# Difference between Rigid3DTransform Euler3DTransform

**URL:** <https://discourse.itk.org/t/difference-between-rigid3dtransform-euler3dtransform/1339>\
**Category:** Uncategorized\
**Created:** [October 8, 2018, 2:12pm UTC](https://discourse.itk.org/t/difference-between-rigid3dtransform-euler3dtransform/1339 "2018-10-08T14:12:52Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![ibr](https://discourse.itk.org/user_avatar/discourse.itk.org/ibr/32/468_2.png) [@ibr](https://discourse.itk.org/u/ibr)\
**Post date:** [October 8, 2018, 2:12pm UTC](https://discourse.itk.org/t/difference-between-rigid3dtransform-euler3dtransform/1339/1 "2018-10-08T14:12:53Z")

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[Rigid3DTransform](https://itk.org/Doxygen/html/classitk_1_1Rigid3DTransform.html) and [Euler3DTransform](https://itk.org/Doxygen/html/classitk_1_1Euler3DTransform.html)  
It is not clear, I appreciate if someone explains the difference.

Another question can someone check my computation bellow is correct:

```
import numpy as np
import sys, math 
print(" ******************************************************")
print (" Points Transformation ")
print(" ******************************************************")
np.set_printoptions(precision=2)
np.set_printoptions(suppress=True)
print("---------------------------------------------------")
print(" Define input ")
print("---------------------------------------------------")
# Input Points 
p=np.array([[-1.1499 , 4.96146 ,-3.73897],[1.12107 , 6.15124 ,-4.09615], [2.56265 , 7.01562 ,-4.0416], [4.1089 , 7.87024 ,-3.37502], [5.23609 , 8.08297 ,-1.92341], [5.37945 , 7.84559 , 0.0562932], [4.72791 , 6.72648 , 1.12291], [3.17306 , 5.88972 , 1.40773], [2.09212 , 5.46035 , 0.298082], [1.39364 , 5.93 ,-0.860716], [1.80359 , 6.98823 ,-1.9811], [2.79237 , 8.16489 ,-1.89504], [3.54371 , 8.4603 ,-0.688138], [3.18755 , 8.01944 , 0.458005], [1.99834 , 7.59242 , 0.468096], [1.50188 , 7.8538 ,-0.489043], [2.28302 , 8.5749 ,-0.594259]])

print(p)
# 3DEuler parameters
ep=np.array([-0.369236, -0.074973, 0.177428, -0.895437, -7.486193, -9.989277])
print("---------------------------------------------------")
print(" Parameters to Rotation Matrix ")
print("---------------------------------------------------")
[rx,ry,rz]=ep[0:3];
[tx,ty,tz]=ep[3:6];
Ms = []
cosx = math.cos(rx)
sinx = math.sin(rx)
Ms.append(np.array([[1, 0 , 0],
                                  [0, cosx, -sinx],
                                  [0, sinx, cosx]]))
cosy = math.cos(ry)
siny = math.sin(ry)
Ms.append(np.array( [[cosy , 0, siny],
                                    [0 , 1, 0],
                                   [-siny, 0, cosy]]))
                                    cosz = math.cos(rz)
sinz = math.sin(rz)
Ms.append(np.array([[cosz, -sinz, 0],
                                   [sinz, cosz, 0],
                                   [0 , 0 , 1] ]))
if Ms:
        mt = reduce(np.dot, Ms[::-1])
        # convert to 4x4
        m= np.zeros((4,4)) ; m[3,3]=1 ; m[0:3,0:3]=mt
        # add translations 
        m[0:3,3]=[tx,ty,tz]
else:    
        m= np.eye(3)
print("---------------------------------------------------")
print(" displacement field ")
print("---------------------------------------------------")
# get the displacement vector = len(p) 
d=np.zeros((len(p),4))
#print(p_dis)
for i in range(0,len(p)):
    dt=np.ones(4); dt[0:3] = dt[0:3] * p[i]
    #print(dt)    
    np.matmul(m,dt,d[i])
    print(d[i])
print("---------------------------------------------------")
print(" new transformed points ")
print("---------------------------------------------------")
# get the new points locations 
pn=np.zeros((len(p),3))
for i in range(0,len(p)):
    pn[i]= p[i] + d[i][0:3]
    print(pn[i])
```

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<div class="post-metadata">

**Author:** ![dzenanz](https://discourse.itk.org/user_avatar/discourse.itk.org/dzenanz/32/1093_2.png) [@dzenanz](https://discourse.itk.org/u/dzenanz)\
**Post date:** [October 8, 2018, 2:26pm UTC](https://discourse.itk.org/t/difference-between-rigid3dtransform-euler3dtransform/1339/2 "2018-10-08T14:26:14Z")

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Those two classes differ in how they internally represent the rotation and translation. This is important when those transforms are optimized, e.g. during image registration.

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<div class="post-metadata">

**Author:** ![ibr](https://discourse.itk.org/user_avatar/discourse.itk.org/ibr/32/468_2.png) [@ibr](https://discourse.itk.org/u/ibr)\
**Post date:** [October 8, 2018, 5:40pm UTC](https://discourse.itk.org/t/difference-between-rigid3dtransform-euler3dtransform/1339/3 "2018-10-08T17:40:58Z")

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Thanks for your quick response. Do they provide the same result? Do they have the same performance?

> The first 9 parameters represents the rotation matrix

Could you please explain how this matrix looks like assuming 3 angles for 3 rotations in each dimension e.g. thetaX , thetaY and theta Z?

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<div class="post-metadata">

**Author:** ![dzenanz](https://discourse.itk.org/user_avatar/discourse.itk.org/dzenanz/32/1093_2.png) [@dzenanz](https://discourse.itk.org/u/dzenanz)\
**Post date:** [October 8, 2018, 6:15pm UTC](https://discourse.itk.org/t/difference-between-rigid3dtransform-euler3dtransform/1339/4 "2018-10-08T18:15:33Z")

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That class uses the classic [4x4 transformation matrix](https://en.wikipedia.org/wiki/Transformation_matrix#Examples_in_3D_computer_graphics).

---

<div class="post-metadata">

**Author:** ![ibr](https://discourse.itk.org/user_avatar/discourse.itk.org/ibr/32/468_2.png) [@ibr](https://discourse.itk.org/u/ibr)\
**Post date:** [October 9, 2018, 7:26am UTC](https://discourse.itk.org/t/difference-between-rigid3dtransform-euler3dtransform/1339/5 "2018-10-09T07:26:41Z")

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Thanks a lot, I will have a look.