Amberly Roberts
Assignment 10: Parametric Equations and Curves
If you are not familiar with parametric equations, I recommend that you visit this website to gain some insight. It provides some general information on what parametric equations are and why they are necessary within the study of mathematical representation. Now, let's investigate.
Consider the parametric equation defined by...
Notice that the parameters of "t" have been set. The result was a circle with radius = 1 whose center is the origin of the coordinate plane. Now, using the same range for "t", we will slightly change the equations and observe the change (if any) in the graph.
We will use the following equations for our investigation.
When a = b, the graphs suggest that the parametric curve is always a circle with radius 1 centered at the origin.
The animations below are very powerful representations. Observe how the graphs change as the value of n varies.
Here we can observe what happens as "a" defined by "n" in the equation above gets increasingly larger than "b." In this animation, the value of "b" is a constant 1. If you slow down the animation, the parametric curve seems to be associated with a particular motion. The curve appears to be marking off a pathway for an object to travel. The circle seems to break when (x, y) reaches (2pi, 0) and begins an upward coiling pattern. The pathway of the coil resembles the oscillating pattern we normally associate with the graphs of trig functions. The more variance we get between the values of "a" and "b," the more coils we have in the graph. The "period" of the coils appears greater when closer to the origin. 
Here we can observe what happens as "b" defined by "n" in the equation above gets increasingly larger than "a." In this animation, the value of "a" is a constant 1. This animation is very similar to the one above. However, all of the motion that seemed vertical in the previous animation appears horizontal in this animation.

Here are a few more graphs to explore. Determine whether they support or negate the claims and observations we made based on the animations.
These curves suggest that the values of "a" and "b" having common factors affects the number of coils. Our conjecture based on the animation did not account for this.
Now, this animation shows that the curve seems to come together and consolidate when "a" and "b" share common factors. 