How do I compute the intersection point of two lines?

Question

I have two lines that intersect at a point. I know the endpoints of the two lines. How do I compute the intersection point in Python?

# Given these endpoints
#line 1
A = [X, Y]
B = [X, Y]

#line 2
C = [X, Y]
D = [X, Y]

# Compute this:
point_of_intersection = [X, Y]

Answer 1

Can't stand aside,

So we have linear system:

A₁ * x + B₁ * y = C₁
A₂ * x + B₂ * y = C₂

let's do it with Cramer's rule, so solution can be found in determinants:

x = D_x/D
y = D_y/D

where D is main determinant of the system:

A₁ B₁
A₂ B₂

and D_x and D_y can be found from matricies:

C₁ B₁
C₂ B₂

and

A₁ C₁
A₂ C₂

(notice, as C column consequently substitues the coef. columns of x and y)

So now the python, for clarity for us, to not mess things up let's do mapping between math and python. We will use array L for storing our coefs A, B, C of the line equations and intestead of pretty x, y we'll have [0], [1], but anyway. Thus, what I wrote above will have the following form further in the code:

for D

L1[0] L1[1]
L2[0] L2[1]

for D_x

L1[2] L1[1]
L2[2] L2[1]

for D_y

L1[0] L1[2]
L2[0] L2[2]

Now go for coding:

line - produces coefs A, B, C of line equation by two points provided,
intersection - finds intersection point (if any) of two lines provided by coefs.

from __future__ import division 

def line(p1, p2):
    A = (p1[1] - p2[1])
    B = (p2[0] - p1[0])
    C = (p1[0]*p2[1] - p2[0]*p1[1])
    return A, B, -C

def intersection(L1, L2):
    D  = L1[0] * L2[1] - L1[1] * L2[0]
    Dx = L1[2] * L2[1] - L1[1] * L2[2]
    Dy = L1[0] * L2[2] - L1[2] * L2[0]
    if D != 0:
        x = Dx / D
        y = Dy / D
        return x,y
    else:
        return False

Usage example:

L1 = line([0,1], [2,3])
L2 = line([2,3], [0,4])

R = intersection(L1, L2)
if R:
    print "Intersection detected:", R
else:
    print "No single intersection point detected"