How to define a cone in spherical coordinates.

My objective is to define a cone in 3d, with it´s point on the axis origin.
I want to be able to orientate the cone in any direction (so im thinking of using spherical coordinates), but the most important part is that i must be able to define the "angle" of the cone with one variable.
So far,i use that variable to change the azimuth and elevation, i get a nice cone, but the problem is that as elevation increases to 90º the cone shape get´s more and more distorted.
Another view as to interpret my objective, is how to draw a perfect circle of radius R on the surface on the earth, at any given latitute/longitude.

I think this can be done using a projective transformation, but it would simplify matters alot if that could be avoided...
Tks