How to define a cone in spherical coordinates.

My objective is to define a cone in 3d, with itīs 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 90š the cone shape getīs 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