Number of points to define cone I am looking for the minimum number of surface points in three-dimensions required to identify an arbitrary cone. i.e. a cone with unknown origin, orientation, and angle. Note that the points must be arbitrarily chosen in the sense that they are not identifiable as being the origin for example.  
Hans Musgrave has come up with a better description of what I want in his answer For an arbitrary fixed cone C, determine the minimal integer n satisfying "if S is a set of surface points on that cone with size at least n then C is the only cone with S as a subset of its surface points" 

Consider an arbitrary line in 3D as the axis of the cone.  The equation of the line is:
$$\frac{x-x_0}{l}= \frac{y-y_0}{m} = \frac{z-z_0}{n}$$
where  $x_0$,$y_0$,$z_0$ are the coordinates of a given point on the line, and $(l, m, n)$ are the direction vector of the line.
We can arbitrarily scale the direction vector so that $l=1$, leaving five arbitrary constants in the equation of the line.
We can pick a point on the line to be the apex of the cone by specifying values of $x$, and using the equation of the line to calculate and $y$ and $z$.  So the choice of apex introduces one more arbitrary constant.
Now we can calculate the distance from a general point to the axis, and the distance from a general point to the apex.
The ratio of these two numbers, line distance over apex distance, for points on the cone, must be a constant, the sine of the apex angle. Yet another arbitrary value.
So if we want to eliminate the seven constants introduced, we would need seven points on the cone substituted into this formula for the cosine.
