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.
3. More Advanced Sine and Cosine
4. Animated
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
![](pics/equation1.jpg)
Figure 2.1: Parametric equations
Now we'll have a look at the effect of a and b on the equations:
![](pics/a1b2.jpg)
Figure 2.2: a = 1 and b = 2
![](pics/a2b1.jpg)
Figure 2.3: a = 2 and b = 1
![](pics/a2b2.jpg)
Figure 2.4: a = 2 and b = 2
![](pics/a2b3.jpg)
Figure 2.5: a = 3 and b = 2
![](pics/a2b3_001.jpg)
Figure 2.6: a = 2 and b = 3
![](pics/a3b3.jpg)
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:
- a = 1,
- b = 1 and
- differering values for n:
![](pics/h0.jpg)
Figure 3.2: h = 0
![](pics/h1.jpg)
Figure 3.3: h = 1
![](pics/h-1.jpg)
Figure 3.4: h = -1
![](pics/h=05.jpg)
Figure 3.5: h = 0.5
![](pics/h-05.jpg)
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:
- Spirographing:
In closing, this is how one artist used spirographing to create some nice graphics:
![](pics/art.jpg)