Assignment # 10- Parametric Curves

The purpose of this assignment is to investigate parametric curves. 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). 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.

First we begin by graphing the parametric equations

x = cos(t)

y = sin(t)

with t ranging from 0 to 2¶. We obtain the following graph.

We can see from this graph that we obtain the unit circle. In order to investigate these equations we will graph the equations

x = cos(at)

y = sin(bt)

where a and b vary simultaneously, a = b = 2, a = b = 3, a = b = 4. We can see from the graph below that if a = b, then we obtain the same graph as before, the unit circle.

Now if we keep a constant (a = 1) and vary b (b = 2, 3, 4), then obtain the following graph.

We can observe from this graph that as b increases y varies more quickly and the amplitude of the curves on the x-axis decreases .

Now if we hold b constant (b = 1) and vary a (a = 2, 3, 4), we obtain the following graph.

This is similar to the previous graph except now as a increases x changes more quickly and the amplitude along the y-axis decreases. We can continue our investigation by graphing

x = a cos(t)

y = b sin(t)

where t ranges from 0 to 2¶. First we will vary a and b simultenously. When we graph the equations with a = b = 2, 3, 4, we obtain the following graph.

We can see from this graph that our original unit circle grows as a and b increase. We can generalize and say that we will always obtain a circle with a radius equal to a and b when a = b.

Now we can examine when b > a. We can hold a constant (a = 1) and vary b (b = 2, 3, 3) to obtain the following graph.

We can see from this graph that as b increases the unit circle becomes an ellipse that stretches along the y-axis.

We can examine when a > b and see the affect of a when we hold b constant (b = 1) and vary a (a = 2, 3, 4).

We can see that a also changes the unit circle into an ellipse, but this time the ellipse is stretched along the x-axis.

We can further investigate parametric curves by graphing

x = a cos(t) + h sin(t)

y = b sin(t) + h cos(t)

as t ranges from 0 to 2¶. We can graph these two equation when a = b = 1 as h varies from -3 to 3 to obtain the following graph.

We can see from this graph that at h = 0 we have the unit circle. The circle degenerates onto the line y = x when h = 1 and onto the line y = -x when h = -1. For -1 > h or h > 1 the curve becomes an ellipse. As h increases the ellipse becomes larger and larger. When h is negative the ellipse is always symmetrical to the y = -x line and when h is positive the ellipse is symmetrical to the y = x line.

Next we graph the equations when a > b (a = 2, b = 1) as h varies from -3 to 3 to obtain the following graph.

At h = 0 we have an ellipse. as |h| increase the ellipse is stretched. When h is positive the ellipse is stretched symmetrically along the y = x line and when h is negative the ellipse is stretched symmetrically along the y = -x line.

Finally we graph the equations when b > a ( a = 1, b = 2) as h varies from -3 to 3 to obtain this graph.

Again at h = 0 we have an ellipse except this ellipse is stretched along the y-axis. Here we also see that as |h| increase the ellipse is stretched. When h is positive the ellipse is stretched symmetrically along the y = x line and when h is negative the ellipse is stretched symmetrically along the y = -x line.