Finding parametric curves on a sphere 
*

*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?
 A: *

*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 $y=\sqrt{1-t^2} \sin (a \pi t)$
A: For your first question, suppose $\gamma$ is any curve.  Then the norm of $\frac{\gamma}{||\gamma||}$ is $\frac{||\gamma||}{||\gamma||} = 1$, so the image of $\frac{\gamma}{||\gamma||}$ lives on the unit sphere in your space.
Your second question can be answered with the help of stereographic projection.  We'll take a curve in the plane and project it onto the unit sphere.

(Image from Wikipedia.)
If we describe the plane with the polar coordinates $(R,\Theta)$, and the sphere with the coordinates $(\varphi,\theta)$, where $\varphi$ is the zenith angle and $\theta$ the azimuth, then the map from the plane to the sphere is given by
$$
\begin{align}
\varphi &= 2 \arctan\left(\frac{1}{R}\right), \\
\theta  &= \Theta.
\end{align}
$$
If you'd like, this can be converted to Cartesian coordinates by
$$
\begin{align}
x &= \cos \theta \sin \varphi, \\
y &= \sin \theta \sin \varphi, \\
z &= \cos \varphi.
\end{align}
$$
As an example, if we take the logarithmic spiral defined in polar coordinates by $R = e^{\Theta/8}$, the Cartesian parametric curve describing its image on the sphere is
$$
\gamma(t) = \left(\begin{array}{c}
\cos t \,\sin \left(2 \arctan \left(e^{-t/8}\right)\right) \\
\sin t \,\sin \left(2 \arctan \left(e^{-t/8}\right)\right) \\
\cos \left(2 \arctan \left(e^{-t/8}\right)\right)
\end{array}\right).
$$

