# Standard to Parametric

Is it possible to transform a standard equation into parametric form? I know it can be done the other way around, but I am curious if it is possible to split it into two. If so, how can it be done? Thank you.

Note by Dacota Sprague
9 months ago

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Can you give an example of an equation that you want to transform?

- 9 months ago

Sure. I will provide a relatively simple one... if there is a formula, I wish to understand why. x^2+y^2=25

- 9 months ago

One way to express this is:

$x = 5 \, \cos(t) \\ y = 5 \, \sin(t)$

- 9 months ago

Ah, this would be because of the definition of a unit circle. I will do one more. What about x^3+2x^2=y^3 . What would be a good way to go about putting this in a parametric equation. I am interesting in learning this so I can see how to differentiate standard equations to parametric.

- 9 months ago

I don't think it is useful to do such a transformation for this case, but here is an arbitrary one:

$x = t^{1/2} \\ y = (t^{3/2} + 2t)^{1/3}$

- 8 months, 4 weeks ago

How were you able to arrive to that conclusion? That is what I'm mainly curious about

- 8 months, 4 weeks ago

I just chose an expression for x at random. Then I plugged it into your equation and isolated y.

- 8 months, 4 weeks ago

But that (isolating y) won't always be possible right??? What then???

- 8 months, 4 weeks ago

If you can't isolate the other variable, it is likely that iterative techniques will be necessary

- 8 months, 4 weeks ago

It all just depends on what you're trying to do

- 8 months, 4 weeks ago

So what you are saying is that the parameters for x can be absolutely anything, as long as the other variables stay consistent with those parameters?

- 8 months, 4 weeks ago

Yes, that's it. If the equation has an implicit form, and the other variable can't be isolated, you're forced to use an iterative technique (like Newton Raphson) anyway, so hand analysis is out the window

- 8 months, 4 weeks ago

Hmm, so if the parameters can be anything you want, is there any value in finding the derivative of the parameters?? :/

- 8 months, 4 weeks ago

Yes, there is. In vector calculus, the standard hand-analysis approach for doing line integrals involves differentiating a parametric form

- 8 months, 4 weeks ago

Actually, I think it is the calculation of the length of a curve that uses it

- 8 months, 4 weeks ago