How to convert $z=\sin(x)\sin(y)$ to a parametric equation?

Question

I want to convert $z=\sin(2\pi x)\sin(2\pi y)$ to a parametric equation such as $x=f(t)$, $y=g(t)$, $z=k(t)$.

Answer 1

Edit: As, $z=\sin(2\pi x)\sin(2\pi y)$ contains three free variable in a single equation, that means we should need atleast two independent variables to connect those variables through the given equation.

So, It's impossible to express those variables in single free variable.

Answer 2

We need two parameters for the parametrization that is for example the trivial

• $x=u$
• $y=v$
• $z=\sin(2\pi u)\sin(2\pi v)$

or also any

• $x= f(u)$
• $y= g(v)$
• $z=\sin(2\pi f(u))\sin(2\pi g(v))$

such that the ranges for $f(u)$ and $g(v)$ agree with the ranges for $x$ and $y$.

For example, using that result, the following

• $x= \frac1{2\pi}\arctan u$
• $y= \frac1{2\pi}\arctan v$
• $z=\sin(2\pi x)\sin(2\pi y)=\frac{uv}{\sqrt{(u^2+1)(v^2+1)}}$

holds with the limitation $x,y \in\left(-\frac14,\frac14\right)$.