python实现三次B样条曲线

B样条曲线是一种用于曲面和曲线的数学工具，常常被应用于CAD，绘图和动画制作领域。Python提供了多种实现B样条曲线的库，如NumPy和SciPy等。以下是使用NumPy库实现三次B样条曲线的代码示例：

import numpy as np
import matplotlib.pyplot as plt

def B_spline(t, i, k, T):
    """三次B样条基函数"""
    if k == 0:
        return 1.0 if T[i] <= t < T[i+1] else 0.0
    else:
        denominator1 = T[i+k] - T[i]
        denominator2 = T[i+k+1] - T[i+1]
        numerator1 = t - T[i]
        numerator2 = T[i+k+1] - t
        result = 0.0
        if denominator1 > 0.0:
            result += (numerator1 / denominator1) * B_spline(t, i, k-1, T)
        if denominator2 > 0.0:
            result += (numerator2 / denominator2) * B_spline(t, i+1, k-1, T)
        return result

def B_spline_curve(points, degree):
    """生成三次B样条曲线"""
    num_points = len(points)
    num_segments = num_points - degree - 1
    T = [0.0] * (num_points + degree + 1)
    for i in range(num_points + degree + 1):
        if i < degree:
            T[i] = 0.0
        elif i >= num_points:
            T[i] = 1.0
        else:
            T[i] = (i - degree) / (num_points - degree)
    curve_points = []
    for t in np.linspace(0, 1, 100):
        x, y = 0.0, 0.0
        for i in range(num_points):
            x += points[i][0] * B_spline(t, i, degree, T)
            y += points[i][1] * B_spline(t, i, degree, T)
        curve_points.append([x, y])
    return np.array(curve_points)

# 测试代码
points = [[0.0, 0.0], [1.0, 1.0], [2.0, -1.0], [3.0, 0.0], [4.0, 1.0], [5.0, 0.0]]
degree = 3
curve_points = B_spline_curve(points, degree)
x_values = [point[0] for point in curve_points]
y_values = [point[1] for point in curve_points]
plt.plot(x_values, y_values, 'b-', label='B-spline Curve')
plt.plot([point[0] for point in points], [point[1] for point in points], 'ro', label='Control Points')
plt.legend(loc='best')
plt.show()

在上述代码中，我们定义了一个`B_spline`函数，该函数用于计算B样条基函数。然后我们用`B_spline_curve`函数生成三次B样条曲线，该函数接受控制点列表和次数作为输入参数，并返回指定次数的B样条曲线点。最后，我们使用Matplotlib库在图形界面上绘制三次B样条曲线和控制点。