How to calculate the intersection point between an infinite line and a line segment?

Question

Basically, a function that fulfills this signature:

function getLineIntersection(vec2 p0, vec2 direction, vec2 p2, vec2 p3) {
   // return a vec2
}

I have looked around at existing solutions, and they all seem to deal with how to find the intersection between two line segments, or between two infinite lines. Is there a solution for this problem where the line has an initial position, an angle, and needs to determine if it intersects with a line segment? Basically, something like this:

There should be one line segment that starts in a location and has a unit direction, and another line segment that is just a line connected by two points. Is this possible, and if so, is there a good way of calculating the intersection point, if it exists?

score 2 · Accepted Answer · edited May 20 '21 at 08:19

2

If you've a endless line which is defined by a point P and a normalized direction R and a second endless line, which is defined by a point Q and a direction S, then the intersection point of the endless lines X is:

alpha ... angle between Q-P and R
beta  ... angle between R and S

gamma  =  180° - alpha - beta

h  =  | Q - P | * sin(alpha)
u  =  h / sin(beta)

t  = | Q - P | * sin(gamma) / sin(beta)

t  =  dot(Q-P, (S.y, -S.x)) / dot(R, (S.y, -S.x))  =  determinant(mat2(Q-P, S)) / determinant(mat2(R, S))
u  =  dot(Q-P, (R.y, -R.x)) / dot(R, (S.y, -S.x))  =  determinant(mat2(Q-P, R)) / determinant(mat2(R, S))

X  =  P + R * t  =  Q + S * u

If you want to detect if the intersection is on the lien, you need to compare the distance of the intersection point with the length of the line.
The intersection point (X) is on the line segment if t is in [0.0, 1.0] for X = p2 + (p3 - p2) * t

vec2 getLineIntersection(vec2 p0, vec2 direction, vec2 p2, vec2 p3) 
{
    vec2 P = p2;
    vec2 R = p3 - p2;  
    vec2 Q = p0;
    vec2 S = direction;

    vec2 N = vec2(S.y, -S.x);
    float t = dot(Q-P, N) / dot(R, N);

    if (t >= 0.0 && t <= 1.0)
        return P + R * t;

    return vec2(-1.0);
}

edited May 20 '21 at 08:19

Ryan Peschel

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answered May 20 '21 at 07:56

Rabbid76

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Does this work if only one of the lines is a segment and the other one is a line with a start point and a direction that extends infinitely? Based on the image it looks like it is for two infinite lines, but in my case I have a line segment and a line ray (with an initial point and a direction) – Ryan Peschel May 20 '21 at 08:03
@RyanPeschel The image is just the explanation for the intersection point. – Rabbid76 May 20 '21 at 08:04
@RyanPeschel What should be returned by the function if the intersection is not on the segment? – Rabbid76 May 20 '21 at 08:05
Hmm, not sure. I'm writing this in GLSL so I don't think they have a concept of a null, so I'd probably just use some arbitrary negative point as a signal that the intersection was not found, because all my lines exist from [0, infinite] and none have any negative coordinates. – Ryan Peschel May 20 '21 at 08:06

score 1 · Answer 2 · answered May 20 '21 at 08:00

Start with the intersection of two infinite lines, expressed in parametric form (e.g., A + tp, where A is the "start point", p is the direction vector and t is a scalar parameter). Solve a system of two equations to get the two parameters of the intersection point.

Now if one of your lines is really a segment AB, and B = A + p (i.e., the direction vector goes from A to B), then if the parameter t is between 0 and 1, the intersection lies on the segment.

How to calculate the intersection point between an infinite line and a line segment?

2 Answers2