†††† 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 at† a = 2 †and† b = 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
These values of b seem to produce the same curves for a, except that they are rotated 90 degrees.
† Now letís see an example for large values of a and b.
† Here a = 19 and b = 29