求二维空间中曲线的平行曲线（Python实现）

import numpy as np


def dx(dis, k):
    return np.sqrt(dis / (k**2 + 1.))


def dy(dis, k):
    return k * dx(dis, k)


def parallel_curve(xs, ys, ks, dis=1.):
    """ 由于对称性, 会返回两条平行曲线上的点
    :param xs: ndarray, 原始曲线上点的x值   [shape:(N,)]
    :param xs: ndarray, 原始曲线上点的y值   [shape:(N,)]
    :param ks: ndarray, 原始曲线上点的斜率  [shape:(N,)]
    :param dis: float, 曲线间的距离
    :return: ndarray, pxs [shape:(2, N)], pys [shape:(2, N)]
    """
    g = np.sign(ks)
    g[g == 0] = 1
    ms = -1. / (ks + 1e-20)
    pxs = np.vstack((xs + dx(dis ** 2, ms) * g, xs - dx(dis ** 2, ms) * g))
    pys = np.vstack((ys + dy(dis ** 2, ms) * g, ys - dy(dis ** 2, ms) * g))
    return pxs, pys    

Example:

x = np.arange(-20, 20, 1)
curve = np.poly1d([0.005, 0.02, -1, 2])     # 可以通过定义或拟合点来找到函数表达式
der = np.polyder(curve)
y = curve(x)
ks = der(x)
ps = parallel_curve(x, y, ks, dis=1.5)

plt.figure(figsize=(17, 12))
plt.plot(x, y, color='crimson', marker='o', linewidth=1.5)
plt.plot(ps[0][0], ps[1][0], color='teal', marker='o', linewidth=1.5)
plt.plot(ps[0][1], ps[1][1], color='mediumpurple', marker='o', linewidth=1.5)
plt.grid()
plt.axis("equal")
plt.xlabel('x', {'fontsize': 'x-large'})
plt.ylabel('y', {'fontsize': 'x-large'})
plt.legend(('original curve', 'parallel curve 1', 'parallel curve 2'), loc=2)
plt.title('distance=1.5', {'fontsize': 'x-large'})
plt.show()

进一步了解平行曲线