Finding when a parametric equation self-intersects

  • Thread starter Mr Davis 97
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In summary, the task is to find the two values of t at which the curve crosses itself, given the parametric equations x=2-πcos(t) and y=2t-πsin(t) on the interval [-π, π). To solve this, one can graph x against t and y against t and use the relationship between t1 and t2 to narrow down the options. It must always be the case that t1 + t2 = 0 because the function for x is even, and the function for y is odd.
  • #1
Mr Davis 97
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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?
 
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  • #2
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##.
 
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  • #3
andrewkirk said:
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##?
 
  • #4
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##?
 

Related to Finding when a parametric equation self-intersects

1. How can I determine if a parametric equation self-intersects?

To determine if a parametric equation self-intersects, you can plot the equation and look for points where the curve crosses itself. Another method is to set the two equations equal to each other and solve for the variable. If the resulting equation has multiple solutions, then the curve self-intersects.

2. Can a parametric equation self-intersect at multiple points?

Yes, a parametric equation can self-intersect at multiple points. This occurs when the curve crosses itself more than once.

3. What does it mean if a parametric equation self-intersects?

If a parametric equation self-intersects, it means that there are points on the curve where the x and y coordinates are the same. This results in the curve crossing over itself, creating a point of intersection.

4. How do I find the points of self-intersection for a parametric equation?

To find the points of self-intersection for a parametric equation, you can set the two equations equal to each other and solve for the variable. This will give you the x-coordinate of the points of intersection. You can then substitute this value into one of the original equations to find the corresponding y-coordinate.

5. Can a parametric equation self-intersect at a point where both x and y are undefined?

Yes, a parametric equation can self-intersect at a point where both x and y are undefined. This occurs when the curve has a vertical or horizontal tangent, which means the slope of the curve is undefined at that point.

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