Can a 2nd degree parametric equation be turned into cartesian?

[mpatryluk]

Let's say i have a parametric equation:

x = t^2
y = t^3 + 4t

Even though this is a 2nd and 3rd degree parametric equation, i can isolate and express in terms of y = f(x) because the parametric equation for x involves only one term for t.

Thus:

t = sqrt(x)

and

y = sqrt(x)^3 + 4(sqrt(x))

But if the parametric equation instead had 2 terms with the variable t, in each equation, of varying degrees:

x = t^2 + 2t
y = t^3 + 4t

Could i still isolate? Because as far as i can see, i would have to express t in terms of itself.

i.e.

t^2 = x - 2t
t = sqrt(x - 2t)Is there any way around this?

 

[kontejnjer]

In this particular case, you could just solve the quadratic equation for the variable t in terms of x. Note that you will get two values for t, hence for each you will get a different function when you plug back into y.

 

[Mute]

To follow up on kontejnjer's comment, you should keep in mind that 2d parametric equations can be used to plot curves in the cartesian plane which are not functions. This means you cannot always write (x(t),y(t)) in the form y = f(x). In fact, as you can see with both of your examples, the curve is not a function - it fails the vertical-line test. In your first example, when you solve for t in terms of x, you need to consider both signs of the square root, just as kontejnjer mentions that you need to do for the second example.

So, in each of your examples, you really need two functions, $y = f_\pm(x)$, to describe your parametric curve.

An alternative is to derive an implicit equation, R(x,y) = 0, which describes the curve.

It can get more complicated than even this: you can have all sorts of crazy parametric curves for which you would need a large set of functions $y = f_n(x)$ to describe all the various segments of the curve. (Though there may be a simple implicit representation).

For example, see the Folium of Descartes. You would need at least three functions to describe that curve (at least four if you want them all to be without kinks).

The Implicit function theorem tells you when you can solve an implicit function R(x,y) = 0 for y, at least over some small interval.