Parametric Equations

Allyson Faircloth

For this activity, we will investigate the parametric equation .

By varying a and b at the same time, we can recognize that the graph  seems to be a circle which is getting filled as a and b both approach 300. The filled in circle then begins to go away as a and b  get larger than 300.  It is likely that the circle would begin to fill in again at 600. 

However if we zoom in, we can see that there are actually just overlapping curves which are providing the illusion of the circl being filled in. (See figure below)

Now let's just vary a.  We now get a graph which looks like a spring sitting upright.  As a gets larger, the spring appears as if it is being squeezed together.  However, the graph is actually adding more curves as a increases.  The spring is not being squeezed together at the top and bottom, but it is rather adding more turns to and curves to the middle.  This is the result of the period of cos(at) getting smaller.  The period is given by .  So as a gets larger the period gets smaller. When the period gets smaller, cos(at) then has more opportunities to complete more cycles.  Thus, we receive more curves on our spring.   


Now let's just vary b.  We now get what looks like a spring on its side.  Once again, as b gets larger, the spring appears to be getting squeezed together.  This is also the result of a change in the period of the graph.  This time the period of sin(bt) or is getting smaller as b increases. Just like above, the larger values of b are resulting in more opportunities for sin(bt) to complete cycles.  That results in there being curves added to the middle of the spring.

Therefore, if we take a look again at when a and b are both varying, we can recognize that these two spring shaped graphs are overlapping to make the illusion that a circle is being filled in.  The reason that parts of the spring which were outside the circle are not in this graph is because a and b are varying together, so we are not getting the points for x and y when a and b are different values.