Testing whether a line segment intersects a sphere
I am trying to determine whether a line segment (i.e. between two points) intersects a sphere. I am not interested in the position of the intersection, just whether or not the segment intersects the sphere surface. Does anyone have any suggestions as to what the most efficient algorithm for this would be? (I'm wondering if there are any algorithms that are simpler than the usual ray-sphere intersection algorithms, since I'm not interested in the intersection position)
Answer
If you are only interested if knowing if it intersects or not then your basic algorithm will look like this...
Consider you have the vector of your ray line, A -> B.
You know that the shortest distance between this vector and the centre of the sphere occurs at the intersection of your ray vector and a vector which is at 90 degrees to this which passes through the centre of the sphere.
You hence have two vectors, the equations of which fully completely defined. You can work out the intersection point of the vectors using linear algebra, and hence the length of the line (or more efficiently the square of the length of the line) and test if this is less than the radius (or the square of the radius) of your sphere.