Numerically compute intersection of infinite line and rectangle

Question

I'm trying to numerically compute where an infinite line defined by two points intersects a (finite) rectangle also defined by two points. Here's a illustration of the various ways that the line can intersect the rectangle: How can I determine the number and location of the places where the infinite line intersects the rectangle? I know how to do this when the line is finite, but I can't figure out how to generalize that to an infinite context.

Answer 1

Idea: find out on which side of the line the four corners lie

• no corners on the line and
  • all four corners on one side: $0$ intersections
  • one corner on one side, three corners on the other side: $2$ intersections
  • two corners on one side, two on the other side: $2$ intersections
• exactly one corner on the line and
  • three on one side: $1$ intersection
  • one on one side, two on the other sides: $2$ intersections
• exactly two corners on the line and
  • they are diagonally opposite ones: $2$ intersections
  • they are adjacent ones: infinitely many intersections