Mathematics
Education
EMAT 6680,
Professor Wilson
Exploration 10,
Parametric Curves by Ursula Kirk
A
parametric curve in the plane is a pair of functions. Where x=f (t) and y=g (t)
and the two continuous functions defined ordered pairs (x, y). The two
equations are called the parametric equation of a curve. The extend of the
curve will depend on the range of t.
My
investigation will be based on exploring how the various values for a
and b affect the parametric curves for the equation below.
When a=b
Observations:
The purple parametric equation yields the purple circle in the picture below. Since a=b=1, the ratio of our circle is also 1. In order to see the whole circle, we must graph our parametric equation from t=0 to t=2pi.
Also, we can convert our original parametric equation by squaring both sides of
the equation when a=b.
By adding both equations we obtain 
which reduces to our unit circle.
Therefore, our purple circle in the
picture below of radius 1 produced by our first parametric equation is our unit
circle.
Similarly, our red equation where a=b=2,
yields a circle of radius 2, our blue equation where a=b=3, yields a circle of radius
3 and finally our green parametric equation where a=b=4, yields a circle of
radius 4.
When a>b
Observations:
In the next case, the value of a is greater than the value of b. As we can see in the image below, these parametric equations yield ellipses. The ellipses are all horizontal with a major vertex equal to a and a minor vertex equal to b.
The center of the ellipse is at (0, 0) and the foci
of the ellipse can be found by solving for c, using the equation a2 – c2 = b2
The general equation is
When a<b
Observations
In
the next case, the value of b is greater than the value of a. As
we can see in the image below, these parametric equations yield ellipses. The
ellipses are all vertical with a major vertex equal to b and a minor
vertex equal to a. The center of the ellipse is at (0, 0) and the foci
of the ellipse can be found by solving for c, using the equation a2 – c2 = b2
The
general equation is
