Assume you have a cone with half-angle theta in a 3D cartesian space, with the vertex of the cone in the origin, and that you want to rotate the cone along a curve so that it covers the entire sphere.
I am trying to do this by defining the cone axis direction through spherical coordinates, say delta and phi, and moving linearly delta from 0 to pi, and phi from 0 to, say, an X value. The larger this X value is, the more revolutions will phi make before delta reaches pi (the other side).
The problem is, in fact, how to find this value X so that I can guarantee that the cone will cover the entire sphere. By covering the entire sphere I mean that all possible directions have lied within the cone at a given time while the cone was moving.