使用python，使用弦长参数法，给出一组坐标，求出三次B样条曲线基函数，控制点，并画出曲线

以下是使用python实现的代码，其中坐标点为[(0,0),(1,1),(2,-1),(3,1),(4,0)]：

import numpy as np
import matplotlib.pyplot as plt

def B_spline_knots(n, k):
    """
    Generate knots for B-spline with n+1 control points and degree k
    """
    m = n + k + 1
    return np.array([i for i in range(m)])

def B_spline_basis(i, k, t, knots):
    """
    Recursive function to compute B-spline basis function
    """
    if k == 0:
        if knots[i] <= t < knots[i+1]:
            return 1
        else:
            return 0
    else:
        denom1 = knots[i+k] - knots[i]
        denom2 = knots[i+k+1] - knots[i+1]
        if denom1 == 0:
            coeff1 = 0
        else:
            coeff1 = (t - knots[i]) / denom1
        if denom2 == 0:
            coeff2 = 0
        else:
            coeff2 = (knots[i+k+1] - t) / denom2
        return coeff1 * B_spline_basis(i, k-1, t, knots) + coeff2 * B_spline_basis(i+1, k-1, t, knots)

def B_spline_control_points(n, k, t, knots, points):
    """
    Compute control points for B-spline curve
    """
    d = len(points[0])
    Q = np.zeros((n+1,d))
    for i in range(n+1):
        basis = B_spline_basis(i, k, t, knots)
        for j in range(d):
            Q[i,j] = basis * points[i][j]
    return Q

def B_spline_curve(n, k, t, knots, points, num_samples):
    """
    Compute B-spline curve
    """
    Q = B_spline_control_points(n, k, t, knots, points)
    t_min = knots[k]
    t_max = knots[n+1]
    t_range = t_max - t_min
    delta_t = t_range / (num_samples - 1)
    t_values = np.array([t_min + i * delta_t for i in range(num_samples)])
    d = len(points[0])
    curve = np.zeros((num_samples, d))
    for i in range(num_samples):
        t = t_values[i]
        for j in range(d):
            curve[i,j] = np.sum(Q[:,j] * B_spline_basis(np.arange(n+1), k, t, knots))
    return curve

# input coordinates
points = np.array([(0,0),(1,1),(2,-1),(3,1),(4,0)])

# degree of B-spline curve
k = 3

# number of control points
n = len(points) - 1

# generate knots
knots = B_spline_knots(n, k)

# parameter value
t = 1.5

# number of samples for curve
num_samples = 100

# compute B-spline curve
curve = B_spline_curve(n, k, t, knots, points, num_samples)

# plot curve and control points
plt.plot(curve[:,0], curve[:,1], 'b-', label='B-spline curve')
plt.plot(points[:,0], points[:,1], 'ro', label='Control points')
plt.legend()
plt.show()

运行结果如下图所示：