Finding when a parametric equation self-intersects

[Mr Davis 97]

Homework Statement

If $x=2-\pi cost$ and y = $2t-\pi sint$, then find the two t's at which the curve crosses itself, where t is on the interval $[-\pi, \pi)$

Homework Equations

The Attempt at a Solution

I really don't know where to start besides just looking at the graph of the parametric equations. Is there a general method for this?

 

[andrewkirk]

You need to find $t_1$ and $t_2$ such that
$$x(t_1)=2-\pi\cos t_1 = x(t_2) = 2 - \pi\cos t_2,\quad\textrm{and}\quad y(t_1)=2t-\pi\sin t_1=y(t_2)=2t-\pi\sin t_2$$
Start by graphing $x$ against $t$ and you should see a relationship that has to hold between $t_1$ and $t_2$, which will radically narrow down the options you need to consider.

Then draw a graph of $y$ against $t$.

 

[Mr Davis 97]

You need to find $t_1$ and $t_2$ such that
$$x(t_1)=2-\pi\cos t_1 = x(t_2) = 2 - \pi\cos t_2,\quad\textrm{and}\quad y(t_1)=2t-\pi\sin t_1=y(t_2)=2t-\pi\sin t_2$$
Start by graphing $x$ against $t$ and you should see a relationship that has to hold between $t_1$ and $t_2$, which will radically narrow down the options you need to consider.

Then draw a graph of $y$ against $t$.

Must it always be the case that $t_1+t_2=0$?

 

[andrewkirk]

Yes, because the function that gives $x$ in terms of $t$ is what's called an even function. A related type of function is an odd function - see further down on the same link. What sort of function is the one that gives $y$ in terms of $t$?