Parametric Curves

by

Lizzy Shaughnessy

Assignment 10

In this write-up, we will look at parametric curves, particularly curves involving sine and cosine.

To start, recall that "a parametric curve in the place is a pair of functions

where the two continuous functions define the ordered pair (x,y). The extent of the curve will depend on 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" (taken from information for Assignment 10). You can also click here for another brief review parametric curves.

We will focus on the following equations:

We will consider these equations when a = b, a < b, and a > b.

To start we will look at the equations. We will start with a = b = 1 and t = 1.

In this table, we have a = b = 1 and varying multiples of t. As the coefficient of t increases, the curve completes a circle with a radius of 1. For the remainder of this write-up we will focus on when the coefficient of t is 2π. We showed in Assignment 1 that if the lead coefficient of sine or cosine is a negative number then the function is simple reflected over the x-axis. In the parametric equations we are created circles so there would be no change in the graph if we used negative values of a or b so we will only consider positive value of a and b.

Now let's see how the parametric curves change if we increase the value of a and b. We will change a and b for each case of a = b, a < b, and a > b.

 a = b Graph a < b Graph a > b Graph

As we can tell from the above table, the value of a and b determines the axes of the parametric curve. The value of a determines the horizontal radius of the curve and the value of b determine the vertical radius of the curve. This is why we appear to have a circle when the a and b equal.

Here are links for a Graphing Calculator 4.0 file to look at these curves with t varying from 0 to 2π.

a = b

a > b

a < b

Now we will consider the parametric equations

Just as we did before, we will look at the cases when a = b, a < b, and a > b. We will only show graphs when t is a multiple of 2π.

 a = b Graph a < b Graph a > b Graph

We first observe that in these equations the curves are actually a line, not a curve like in the first case. The value of a and b affect the slope of the line created by our equations. In the case of a = b, the slope of the line is -1. When a < b the line becomes steeper and the slope becomes -b/a. When the a > b, the line flattens out and the ratio for the slope remains -b/a.

Here are links for a Graphing Calculator 4.0 file to look at these curves with t varying from 0 to 2π.

a = b

a > b

a < b

Finally, let's look at the last set of parametric equations:

 a = b Graph a < b Graph a > b Graph

Just as in the previous cases, the value of a and b determines where the curve touches the x-axis and y-axis, respectively. We also notice that when the exponent of the sine and cosine equations is an odd number the curve exists in all four quadrants but when the exponent is even the function only exists in the first quadrant.

Click here for Graphing Calculator 4.0 file to look at these curves varying from o to 2π.

a = b

a > b

a < b

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