TITLE: Parametric Equation for Great Circle
QUESTION [10 upvotes]: So I've been doing a lot of searching and haven't found exactly what I'm looking for.  My math skills are a bit rusty, so I haven't had luck deriving this on my own.
What I'm looking for is an equation (or set of equations) where I can plug in starting and ending spherical coordinates, plus a percentage (ie [0,1]) and output spherical coordinates of some point in between (ie, progress along a great circle).
The idea is basically to chart the progress of a plane between two cities and draw it on a globe or map.
Inputs:

lat1,lon1
lat2,lon2
r (radius)
p (progress, from 0->1)

Output:

lat_x,lon_x (point in-between)

REPLY [8 votes]: Perhaps the most succinct and easiest answer is (as quoted from here)

You are given two points $u$ and $v$ on the unit sphere. Think of them as position vectors $\vec u$ and $\vec v$ . It is easy enough to calculate cross products. Calculate $\vec w = (\vec u \times \vec v)\times\vec u$. Then $\vec w$ and $\vec u$ are unit vectors perpendicular to each other and in the plane of the circle. So a parameterization of the circle is $$\vec R (t)=\vec u \cos t+\vec w \sin t.$$

Correction: $\vec w$ is only a unit vector if $\vec u$ and 
$\vec v$ are perpendicular.