在由 numpy 数组创建的曲线中查找弧长

Question

我有一个 NumPy 数组，可用于创建一条曲线，数组如下所示。

curve=np.asarray([0,0,0], [0.5,0,0], [0.8,0.2,0], [1,0.5,0], [1.2,0.7,0])

如果我把它们连起来就会像下图一样

点是曲线上的顶点。

有什么方法可以找到这条曲线上任意两个给定点之间的弧长，例如两个蓝点？

Answer 1

要获得曲线任意两点之间的曲线长度，您需要将这些点定义为该曲线的顶点。

在下面的例子中，我在需要的位置插入两个蓝色点，然后求和这两个点之间所有顶点的欧氏距离：

import numpy as np

curve = np.asarray([[0,0,0], [0.5,0,0], [0.8,0.2,0], [1,0.5,0], [1.2,0.7,0]])
points = np.array([[.65,.1,0],[.9,.35,0]])

#insert points in curve 
start = np.searchsorted(curve[:,0], points[0,0])
curve = np.insert(curve, start, points[0], axis=0)
end = np.searchsorted(curve[:,0], points[1,0])
curve = np.insert(curve, end, points[1], axis=0)
print(curve)
#[[0.   0.   0.  ]
# [0.5  0.   0.  ]
# [0.65 0.1  0.  ]
# [0.8  0.2  0.  ]
# [0.9  0.35 0.  ]
# [1.   0.5  0.  ]
# [1.2  0.7  0.  ]]

def curve_length(curve):
    """ sum of Euclidean distances between points """
    return np.sum(np.sqrt(np.sum((curve[:-1] - curve[1:])**2,axis=1)))

print(curve_length(curve[start:end+1]))
#0.3605551275463989

更新

根据评论：为了找到曲线上距离原点 len 个单位的点 point 的坐标（curve[0]），我们首先累加起来曲线段的长度以找到我们的点所在的段。设此段为 curve[i] 和 curve[i+1] 之间的线。然后我们根据以下草图找到该段的部分 lambda (l)：
有了这个 lambda，我们就可以计算出搜索点：

import numpy as np
import matplotlib.pyplot as plt

curve = np.asarray([[0,0,0], [0.5,0,0], [0.8,0.2,0], [1,0.5,0], [1.2,0.7,0]])
len = 1

#len is reached between curve[i] and curve[i+1]
cs = np.cumsum(np.sqrt(np.sum((curve[:-1] - curve[1:])**2, axis=1)))
i = np.argmin(cs < len)
dist = len - cs[i-1]

l = dist / np.sqrt(np.sum((curve[i+1] - curve[i])**2))
point = l * (curve[i+1] - curve[i]) + curve[i]
print(point)
#[0.8773501  0.31602515 0.        ]

可视化：

plt.plot(curve[:,0], curve[:,1])
plt.plot(point[0], point[1], 'o')

验证：

end = np.searchsorted(curve[:,0], point[0])
curve = np.insert(curve, end, point, axis=0)
print(curve_length(curve[:end+1]))
#1.0

在由 numpy 数组创建的曲线中查找弧长

finding arc length in a curve created by numpy array

python

arrays

numpy

open3d

更新