Parameterization of a closed curve on a sphere

Question

I'm looking for a parameterization of a closed curve C on a sphere.

assume the projections of C on y-z, x-z, x-y plane are f(x), g(y), h(z), respectively, and

${\oint}f(x)dx={\oint}g(y)dy=0$, and ${\oint}h(z)dz=z_0, 0<z_0<1$.

My question is the following:

1. Is there any restriction on f(x), g(y), h(z) so that such a closed curve exists on a sphere?

2. How to obtain the parametrization of such a curve using spheric coordinates?

Thanks!