Cone-sphere intersection test

The method takes the perpendicular distance from the axis of the cone to the centre of the sphere and compares this to the maximum distance for intersection. The maximum distance (given the perpendicular through the centre of the sphere) is related to:

1. the cone's interior angle, θ
2. the distance of the perpendicular from the cone's vertex, p
3. the radius of the sphere, r

using the formula

r + p sin(θ) ------------ cos(θ)

Demonstration

In the diagram, the axis of the cone is VA, with vertex V. The perpendicular from the axis through the centre of the sphere (C) intersects VA at W, is of length |CW| and lies a distance of |VW| from the vertex of the cone (distance p on the diagram). The distance |WX| to the edge of the cone, also marked q on the diagram, is p by tan(θ), where θ is the interior angle of the cone.

When the sphere and cone touch at exactly one point, marked Y on the diagram, the line VY is tangential to the sphere, and forms a right angle with the radius YC. This radius forms an angle with the perpendicular CW that is identical to the cone's interior angle θ (the opposite angles at X being equal). This gives a distance (marked m on the diagram) of r/cos(θ) from C along CW to the edge of the cone. The total distance q+m is ptan(θ)+(r/cos(θ)), which simplifies to:

 r + p sin(θ)
 ------------
    cos(θ)