Points of intersection of vector with cone. I have a Vector $\vec A$ defined as : $(A_o+t*A_d)$
I also have a Cone with vertex (cone tip) V and axis direction $\vec D$, base radius R and height H. The cone angle can be computed via $θ=2{tan^-}^1(R/H)$.
How can I find the 2 points of intersection between the vector and cone?
Here's a simple illustration:


to get a handle on the problem, it makes sense to set the vertex of the cone to be the origin of co-ordinates. then the axis direction can be specified by a unit vector $\bar v$, and the test line is defined by a single real parameter $t$ as
$$
\bar r = \bar a + \bar b t
$$
a point on the cone is distinguished by the fact that its distance from the axis is a constant fraction of its distance from the vertex. let us just call this $\lambda$ - it is, as you say, determined by the dimensions of the cone.
so this condition may be expressed as:
$$
(\bar r \cdot \bar r)-(\bar r \cdot \bar v)^2 = \lambda^2 (\bar r \cdot \bar r) 
$$
substituting for $r$ will give you a quadratic equation in $t$, which may have $0,1$ or $2$ solutions
