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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.
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Let’s look at the
following parametric equations and vary a and b:
for 0 <
t < 2pi
What happens when
we set a and b equal?
Let’s 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.
Let’s see what
happens when a and b aren’t equal.
We’ll 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 let’s see what
happens when we hold b constant and vary a.
Let’s 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.
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Now let’s see an
example for large values of a and b.
Here a = 19
and b = 29