3D Least Squares Plane

Question

What\'s the algorithm for computing a least squares plane in (x, y, z) space, given a set of 3D data points? In other words, if I had a bunch of points like (1, 2, 3), (4, 5

Answer 1

The equation for a plane is: ax + by + c = z. So set up matrices like this with all your data:

    x_0   y_0   1  
A = x_1   y_1   1  
          ... 
    x_n   y_n   1  

And

    a  
x = b  
    c

And

    z_0   
B = z_1   
    ...   
    z_n

In other words: Ax = B. Now solve for x which are your coefficients. But since (I assume) you have more than 3 points, the system is over-determined so you need to use the left pseudo inverse. So the answer is:

a 
b = (A^T A)^-1 A^T B
c

And here is some simple Python code with an example:

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

N_POINTS = 10
TARGET_X_SLOPE = 2
TARGET_y_SLOPE = 3
TARGET_OFFSET  = 5
EXTENTS = 5
NOISE = 5

# create random data
xs = [np.random.uniform(2*EXTENTS)-EXTENTS for i in range(N_POINTS)]
ys = [np.random.uniform(2*EXTENTS)-EXTENTS for i in range(N_POINTS)]
zs = []
for i in range(N_POINTS):
    zs.append(xs[i]*TARGET_X_SLOPE + \
              ys[i]*TARGET_y_SLOPE + \
              TARGET_OFFSET + np.random.normal(scale=NOISE))

# plot raw data
plt.figure()
ax = plt.subplot(111, projection='3d')
ax.scatter(xs, ys, zs, color='b')

# do fit
tmp_A = []
tmp_b = []
for i in range(len(xs)):
    tmp_A.append([xs[i], ys[i], 1])
    tmp_b.append(zs[i])
b = np.matrix(tmp_b).T
A = np.matrix(tmp_A)
fit = (A.T * A).I * A.T * b
errors = b - A * fit
residual = np.linalg.norm(errors)

print "solution:"
print "%f x + %f y + %f = z" % (fit[0], fit[1], fit[2])
print "errors:"
print errors
print "residual:"
print residual

# plot plane
xlim = ax.get_xlim()
ylim = ax.get_ylim()
X,Y = np.meshgrid(np.arange(xlim[0], xlim[1]),
                  np.arange(ylim[0], ylim[1]))
Z = np.zeros(X.shape)
for r in range(X.shape[0]):
    for c in range(X.shape[1]):
        Z[r,c] = fit[0] * X[r,c] + fit[1] * Y[r,c] + fit[2]
ax.plot_wireframe(X,Y,Z, color='k')

ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
plt.show()