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
3 months, 1 week ago

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

Steven Chase - 3 months, 1 week ago

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

Dacota Sprague - 3 months, 1 week ago

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One way to express this is:

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

Steven Chase - 3 months, 1 week ago

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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.

Dacota Sprague - 3 months, 1 week ago

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

x=t1/2y=(t3/2+2t)1/3x = t^{1/2} \\ y = (t^{3/2} + 2t)^{1/3}

Steven Chase - 3 months, 1 week ago

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

Dacota Sprague - 3 months, 1 week ago

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

Steven Chase - 3 months, 1 week ago

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

Aaghaz Mahajan - 3 months, 1 week ago

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

Steven Chase - 3 months, 1 week ago

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

Steven Chase - 3 months, 1 week ago

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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?

Dacota Sprague - 3 months, 1 week ago

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@Dacota Sprague 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

Steven Chase - 3 months, 1 week ago

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

Dacota Sprague - 3 months, 1 week ago

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

Steven Chase - 3 months, 1 week ago

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

Steven Chase - 3 months, 1 week ago

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