3-D  Rotational Matrix

 

               
    ROTATION AROUND Z-axis in X-Y plane, from positive x to y          
    Angle α (in Degree ) Rz:        
    matrix X1(anti-clockwise)        
    x11 x12 x13
    x21 x22 x23
    x31 x32 x33
    matrix X2(clockwise)        
    ax11 ax12 ax13
    ax21 ax22 ax23
    ax31 ax32 ax33
    ( for reflection)          
    ROTATION AROUND X-axis in Y-Z plane, from positive y to z          
    Angle β(in Degree )Rx :        
    matrix Y1(anti-clockwise)        
    y11 y12 y13
    y21 y22 y23
    y31 y32 y33
    matrix Y2(clockwise)        
    ay11 ay12 ay13
    ay21 ay22 ay23
    ay31 ay32 ay33
               
    ROTATION AROUND Y-axis in Z-X plane, from positive z to x          
    Angle  γ (in Degree )Ry :        
    matrix Z1(anti-clockwise)        
    z11 z12 z13
    z21 z22 z23
    z31 z32 z33
    matrix Z2(clockwise)        
    az11 az12 az13
    az21 az22 az23
    az31 az32 az33
               
    General 3x3 matrix            
   

(for reflecion)

         
    (Put Value of A either x1.x2,y1,y2,z1,z2 or 0. If 0 is put , write down the value of matrix.)A        
    a11 a12 a13
    a21 a22 a23
    a31 a32 a33
    det A        
    (Put Value of B either x1.x2,y1,y2,z1,z2 or 0. If 0 is put , write down the value of matrix)B        
    b11 b12 b13
    b21 b22 b23
    b31 b32 b33
    det B        
    (Put Value of C either x1.x2,y1,y2,z1,z2 or 0.If 0 is put , write down the value of matrix )C        
    c11 c12 c13
    c21 c22 c23
    c31 c32 c33
    det C

       
   

       ABC

         
    d11 d12 d13
    d21 d22 d23
    d31 d32 d33
    Det ABC        
    Trace ABC (d11+d22+d33)        
    Σii≠jjdii*djj(d11*d22+d22*d33+d33*d11)        
    Σi≠jdij*dji(d12*d21+d23*d32+d13*d31)        
    Point P represented by vector before rotation of axis  i  j k
    Point P' represented by vector after rotation of axis  i  j k
               
    Point Q represented by vector before rotation of axis  i 

j

k

    Point Q' represented by vector after rotation of axis  i 

j

k

    Length of vector OP        
    Length of Vector OP'        
    Length of vector OQ        
    Length of Vector OQ'        
    Angle between OP & OQ in degree        
    Angle between OP' & OQ' in degree        
    Unit vector along OP  i j k
    Unit vector along OP'  i j k
    D1=(d11)2+(d21)2+(d31)2 norm    
    D2=(d12)2+(d22)2+(d32)2 norm    
    D3=(d13)2+(d23)2+(d33)2 norm    
    Euclidian distance between D1 & D2:d(D1,D2)        
    Euclidian distance between D2 & D3:d(D2,D3)        
    Euclidian distance between D1 & D3:d(D1,D3)        
    Dot Product of the column vectors column-1 & 2 of matrix ABC :d11*d12+d21*d22+d31*d32

       
    Dot Product of the column vectors column-2 & 3 of matrix ABC :d12*d13+d22*d23+d32*d33        
    Dot Product of the column vectors column-1 & 3 of matrix ABC :d11*d13+d21*d23+d31*d33        
             
             
             
   

the following 48 3-D rotational matrices can be tried:- (x1y1z1,x1y1z2,x1y2z1,x1y2z2),(x2y1z1,x2y1z2,x2y2z1,x2y2z2)

                                                                                       (y1x1z1,y1x1z2,y1x2z1,y1x2z2),(y2x1z1,y2x1z2,y2x2z1,y2x2z2)

                                                                                       (z1x1y1,z1x1y2,z1x2y1,z1x2y2),(z2x1y1,z2x1y2,z2x2y1,z2x2y2)

                                                                                       (z1y1x1,z1y1x2,z1y2x1,z1y2x2),(z2y1x1,z2y1x2,z2y2x1,z2y2x2)

                                                                                       (y1z1x1,y1z1x2,y1z2x1,y1z2x2),(y2z1x1,y2z1x2,y2z2x1,y2z2x2)

                                                                                       (x1z1y1,x1z1y2,x1z2y1,x1z2y2),(x2z1y1,x2z1y2,x2z2y1,x2z2y2)

         
   

How to know that there are 48 rotational matrices  ? The following example shall illustrate the thing. Suppose there is 1 coin with 2 numbers. How many possible outcomes due to a toss ? It is 21 * 1! =2.

