如何使用 numpy.linalg.solve 给定点坐标找到两条线相交的位置？

Question

所以我尝试使用 numpy.linalg.solve() 仅使用一些端点坐标来查找两条线彼此相交的位置。如果一行的坐标是：(x1, y1)，(x2, y2)。我试过了：

import numpy as np
a = np.array([[y2-y1],[x1-x2]])
b = np.array([(x1*y2)-(y1*x2)])

np.linalg.solve(a,b)

但是我认为方程式不正确，它返回以下错误：

numpy.linalg.LinAlgError: Last 2 dimensions of the array must be square

所以我不太确定该怎么做，有人可以帮我解决这个问题吗？

Answer 1

已按要求关注 these answers which gives clear explanations about the equations behind this problem and its well-known analytical resolution (based on Cramer's rule and determinants), it is possible to construct a simple linear system A x = b in order to use np.linalg.solve：

import numpy as np

# Given these endpoints coordinates
# Line 1 passing through points p1 (x1,y1) and p2 (x2,y2)
p1 = [0, 0]
p2 = [1, 1]

# Line 2 passing through points p3 (x3,y3) and p4 (x4,y4)
p3 = [0, 1]
p4 = [1, 0]

# Line 1 dy, dx and determinant
a11 = (p1[1] - p2[1])
a12 = (p2[0] - p1[0])
b1 = (p1[0]*p2[1] - p2[0]*p1[1])

# Line 2 dy, dx and determinant
a21 = (p3[1] - p4[1])
a22 = (p4[0] - p3[0])
b2 = (p3[0]*p4[1] - p4[0]*p3[1])

# Construction of the linear system
# coefficient matrix
A = np.array([[a11, a12],
              [a21, a22]])

# right hand side vector
b = -np.array([b1,
               b2])
# solve
try:
    intersection_point = np.linalg.solve(A,b)
    print('Intersection point detected at:', intersection_point)
except np.linalg.LinAlgError:
    print('No single intersection point detected')

它给出了那些给定点的预期输出：

>>> Intersection point detected at: [0.5 0.5]

如何使用 numpy.linalg.solve 给定点坐标找到两条线相交的位置？

How to find where two lines intersect using numpy.linalg.solve given points coordinates?

python

numpy

linear-algebra