Converting Equations to Parametric Form: Exploring the Purpose and Process

[Fuz]

How do you take an equation and turn it into a parametric one? Eliminating the parameter is straightforward and easy; but I'm trying to go the other way.

e.g. what steps would one take to convert some curve like

y = x²

or

x² + y² = 1

into parametric equations with 't' as the independent variable?

 

[Some Pig]

There are number of ways, for conics, some traditional ways are:
For y=x²; x=t, y=t².
For x²+y²=1; x=cost, y=sint.

 

[Fuz]

What was your motivation for setting x = cos(t) and y = sin(t)? Is there a method for turning an equation into a set of parametric equations?

 

[Dickfore]

actually, eliminating the parameter is equally hard. If the equation is in an explicit form $y = f(x)$, then, whatever you take as a parametric representation of x, $x = \phi(t)$, you can find $y = y \left[ \phi(t) \right] = \psi(t)$. In other cases, there is no general rule. For example, eliminate the parameter in:
$$x = t \, \cos t, \ y = t \, \sin t$$
describing an Archimedian spiral.

 

[HallsofIvy]

What was your motivation for setting x = cos(t) and y = sin(t)? Is there a method for turning an equation into a set of parametric equations?

You hopefully have learned that $cos^2(t)+ sin^2(t)= 1$. Comparing that to $x^2+ y^2= 1$ should make the parameterization obvious. But there is no general way of parameterizing (except that if y= f(x), you can always use x= t, y= f(t)).

 

[chiro]

Hey Fuz.

On top of what the other posters have said, it does help immensely if you know the dimension of the system.

If you are dealing with a one-dimensional system (like a line), then there are techniques that you can do to make a move towards getting a complete analytic parametrization.

 

[Fuz]

You hopefully have learned that $cos^2(t)+ sin^2(t)= 1$. Comparing that to $x^2+ y^2= 1$ should make the parameterization obvious. But there is no general way of parameterizing (except that if y= f(x), you can always use x= t, y= f(t)).

Yes I have learned this, and that basically answered my question, but then what is the point in just setting x equal to t?