Suppose there are 2 coins with 4 different numbers. How many possible outcomes due to a toss ? It is 22 * 2! =8

Suppose there are 3 coins with 6 different numbers. How many possible outcomes due to a toss ? It is 23 * 3! =48

Suppose there are 4 coins with 8 different numbers. How many possible outcomes due to a toss ? It is 24 * 4! =384

         
               

 

 

48 3-D Rotational Matrices

  Remarks
1 x2y2z2 cosαcosγ +sinαsinβsinγ -sinαcosβ -cosαsinγ+sinαsinβcosγ 01=(10)T
    -sinαcosγ-cosαsinβsinγ cosαcosβ sinαsinγ-cosαsinβcosγ  
    cosβsinγ sinβ cosβcosγ  
           
2 x1y1z1 cosαcosγ -sinαsinβsinγ sinαcosβ cosαsinγ+sinαsinβcosγ 02=(09)T
    -sinαcosγ-cosαsinβsinγ cosαcosβ -sinαsinγ+cosαsinβcosγ  
    -cosβsinγ -sinβ cosβcosγ  
           
3 y2z2x2 cosαcosγ -sinαcosγ -sinγ 03=(08)T
    -cosαsinβsinγ+sinαcosβ sinαsinβsinγ+cosαcosβ -sinβcosγ  
    cosαcosβsinγ+sinαsinβ -sinαcosβsinγ+cosαsinβ cosβcosγ  
           
4 y1z1x1 cosαcosγ sinαcosγ sinγ 04=(07)T
    -cosαsinβsinγ-sinαcosβ -sinαsinβsinγ+cosαcosβ sinβcosγ  
    -cosαcosβsinγ+sinαsinβ -sinαcosβsinγ-cosαsinβ cosβcosγ  
           
5 z2x2y2 cosαcosγ -sinαcosβcosγ-sinβsinγ sinαsinβcosγ-cosβsinγ 05=(12)T
    sinα cosαcosβ -cosαsinβ  
    cosαsinγ -sinαcosβsinγ+sinβcosγ sinαsinβsinγ+cosβcosγ  
           
6 z1x1y1 cosαcosγ sinαcosβcosγ-sinβsinγ sinαsinβcosγ+cosβsinγ 06=(11)T
    -sinα cosαcosβ cosαsinβ  
    -cosαsinγ -sinαcosβsinγ-sinβcosγ -sinαsinβsinγ+cosβcosγ  
           
7 x2z2y2 cosαcosγ -sinαcosβ-cosαsinβsinγ sinαsinβ-cosαcosβsinγ x2z2y2=(y1z1x1)T
    sinαcosγ cosαcosβ-sinαsinβsinγ -cosαsinβ-sinαcosβsinγ 07=(04)T
    sinγ sinβcosγ cosβcosγ

 

 
           
8 x1z1y1 cosαcosγ sinαcosβ-cosαsinβsinγ sinαsinβ+cosαcosβsinγ x1z1y1=(y2z2x2)T
    -sinαcosγ cosαcosβ+sinαsinβsinγ cosαsinβ-sinαcosβsinγ 08=(03)T
    -sinγ -sinβcosγ cosβcosγ  
           
9 z2y2x2 cosαcosγ-sinαsinβsinγ -sinαcosγ-cosαsinβsinγ -cosβsinγ z2y2x2=(x1y1z1)T
    sinαcosβ cosαcosβ -sinβ 09=(02)T
    cosαsinγ+sinαsinβcosγ -sinαsinγ+cosαsinβcosγ cosβcosγ  
           
10 z1y1x1 cosαcosγ+sinαsinβsinγ sinαcosγ-cosαsinβsinγ cosβsinγ z1y1x1=(x2y2z2)T
    -sinαcosβ cosαcosβ sinβ 10=(01)T
    -cosαsinγ+sinαsinβcosγ sinαsinγ-cosαsinβcosγ cosβcosγ  
           
11 y2x2z2 cosαcosγ -sinα -cosαsinγ y2x2z2=(z1x1y1)T
    sinαcosβcosγ-sinβsinγ cosαcosβ -sinαcosβsinγ-sinβcosγ 11=(06)T
    sinαsinβcosγ+cosβsinγ cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
12 y1x1z1 cosαcosγ sinα cosαsinγ y1x1z1=(z2x2y2)T
    -sinαcosβcosγ-sinβsinγ cosαcosβ -sinαcosβsinγ+sinβcosγ 12=(05)T
    sinαsinβcosγ-cosβsinγ -cosαsinβ sinαsinβsinγ+cosβcosγ  
           
13 x1y1z2 cosαcosγ+sinαsinβsinγ sinαcosβ -cosαsinγ+sinαsinβcosγ 13=(16)T
    -sinαcosγ+cosαsinβsinγ cosαcosβ sinαsinγ+cosαsinβcosγ  
    cosβsinγ -sinβ cosβcosγ  
           
14 x2y2z1 cosαcosγ-sinαsinβsinγ -sinαcosβ cosαsinγ+sinαsinβcosγ 14=(15)T
    sinαcosγ+cosαsinβsinγ cosαcosβ sinαsinγ-cosαsinβcosγ  
    -cosβsinγ sinβ cosβcosγ  
           
