Find if point lies on line segment

Question

I have line segment defined by these two points: A(x1,y1,z1) and B(x2,y2,z2). I have point p(x,y,z). How can I check if the point lies on the line segment?

Solution 1

If the point is on the line then:

(x - x1) / (x2 - x1) = (y - y1) / (y2 - y1) = (z - z1) / (z2 - z1)

Calculate all three values, and if they are the same (to some degree of tolerance), your point is on the line.

To test if the point is in the segment, not just on the line, you can check that

x1 < x < x2, assuming x1 < x2, or
y1 < y < y2, assuming y1 < y2, or
z1 < z < z2, assuming z1 < z2

Solution 2

Find the distance of point P from both the line end points A, B. If AB = AP + PB, then P lies on the line segment AB.

AB = sqrt((x2-x1)*(x2-x1)+(y2-y1)*(y2-y1)+(z2-z1)*(z2-z1));
AP = sqrt((x-x1)*(x-x1)+(y-y1)*(y-y1)+(z-z1)*(z-z1));
PB = sqrt((x2-x)*(x2-x)+(y2-y)*(y2-y)+(z2-z)*(z2-z));
if(AB == AP + PB)
    return true;

Solution 3

First take the cross product of AB and AP. If they are colinear, then it will be 0.

At this point, it could still be on the greater line extending past B or before A, so then I think you should be able to just check if pz is between az and bz.

This appears to be a duplicate, actually, and as one of the answers mentions, it is in Beautiful Code.

Solution 4

in case if someone looks for inline version:

public static bool PointOnLine2D (this Vector2 p, Vector2 a, Vector2 b, float t = 1E-03f)
{
    // ensure points are collinear
    var zero = (b.x - a.x) * (p.y - a.y) - (p.x - a.x) * (b.y - a.y);
    if (zero > t || zero < -t) return false;

    // check if x-coordinates are not equal
    if (a.x - b.x > t || b.x - a.x > t)
        // ensure x is between a.x & b.x (use tolerance)
        return a.x > b.x
            ? p.x + t > b.x && p.x - t < a.x
            : p.x + t > a.x && p.x - t < b.x;

    // ensure y is between a.y & b.y (use tolerance)
    return a.y > b.y
        ? p.y + t > b.y && p.y - t < a.y
        : p.y + t > a.y && p.y - t < b.y;
}

Solution 5

Your segment is best defined by parametric equation

for all points on your segment, following equation holds: x = x1 + (x2 - x1) * p y = y1 + (y2 - y1) * p z = z1 + (z2 - z1) * p

Where p is a number in [0;1]

So, if there is a p such that your point coordinates satisfy those 3 equations, your point is on this line. And it p is between 0 and 1 - it is also on line segment

View more solutions