Assignment #5

Parametric Equations

By: Damarrio C. Holloway

A parametric curve in the plane is a pair of functions **x
= f(t)**

** y = g(t)**

where the two continuous functions define ordered pairs **(x,y)**. These
two functions are called the parametric equations of the curve that they
form. The degree of the curve will
depend on the range of **t**, in
which in this exploration, we will denote **t** as the angle of rotation that some line makes from
an initial location. The functions
of **x** and **y** will vary with this time **t**.

Let us explore the different variations of graphs using the
base equation of a cycloid:

x = (a + cos(3t) cos(t)

y = (a + cos(3t) sin (t). In the following graphs we will first set numbers for **a** and we will vary **t** to explore the angles of rotation.

In this initial graph, we have the base equation of our cycloid with a=1 and the rotation of our graph varies over t ranging from 0É..1. In this graph, we have one curve, or as we will see later, we will have half a flower leaf.

LetÕs explore different ranges of **t** shall we.

*
Figure 2 Figure
3*

Figure 2 shows the range of **t**: 0É.2, while Figure 3 shows the range **t**: 0É.5.
In these images, we see that the set range for **t** determines the number of curves the figure will
make.

A complete look at the rotation:

yields a three leaf rose.

Let us now explore variations of **a** in our equation.

This figure displays the rotations when a=0.5 and t has rotations ranging from 0É.8. We see that it has two sets of complete rotations when a = 0.5.

As the range of t is increased by a multiple of 10, the rotations of the graph increase, giving the graph a bold look.

Even with a multiplication of 5 from the previous graph, the rotation of the curve has a drastic increase.

*Figure 5 Figure
6*

When a = 2, we have a quite different graph. The rotation of the graph does not go
through the origin as did the original graphs.

We have seen what varying **a** and **t** will do to the
graph, now letÕs take a quick look at a change in Òleaves.Ó

Figure 7

In Figure 7, we now have a 4 leaf rose because of the increase from 3 to 4 for the ÔtÕ coefficient.

or even

*Figure
8*

Figure 8 yields a 5 leaf rose with the increase of 1 from
figure 4. The rose rotates through
the origin because a = 1 as in the original equation. The shapes and curves for this particular parametric
equation are endless. As you can
see, the higher you set your t-values, the more rotations you can create. Also, with more rotations and any
increase in youÕre a-values, the closer your graph reaches the origin.

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