# Convert to parametric equation from implicit equation?

Given an implicit equation such as $$x^2+y^2=1$$ , I know it corresponds to the parametric equations $$\begin{cases} x=\cos t\\ y=\sin t \end{cases}$$.

But I don't know how to get from the implicit equations to the parametric one, only the other way around.

I'm interested in a general algorithm in solving for the parametric equations given the implicit equation. If such algorithm doesn't exist I want to know how to approach this problem given different types of implicit equations:

• Polynomial
• Trigonometric
• Exponential
• Other types and/or a combination of any of the above

Maybe some of them can be transformed and some others can't (How can I tell?).

To illustrate such an algorithm, please show me how to get parametric equations for $$y\sin x+x\cos y=1$$.

Thank you!

• If you find such algorithm, invite a round of beers for the Field medal. Dec 11, 2022 at 22:52
• Then I guess this question shows how little do I know about this stuff @ajotatxe. Dec 11, 2022 at 22:53
• @akujack There are a few tricks that work on some curves. For instance, for $x^2+y^2=1$, let $y=tx$, then $x^2+t^2x^2=1$ hence $x=\dfrac{1}{\sqrt{1+t^2}}$ and $y=\dfrac{t}{\sqrt{1+t^2}}$ is a parameterization. Dec 11, 2022 at 22:55
• Indeed, it's often the case with a parameterization. Of course here you get the other half with $x=-\dfrac{1}{\sqrt{1+t^2}}$, $y=-\dfrac{t}{\sqrt{1+t^2}}$ (when solving for $t$ I picked only the + sign above, you get another part with the - sign). The same applies to $y=\sqrt{1-x^2}$, that you can write $x=t, y=\sqrt{1-t^2}$. Dec 11, 2022 at 23:03
• This one is easy: solve for $x$ as a function of $y$, and let $y=t$. It gets complicated when you can't solve easily for one of the variables. Dec 11, 2022 at 23:07

The parametric equation $$\begin{cases}x=\cos t\\ y=\sin t\end{cases}$$ you wrote, is actually the entire solution set of the implicit equation $$x^2+y^2=1$$. In this sense, if we cannot solve the implicit equation in closed-form using standard mathematical functions with respect to any variable, then we cannot obtain the parametric equation either.

Let's consider the variable $$y$$, as a fixed number. We want to see the solution of the equation $$y\sin x+x\cos y=1$$, with respect to $$x:$$

\begin{align}&\sin x=\frac {1-x\cos y}{y}\\ \implies &\sin x=-\frac {\cos y}{y}x+\frac 1y\end{align}

Then, note that the equation $$\sin x=kx,~k≠0$$ (except $$x=0$$) and $$\sin x=ax+b$$, where $$a,b≠0$$ are types of transcendental equations that do not have a general closed-form solution.

Therefore, the implicit equation $$y\sin x+x\cos y=1$$ cannot be expressed by means of closed - form parametric equations, in general.

However, the equations we noted as transcendental, do not have only general closed-form solutions. For instance, the specific transcendental equation $$\sin x=\frac 3\pi x$$ have closed-form solutions. Indeed, $$x_1=0,~x_2=\frac \pi 6,~x_3=-\frac \pi 6$$ are only possible solutions.

To summarize, the implicit equation $$y\sin x+x\cos y=1$$, cannot be expressed with parametric equations, even through any special functions defined in mathematics.