By Jaepil Han
2. For various a and b, investigate
(1) Varying a from 1 to 5 for integer values
If a goes large, then the curve of the parametric equation has more frequency than before. However, the curves located only between y=-1 and y=1. As we may know, when the parameter a is 1, x2+y2=1 because (cost)2+(sint)2=1. Also, When the parameter a is 2, x=1-2y2. And so on. Interstingly, when the parameter a is even the graph has a continuous loop. When the parameter a is odd the graph hasn't a continuous loop.
Here's the graph of the parametric equation when a=50.
(2) Varying a from 1 to 5 for integer values
Similarly, the value of b getting greater, the curve of the parametric equation has more frequency.
Here's the graph of the parametric equation when b=50.
Even we change the values of a and b, it only changes the frequency of the curves. All the curves generated by the values of a and b make a sort of square.
Here's the graph.
Here's useful links releated to Lissajous-curves.
Lissajous-curves by WolframMathworld
Lissajous-curves by MacTutor
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