# Thread: URGENT Parametrics

1. ## URGENT Parametrics

Let be the graph of the parametric equations

What is the length of the smallest interval
such that the graph of these equations for all produces the entire graph ?

2. ## Re: URGENT Parametrics

Originally Posted by orange
Let be the graph of the parametric equations

What is the length of the smallest interval
such that the graph of these equations for all produces the entire graph ?
Cos[4t] needs (2pi)/4 = pi/2

Sin[6t] needs (2pi)/6 = pi/3

what's the least common multiple of 1/2 and 1/3 ? So what's the minimum interval needed for a full cycle?

3. ## Re: URGENT Parametrics

1 is their LCM. So I guess the minimum interval is just pi?

4. ## Re: URGENT Parametrics

give that man a kewpie doll!

5. ## Re: URGENT Parametrics

Hello, orange!

I "eyeballed" the problem.
I think I have the solution.

Let $\displaystyle G$ be the graph of the parametric equations: .$\displaystyle \begin{Bmatrix}x &=& \cos(4t) \\ y &=& \sin(6t)\end{Bmatrix}$

What is the length of the smallest interval $\displaystyle I$ such that the graph
of the equations for all $\displaystyle t \in I$ produces the entire graph of $\displaystyle G$ ?

When $\displaystyle t = 0\!:\;\begin{Bmatrix}x &=& \cos(0) &=& 1 \\ y &=& \sin(0) &=& 0\end{Bmatrix}$

At the "start", the graph is at $\displaystyle (1,0).$

Here is what I found:

. . $\displaystyle \begin{array}{c|c|c|} t & \cos(4t) & \sin(6t) \\ \hline 0 & 1 & 0 \\ \frac{\pi}{4} & \text{-}1 & \text{-}1 \\ \frac{\pi}{2} & 1 & 0 \\ \frac{3\pi}{4} & \text{-}1 & 1 \\ \hline \pi & 1 & 0 \\ \vdots & \vdots & \vdots\end{array}$

. . and the cycle repeats.

Therefore: .$\displaystyle I \,=\,[0,\,\pi]$

6. ## Re: URGENT Parametrics

Hi,
If a given parametric curve is periodic, finding its period is a mystery to me. Here's a problem that I tackled some time ago with its solution; you might try your hand at it (my proof was somewhat long and involved). I wish I knew some general procedure to find the period of such periodic functions.