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 cos2t + sin2t = 1. So it follows that we have
cos(t) = -2x and sin(t) = y
(-2x)2 + (y)2 = 1
4x2 + y2 = 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
x2/a2 + y2/b2 = 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 x2 + y2 = 1. It is important to note here that in the case when a = b > 0 , the resulting graph would be the same as in the negative case which was already discussed.
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
x2 + (2y)2 = 1
x2 + 4y2 = 1
x2 + (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|.
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