We get a lot more intersection points, or points where the curve overlaps.
Let us now compare these Lissajous curves with
With these graphs our domain and range is -1 to 1, 1 also being the coefficient of our sin equations. The graphs are identical except for when a=12 and b=13. Instead of a smooth curve, it is made of multiple jagged lines.