Parametric Equations

Assignment 10

A parametric curve in the plane is a pair of functions

where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention 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.

Explorations

Graph :

How would the graph change for various a and b where 0 ≤ t ≤ 2π?

Up above a and b equaled one.  Will the graph change if a and b were still equal however neither equaled one?  Let's try it.

Graph: x = cos (at) and  y = sin (bt) for a = b = 2  and a = b = -1.

As you can see as long as a and b are equal the graph looks the same as our 1st graph.  What about when a>b?

I noticed when b is 1/2 of a , a sideways parabola occurs!  Let's explore some more.

As you can see, some interesting shapes are occurring.  Let's see what happens when a<b.

When a is 1/2 of b the same shape occurs again, however, it is different than when b is 1/2 of a.  Let's explore some more.

When a<b, the odd numbers take on a slightly different shape than the even.  The number of loops in the graph increases as the b increases as well.  The graph is also more horizontal now, where as, when a >b, the graphs were more vertical.