Question

I have two rays on a 2D plane that extend to infinity, but both have a starting point. They are both described by a starting point and a vector in the direction of the ray extending to infinity. I want to find out if the two rays intersect, but I don’t need to know where they intersect (it’s part of a collision detection algorithm).

Everything I have looked at so far describes finding the intersection point of two lines or line segments. Is there a fast algorithm to solve this?

Answer 1

Given: two rays a, b with starting points (origin vectors) as, bs, and direction vectors ad, bd.

The two lines intersect if there is an intersection point p:

p = as + ad * u
p = bs + bd * v

If this equation system has a solution for u>=0 and v>=0 (the positive direction is what makes them rays), the rays intersect.

For the x/y coordinates of the 2d vectors, this means:

p.x = as.x + ad.x * u
p.y = as.y + ad.y * u
p.x = bs.x + bd.x * v
p.y = bs.y + bd.y * v

Further steps:

as.x + ad.x * u = bs.x + bd.x * v
as.y + ad.y * u = bs.y + bd.y * v

Solving against v:

v := (as.x + ad.x * u - bs.x) / bd.x

Inserting and solving against u:

as.y + ad.y * u = bs.y + bd.y * ((as.x + ad.x * u - bs.x) / bd.x) 
u := (as.y*bd.x + bd.y*bs.x - bs.y*bd.x - bd.y*as.x ) / (ad.x*bd.y - ad.y*bd.x)

Calculate u, then calculate v, if both are positive the rays intersect, else not.