in the euclidean plane the distance from the origin to a point is
$s^2 = x^2 + y^2 $
I am reading a paper which say that this could be called an algabraic metric for the plane.
the paper then states that the algebraic metric on the sphere is this:
$ s^2 = \alpha x^2 + \beta y^2 + \gamma xy $
however if we choose a sphere with constant radius R how exactly do we come about finding this expression for the great circle distance, and what are those constants? I would appreciate some pointers as to how to derive this equation and what those constants are. I note that there is no z term- is this because it is eliminated using the equation for a sphere e.g. $ z = \sqrt {R^2 -x^2 -y^2} $ does that make any sense?
NOTE: I am talking about distances on THE SURFACE of a shere