Pierre Sutherland

Parametric Equations

 November 13, 2010

This investigation makes use of Graphing Calculator. The file used can be downloaded here.

Please note: you might need a plugin to play the animations in this page.

1. Quick introduction

2. Basic Sine and Cosine

3. More Advanced Sine and Cosine

4. Animated

5. Interesting application

1. Quick introduction

Here is a quick introduction to parametric equations from Brightstorm (if it's not available, just wait a few seconds and click play again):

2. Basic Sine and Cosine

Given the following parametric equations, we have a look at the effect of the parameter t as it goes from the value 0 up to 6.28 (or approximately twice the value of pi).

Figure 2.1: Parametric equations

Notice how the the graph progresses as the parameter t increases (The values of a and b are both equal to 1).

Now we'll have a look at the effect of a and b on the equations:

Figure 2.2: a = 1 and b = 2

Figure 2.3: a = 2 and b = 1

Figure 2.4: a = 2 and b = 2

Figure 2.5: a = 3 and b = 2

Figure 2.6: a = 2 and b = 3

Figure 2.7: a = 3 and b = 3

In the next section we look at more advanced parametric equations.

3. Advanced Sine and Cosine

We expand on the previous set of equations:

Figure 3.1: Advanced parametric equations

Now we'll have a look at the effect on these equations with:

  • a = 1, 
  • b = 1 and 
  • differering values for n:

Figure 3.2: h = 0

Figure 3.3: h = 1

Figure 3.4: h = -1

Figure 3.5: h = 0.5

Figure 3.6: h = -0.5

4. Animated

This is how the graphs would look by moving the the value h from -2.5 to 2.5 in a 100 steps:

5. Interesting Application

  • Locus of a triangle:
The green parametric equations describe the locus of the vertex (x,y) of a triangle with altitude n whose other two vertices are moved, one along the x-axis and the other along the y-axis.

  • Spirographing:
Parametric equations can be used to describe the curve of a rolling circle, try this applet to see how you can create some interesting graphs using parametric equations.

In closing, this is how one artist used spirographing to create some nice graphics: