By:Lauren Lee




†††† A parametric curve in the plane is a pair of functions

(x = f(t) and y = g(t)), where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention the range of t.In many applications, we think of x and y "varying with time t " or the angle of rotation that some line makes from an initial location.





Letís look at the following parametric equations and vary a and b:



for 0 < t < 2pi




What happens when we set a and b equal?


Letís look ata = 2 andb = 2











You will notice that we get a circle with radius one.



Further investigations revealed to me that this will always be the case††††† when a = b.







Letís see what happens when a and b arenít equal.

Weíll let a = 2 and vary the values of b.



What happens when b = 4?












What happens when b = 6?












And when b = 10 :










Notice in these examples that the number of circular shapes created is equal to the value of b divided by the value of a.






Now letís see what happens when we hold b constant and vary a.

Letís make b = 2.




Let a = 4













Let a = 6













Let a = 10













These values of b seem to produce the same curves for a, except that they are rotated 90 degrees.






Just for fun


Now letís see an example for large values of a and b.


Here a = 19 and b = 29