CODEWITHSUNDEEP
pythonnumpymath3dgeometry
Bolo
Bolo

Reputation: 61

Function to transform 3d points to a new coordinate system with numpy

I have n points in space: points.shape == (n,3)

I have a new coordinate system defined by a point O = [ox, oy, oz] and 3 orthogonal vectors of different lengths: Ox = [oxx, oxy, oxz], Oy = [oyx, oyy, oyz], Oz = [ozx, ozy, ozz].

How can I write a function like that?

def change_coord_system(points, O, Ox, Oy, Oz)
    return # points in new coordinate system

Upvotes: 0

Views: 5974

Answers (2)

MBo
MBo

Reputation: 80107

You have 4 non-coplanar points in original system (where lx is length of the first vector and so on):

(0,0,0), (lx,0,0), (0,ly,0), (0,0,lz)

and their twins in new system

 [ox, oy, oz]
 [oxx + ox, oxy + oy, oxz + oz]
 [oyx + ox, oyy + oy, oyz + oz]
 [ozx + ox, ozy + oy, ozz + oz]

Affine transformation matrix A should transform initial points into their pair points

   A * P = P'

make matrix with point column vectors:

      |x1  x2  x3  x4|    |x1' x2' x3' x4'|
   A *|y1  y2  y3  y4| =  |y1' y2' y3' y4'|  
      |z1  z2  z3  z4|    |z1' z2' z3' z4'|
      |1   1   1    1|    |1   1    1    1|

      |0  lx  0  0|    |ox oxx + ox . .|
   A *|0  0  ly  0| =  |oy oxy + oy . .| // lazy to make last columns  
      |0  0  0  lz|    |oz oxz + oz . .|
      |1  1  1   1|    |1   1    1    1|

To calculate A, it is needed to multiply both sudes by inverse of P matrix

A * P * P-1 = P' * Pinverse
A * E = P' * Pinverse
A = P' * Pinverse

So calculate inverse matrix for P and multiply it with right-side matrix.

Edit: inverse matrix calculated by Maple is

 [[-1/lx, -1/ly, -1/lz, 1], 
  [1/lx, 0, 0, 0], 
  [0, 1/ly, 0, 0], 
  [0, 0, 1/lz, 0]]

And resulting affine transformation matrix is

[[-ox/lx+(oxx+ox)/lx, -ox/ly+(oyx+ox)/ly, -ox/lz+(ozx+ox)/lz, ox],
 [-oy/lx+(oxy+oy)/lx, -oy/ly+(oyy+oy)/ly, -oy/lz+(ozy+oy)/lz, oy], 
 [-oz/lx+(oxz+oz)/lx, -oz/ly+(oyz+oz)/ly, -oz/lz+(ozz+oz)/lz, oz], 
 [0, 0, 0, 1]]

Maple sheet view for reference

Edit:
Just have noticed: Maple did not remove excessive summands, so result should be simpler:

[[(oxx)/lx, (oyx)/ly, (ozx)/lz, ox],
 [(oxy)/lx, (oyy)/ly, (ozy)/lz, oy], 
 [(oxz)/lx, (oyz)/ly, (ozz)/lz, oz], 
 [0, 0, 0, 1]]

Upvotes: 3

Jacob Fuchs
Jacob Fuchs

Reputation: 357

Suppose that we have two points, P=[2, 4, 5] and Q=[7, 2, 5]. First you have to find the matrix for the rotation transformation A and the matrix B for transport and apply the below equation

enter image description here

The code using numpy is

import numpy as np
# points P and Q
points = np.array([[2,4,5], [7,2,5]])

# suppose that the matrices are
rotation_matrix = np.matrix('1 2 1; 1 2 1; 1 2 1')
b = np.array([1, 1, 1])

def transformation(points, rotation_matrix, b):
    for n in range(points.shape[0]):
    points[n,0] = rotation_matrix[0,0] * points[n, 0] + rotation_matrix[0,1] * points[n, 1] + rotation_matrix[0,2] * points[n, 2] + b[0]
    points[n,1] = rotation_matrix[1,0] * points[n, 0] + rotation_matrix[1,1] * points[n, 1] + rotation_matrix[1,2] * points[n, 2] + b[1]
    points[n,2] = rotation_matrix[2,0] * points[n, 0] + rotation_matrix[2,1] * points[n, 1] + rotation_matrix[2,2] * points[n, 2] + b[2]


Output:  array([[16, 30, 82],
                [17, 27, 77]])

I think that the above function gives the new points. You can check it. Of course, you can perform matrix multiplication with numpy although, you need to reshape the np.arrays.

Upvotes: 1

Related Questions