计算 D 维圆上 2 点之间的欧氏距离 space

Question

计算两点之间的直接欧氏距离相当简单：

import torch
p1 = torch.tensor([1.0, 3.5])
p2 = torch.tensor([5.0, 9.2])

dis = torch.sum(torch.square(p1-p2))
dis
>>> tensor(48.4900)

但是，如何在不穿过圆的情况下计算圆上两点之间的距离？也就是说，圆周长的距离，D维space.

显然，在 2D 中，圆只是一个圆：

import numpy as np
import matplotlib.pyplot as plt

def circle_points(r, n):
    circles = []
    for r, n in zip(r, n):
        t = np.linspace(0, 2*np.pi, n, endpoint=False)
        x = r * np.cos(t)
        y = r * np.sin(t)
        circles.append(np.c_[x, y])
    return circles

r = [2]
n = [20]
circles = circle_points(r, n)

fig, ax = plt.subplots()
for circle in circles:
    ax.scatter(circle[:, 0], circle[:, 1])
ax.set_aspect('equal')
plt.show()

point_1 = circles[0][0]
point_2 = circles[0][11]
print('point_1: ', point_1, 'point_2: ', point_2)
>>> point_1:  [2. 0.] point_2:  [-1.90211303 -0.61803399]

而在 3D 中它将是球体、4D 超球体等

Answer 1

让 center 成为你的圆心。

然后你可以用点积公式计算两个中心到点向量之间的角度，它将点积与角度的余弦和向量的范数联系起来（寻找几何定义dot product)

normalize = lambda vect: vect/vect.norm()
v1 = normalize(point_1 - center)
v2 = normalize(point_2 - center)
angle = torch.arccos(v1.dot(v2))

那么，圆弧的长度就是圆的半径乘以角度，即

distance = angle*r

计算 D 维圆上 2 点之间的欧氏距离 space

Calculating the Euclidean distance between 2 points on a circle in D-dimensional space

python

geometry

pytorch