Parametric Curves

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What is a parametric curve?

A parametric curve in the plane is a pair of functions, such as:

x = f(t)

y = g(t)

where the two continuous functions define ordered pairs (x,y). The two functions (or equations) are usually called the parametric equations of a curve.

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What does a parametric curve look like geometrically?

Well, each distinct pair of functions (or equations) will produce a unique geometric interpretation. We will use Graphing Calculator 3.2 software to investigate the following parametric curve.

In this software, equations for parametric curves must be entered in vector form. Let's investigate the following two equations:

, where 0 <= t <= 2pi

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What happens when these equations or their respective angles are magnified?

Let's consider the following general examples when a = b, respectively:

Example 1 (when a and b are multiplied with the functions, respectively)

, where 0 <= t <= 2pi

There is a lot going on here that one may not realize upon first glance. For example, where a = b and a, b are equal to or greater than zero, the parametric curve will simply expand its "radius" equal to a, b. If a = 1, b = 1, then the parametric curve will be a circle centered at (0,0) with radius = 1. If a = 2, b = 2, then the parametric curve will be a circle centered at (0,0) with radius = 2, as observed above. And so on...

What would happen if a did not equal b? Let's hold a = 1 and vary b from 1 to 3:

Now, hold b = 1 and vary a from 1 to 3:

....how interesting!!!

Now, let's consider the parametric curves when a, b magnify the respective angles of each function.

Example 2 (when a and b are multiplied with the angles of the functions, respectively)

, where 0 <= t <= 2pi

When a = b, the graph will always look like the one above! On the contrary, when a is held constant and b varies from 1 to 10, the parametric curve changes in this way. Likewise, when b is held constant and a varies from 1 to 10, the parametric curve changes in this way.

All of the above observations are general manipulations of the parametric curve created by the cos (t) and sin (t) functions.

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