Finding parametric curves on a sphere 
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*Is there some general method for finding such curves? Let's say I have a planar curve, how can I project it onto a sphere?

*I am interested in a curve that starts at the south pole of a sphere, then wraps it in spiral motion and ends at the north pole. Is it possible to construct?

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*Do you really mean circle, rather than sphere?

*A parametrization with constant rate of increase along the $z$ axis is
$(x,y,z)=\left( \sqrt{1-t^2}\cos (a \pi t), \: \sqrt{1-t^2} \sin (a \pi t), \: t \right), \: 0<t<1$, where the $a$ parameter controls the number of revolutions. This is with $a=5$:

You can derive this by noting that $x \propto \cos(\cdot) $ and $y \propto \sin (\cdot)$. If $z$ is to increase in a linear fashion from -1 to +1 we can let $z=t$ for $t=-1 \dots 1$. Finally, since the spiral is to lie on the surface of a sphere $x^2+y^2+z^2=1$, so it follows that $x=\sqrt{1-t^2}\cos (a \pi t)$ and $y=\sqrt{1-t^2} \sin (a \pi t)$
