需要帮助提高曲面拟合法线的准确性

Question

我正在尝试将不规则曲面拟合到具有各自法线的三个点。该表面必须垂直于每条法线。 我能得到的最接近的是通过以下：

    import numpy as np
    import matplotlib.pyplot as plt
    from scipy.interpolate import Rbf
    from scipy.optimize import minimize
    
    # Data
    #positions
    A = np.array([1054.74352286, -1439.33862908, -1.50677694])
    W = np.array([895.76063743, 61.69720144, -7.4430564])
    B = np.array([869.90944818, 6.61926443, 5.88590894])
    positions = np.vstack([A, W, B])
    # normals
    N_A = np.array([0.86465823, -0.02097322, 0.50192258])
    N_W = np.array([0.82339849, -0.43041935, -0.3698028])
    N_B = np.array([0.8991198, -0.14663834, -0.41240851])
    normals = np.vstack([N_A, N_W, N_B])
    
    def objective_function(points, positions, normals, w=1000):
        points = points.reshape(-1, 3)
        rbf = Rbf(*points.T, function='multiquadric')
    
        def rbf_gradient(pos_2d, delta=1e-6):
            x, y = pos_2d
            grad_x = (rbf(x + delta, y) - rbf(x - delta, y)) / (2 * delta)
            grad_y = (rbf(x, y + delta) - rbf(x, y - delta)) / (2 * delta)
            return grad_x, grad_y
    
        error = 0
        for pos, normal in zip(positions, normals):
            pos_2d = np.array([pos[0], pos[1]])
            interpolated_grads = rbf_gradient(pos_2d)
            interpolated_normal = np.array([interpolated_grads[0], interpolated_grads[1], -1])
            interpolated_normal /= np.linalg.norm(interpolated_normal)
            error += w * (1 - np.dot(interpolated_normal, normal)) ** 2
        return error
    
    result = minimize(objective_function, positions, args=(positions, normals, 10000), method='L-BFGS-B')
    
    # Compute the surface using RBF interpolation
    points = result.x.reshape(-1, 3)
    rbf = Rbf(*points.T, function='multiquadric')
    
    # Create a grid and evaluate the surface
    X, Y = np.meshgrid(np.linspace(800, 1100, 100), np.linspace(-1500, 100, 100))
    Z = rbf(X, Y)
    
    # Plot the surface and normals
    fig = plt.figure()
    ax = fig.add_subplot(111, projection='3d')
    ax.plot_surface(X, Y, Z, alpha=0.5, color='b', label='Interpolated Surface')
    
    for pos, normal in zip(positions, normals):
        ax.scatter(*pos, color='r', marker='o', s=50)
        ax.quiver(pos[0], pos[1], pos[2], *normal, color='r', length=50, normalize=True)
    
    ax.set_xlabel('X')
    ax.set_ylabel('Y')
    ax.set_zlabel('Z')
    ax.set_title('Interpolated Surface and Normals')
`   plt.show()

结果看起来像这样。问题是垂直度不好。我想要一个可以垂直于每个法线的表面。

我的问题是这种方法是否正确，还有其他建议适合的方法或如何增加垂直度吗？