Creating a Plane of best fit from points

Is there a way to make a plane of best fit using matplotlib? I'm trying to get a smooth curved plane, but I'm unsure on how to do so.

My points are arranged as shown in the following image: They are quite smooth, except for a few exceptions, which are mostly clear.

My current code is:

from mpl_toolkits import mplot3d
import matplotlib.pyplot as plt
from sklearn import linear_model
plt.style.use('seaborn-poster')

x = np.array([12.5, 10, 9.5, 7.5, 6])
y = np.array([30, 45, 50, 55, 60, 65])
z = np.array([
    [62.13, 55.41, 54.49, 46.46, 42.13],
    [67.11, 59.43, 56.39, 52.64, 41.89],
    [82.89, 61.13, 57.30, 50.75, 43.02],
    [73.31, 60.57, 57.17, 52.64, 41.73],
    [78.11, 62.92, 63.40, 58.08, 48.69],
    [83.96, 65.19, 60.22, 53.57, 44.22]
])


X, Y = np.meshgrid(x, y)
Z = z

x1, y1, z1 = X.flatten(), Y.flatten(), Z.flatten()

X_data = np.array([x1, y1]).reshape((-1, 2))
Y_data = z1

reg = linear_model.LinearRegression().fit(X_data, Y_data)
a1, a2, c = float(reg.coef_[0]), float(reg.coef_[1]), float(reg.intercept_)

fig = plt.figure(figsize = (9,9))
ax = plt.axes(projection='3d')
ax.grid()

ax.plot_surface(X, Y, z)

ax.scatter(X, Y, z, c = 'r', s = 50)
ax.set_title('Figure 1.21 - Plot of Final results')

ax.set_xlabel('Radius of Ball (mm)', labelpad=20)
ax.set_ylabel('Height from which ball was dropped (cm)', labelpad=20)
ax.set_zlabel('Diameter of ripple (mm)', labelpad=20)

plt.show()

I have the a1, a2 and c values using linear regression but how do I plot them? Is linear regression going to give the right sort of result for this graph? I'm quite new to matplotlib, so sorry if this seems obvious.

source https://stackoverflow.com/questions/74910574/creating-a-plane-of-best-fit-from-points

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