Investigation of Parametric Curves

By: Denise Natasha Brewley

Consider the following parametric equations of
the following form for *0 < t < 2p* :

*x = a cos(t)*

*y = b sin(t)*

We will begin this investigation by considering
different values of *a* and *b*. First let us look at the case when we have
*a
< 0 < b*. Namely when *a = -1/2* and *b = 1*.

Notice that the parametric equations give us what
appears to be an ellipse centered at the origin. An ellipse is the set of
all points *P *in a plane such that the sum of the distances of *P*
from two fixed points in the plane is constant. What we now want to do is
verify that this is an ellipse. Recall that each value of *t*
determines a value of *x* and *y* in rectangular
coordinates. To eliminate the
parameter *t*, we solve the system of parametric equations and substitute
our respective values of *cos(t)* and *sin(t)* into the Pythagorean identity
*cos ^{2}t
+ sin^{2}t = 1*. So it follows that we have

*cos(t) = -2x and sin(t) = y*

*(-2x) ^{2} + (y)^{2} = 1*

*4x ^{2} + y^{2} = 1*

*(x/(1/2)) ^{2} + (y/1)^{2} = 1*

The result is an ellipse in rectangular form with
major axis of length *= 1* and minor axis of length *= 1/2* just as we
conjectured. The major axis is longest distance found through the center
and foci of an ellipse. Similarly the minor axis is the shortest distance
found through the center of the ellipse. This means that
the distance from the origin to the major vertices is *1* and the distance from
origin to the co-vertices is *1/2*. In general we can determine the
length of the major axis and minor axis of the ellipse by simply looking at our
parametric equations. The major axis is the the longest length between a
and b. And the minor is the shortest length between *a* and *b*.
Since we are given that *[x = -0.5 cost and y = sint]* with *a = -1/2*
and *b = 1*. It follows that the length of our minor axis is *|a|=
1/2* and our major axis is* |b|= 1; 1/2 < 1*. This also tells us
if the major and minor axis will be vertical or horizontal. Notice that
for this parametric curve it has a major axis that is vertical and a minor axis that is horizontal.

Because we have the equation of an ellipse, as
indicated above, we can make a conjecture. But before that, let's ask some
questions. What will happen if we have the
case *a < 0 < b* and *0 < a < b*? Will this give us the same
parametric curve? Let us see what will happen. If we are given the following, *x = a cos(t)* and *y = b sin(t),* it follows that

*cos(t) = x/a and sin(t) = y/b*

*(x/a) ^{2} + (y/b)^{2} = 1*

*x ^{2}/a^{2 }+ y^{2}/b^{2
}= 1*

So we can say for values *a < 0 < b* and *0
< a < b* such that*, |a|< |b|, *we will have the same parametric curves in *t* and equation in
rectangular form in *x* and *y*. Try this for the parametric
equations *[x = -0.5 cos(t) and y = sin(t)]* and *[x = 0.5 cos(t) and y = sin(t)]. *
Now let's look at a family of parametric curves that satisfies this claim.

Using this model we will continue our exploration
of
other values of a and b. We will now consider when *a = b*,
specifically when *a = b = -1 < 0*.
If we continue this investigation as we did in
the previous example, we will obtain an ellipse, where the major and minor axis
are the same. We can also say that the distance from origin to the
vertices and co-vertices are also the same. But this is just a circle centered at the origin
with radius *= 1*, namely *x ^{2} + y^{2} = 1*.
It is important to note here that in the
case when

So for values of a and b such that *|a|=|b|*, we can consider the
family of the parametric curves. Can you guess what the
equation of each curve is based on our discussion?

Now let us look at the case when *0 < b < a*
and *b < 0 < a*. We will consider case *a = 1* and *b = 1/2*
and generalize. Repeating the steps above, we obtain the
following:

*cos(t) = x and sin(t) = 2y*

*x ^{2} + (2y)^{2} = 1*

*x ^{2} + 4y^{2} = 1*

*x ^{2} + (y/(1/2))^{2} = 1*

In this case the major axis is horizontal and the
minor axis is vertical. Which follows since *|b|= 1/2 < |a|= 1.*

So we can generalize for this case as well. That is, when |b|<|a|.