Orthogonal Projection

In [1]: 
import numpy as np
import numpy.linalg as la
In [2]: 
# for in-line plots
%matplotlib inline

# for plots in a window
# %matplotlib qt

import matplotlib.pyplot as pt
from mpl_toolkits.mplot3d import Axes3D

Make two random 3D vectors:

In [3]: 
np.random.seed(13)
x = np.random.randn(3)
y = np.random.randn(3)

Make them orthonormal:

In [4]: 
y = y - y.dot(x)/x.dot(x)*x
x = x/la.norm(x)
y = y/la.norm(y)

Check:

In [5]: 
print(y.dot(x))
print(la.norm(x))
print(la.norm(y))

7.6327832943e-17
1.0
1.0

Plot the two vectors:

In [6]: 
fig = pt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.set_xlim3d([-3, 3])
ax.set_ylim3d([-3, 3])
ax.set_zlim3d([-3, 3])

xy = np.array([x, y]).T
ax.quiver(
   0, 0, 0,
   xy[0], xy[1], xy[2],)
Out[6]:
<mpl_toolkits.mplot3d.art3d.Line3DCollection at 0x7fb167041cf8>

Make an array with the cornerpoints of a cube:

In [7]: 
points = np.array([
    [-1,-1,-1],
    [-1,-1,1],
    [-1,1,-1],
    [-1,1,1],

    [1,-1,-1],
    [1,-1,1],
    [1,1,-1],
    [1,1,1],
])

Plot them:

In [8]: 
fig = pt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(points[:,0], points[:, 1], points[:, 2])
Out[8]:
<mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x7fb166f61278>

Construct the projection matrix:

In [9]: 
Q = np.array([
   x,y,np.zeros(3)
]).T
print(Q)

[[-0.68624693  0.65133881  0.        ]
 [ 0.72610421  0.63968157  0.        ]
 [-0.04286988  0.40812404  0.        ]]
In [10]: 
P = Q.dot(Q.T)
print(P)

[[ 0.89517709 -0.08163735  0.29524635]
 [-0.08163735  0.93641985  0.22994143]
 [ 0.29524635  0.22994143  0.16840306]]

Check that $P^2=P$:

In [11]: 
la.norm(P.dot(P)-P)
Out[11]:
1.7880278822442372e-16

Project the points, assign to proj_points:

In [12]: 
proj_points = np.einsum("ij,nj->ni", P, points)
In [13]: 
fig = pt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(points[:,0], points[:, 1], points[:, 2])
ax.scatter(proj_points[:,0], proj_points[:, 1], proj_points[:, 2], color="red")

xy = np.array([x, y]).T
ax.quiver(
   0, 0, 0,
   xy[0], xy[1], xy[2],)
Out[13]:
<mpl_toolkits.mplot3d.art3d.Line3DCollection at 0x7fb166f94080>