Numpy and line intersections

Question

How would I use numpy to calculate the intersection between two line segments?

In the code I have segment1 = ((x1,y1),(x2,y2)) and segment2 = ((x1,

Answer 1

import numpy as np

def get_intersect(a1, a2, b1, b2):
    """ 
    Returns the point of intersection of the lines passing through a2,a1 and b2,b1.
    a1: [x, y] a point on the first line
    a2: [x, y] another point on the first line
    b1: [x, y] a point on the second line
    b2: [x, y] another point on the second line
    """
    s = np.vstack([a1,a2,b1,b2])        # s for stacked
    h = np.hstack((s, np.ones((4, 1)))) # h for homogeneous
    l1 = np.cross(h[0], h[1])           # get first line
    l2 = np.cross(h[2], h[3])           # get second line
    x, y, z = np.cross(l1, l2)          # point of intersection
    if z == 0:                          # lines are parallel
        return (float('inf'), float('inf'))
    return (x/z, y/z)

if __name__ == "__main__":
    print get_intersect((0, 1), (0, 2), (1, 10), (1, 9))  # parallel  lines
    print get_intersect((0, 1), (0, 2), (1, 10), (2, 10)) # vertical and horizontal lines
    print get_intersect((0, 1), (1, 2), (0, 10), (1, 9))  # another line for fun

---

Explanation

Note that the equation of a line is ax+by+c=0. So if a point is on this line, then it is a solution to (a,b,c).(x,y,1)=0 (. is the dot product)

let l1=(a1,b1,c1), l2=(a2,b2,c2) be two lines and p1=(x1,y1,1), p2=(x2,y2,1) be two points.

Finding the line passing through two points:

let t=p1xp2 (the cross product of two points) be a vector representing a line.

We know that p1 is on the line t because t.p1 = (p1xp2).p1=0. We also know that p2 is on t because t.p2 = (p1xp2).p2=0. So t must be the line passing through p1 and p2.

This means that we can get the vector representation of a line by taking the cross product of two points on that line.

Finding the point of intersection:

Now let r=l1xl2 (the cross product of two lines) be a vector representing a point

We know r lies on l1 because r.l1=(l1xl2).l1=0. We also know r lies on l2 because r.l2=(l1xl2).l2=0. So r must be the point of intersection of the lines l1 and l2.

Interestingly, we can find the point of intersection by taking the cross product of two lines.