15 z2y1x1 cosαcosγ-sinαsinβsinγ sinαcosγ+cosαsinβsinγ -cosβsinγ z2y1x1=(x2y2z1)T
    -sinαcosβ cosαcosβ sinβ 15=(14)T
    cosαsinγ+sinαsinβcosγ sinαsinγ-cosαsinβcosγ cosβcosγ  
           
16 z1y2x2 cosαcosγ+sinαsinβsinγ -sinαcosγ+cosαsinβsinγ cosβsinγ z1y2x2=(x1y1z2)T
    sinαcosβ cosαcosβ -sinβ 16=(13)T
    -cosαsinγ+sinαsinβcosγ sinαsinγ+cosαsinβcosγ cosβcosγ  
           
17 z2x1y1 cosαcosγ sinαcosβcosγ+sinβsinγ -cosβsinγ+sinαsinβcosγ 17=(20)T
    -sinα cosαcosβ cosαsinβ  
    cosαsinγ sinαcosβsinγ-sinβcosγ sinαsinβsinγ+cosβcosγ  
           
18 z1x2y2 cosαcosγ -sinαcosβcosγ+sinβsinγ cosβsinγ+sinαsinβcosγ 18=(19)T
    sinα cosαcosβ -cosαsinβ  
    -cosαsinγ sinαcosβsinγ+sinβcosγ -sinαsinβsinγ+cosβcosγ  
           
19 y1x1z2 cosαcosγ sinα -cosαsinγ y1x1z2=(z1x2y2)T
    -sinαcosβcosγ+sinβsinγ cosαcosβ sinαcosβsinγ+sinβcosγ 19=(18)T
    cosβsinγ+sinαsinβcosγ -cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
20. y2x2z1 cosαcosγ -sinα cosαsinγ y2x2z1=(z2x1y1)T
    sinαcosβcosγ+sinβsinγ cosαcosβ sinαcosβsinγ-sinβcosγ 20=(17)T
    -cosβsinγ+sinαsinβcosγ cosαsinβ sinαsinβsinγ+cosβcosγ  
           
21 z1x1y2 cosαcosγ sinαcosβcosγ+sinβsinγ cosβsinγ-sinαsinβcosγ 21=(23)T
    -sinα cosαcosβ -cosαsinβ  
    -cosαsinγ sinβcosγ-sinαcosβsinγ sinαsinβsinγ+cosβcosγ  
           
22 z2x2y1 cosαcosγ -sinαcosβcosγ+sinβsinγ -sinαsinβcosγ-cosβsinγ 22=(24)T
    sinα cosαcosβ cosαsinβ  
    cosαsinγ -sinβcosγ-sinαcosβsinγ -sinαsinβsinγ+cosβcosγ  
           
23 y1x2z2 cosαcosγ -sinα -cosαsinγ y1x2z2=(z1x1y2)T
    sinαcosβcosγ+sinβsinγ cosαcosβ sinβcosγ-sinαcosβsinγ 23=(21)T
    cosβsinγ-sinαsinβcosγ -cosαsinβ sinαsinβsinγ+cosβcosγ  
           
24 y2x1z1 cosαcosγ sinα cosαsinγ y2x1z1=(z2x2y1)T
    -sinαcosβcosγ+sinβsinγ cosαcosβ -sinβcosγ-sinαcosβsinγ 24=(22)T
    -sinαsinβcosγ-cosβsinγ cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
25 x1y2z2 cosαcosγ-sinαsinβsinγ sinαcosβ -cosαsinγ-sinαsinβcosγ 25=(26)T
    -sinαcosγ-cosαsinβsinγ cosαcosβ sinαsinγ-cosαsinβcosγ  
    cosβsinγ sinβ cosβcosγ  
           
26 z1y1x2 cosαcosγ-sinαsinβsinγ -sinαcosγ-cosαsinβsinγ cosβsinγ z1y1x2=(x1y2z2)T
    sinαcosβ cosαcosβ sinβ 26=(25)T
    -cosαsinγ-sinαsinβcosγ sinαsinγ-cosαsinβcosγ cosβcosγ  
           
27 z2y2x1 cosαcosγ+sinαsinβsinγ sinαcosγ-cosαsinβsinγ -cosβsinγ 27=(28)T
    -sinαcosβ cosαcosβ -sinβ  
    cosαsinγ-sinαsinβcosγ sinαsinγ+cosαsinβcosγ cosβcosγ  
           
28 x2y1z1 cosαcosγ+sinαsinβsinγ -sinαcosβ cosαsinγ-sinαsinβcosγ x2y1z1=(z2y2x1)T
    sinαcosγ-cosαsinβsinγ cosαcosβ sinαsinγ+cosαsinβcosγ 28=(27)T
    -cosβsinγ -sinβ cosβcosγ  
           
29 y1z1x2 cosαcosγ -sinαcosγ sinγ 29=(30)T
    sinαcosβ-cosαsinβsinγ cosαcosβ+sinαsinβsinγ sinβcosγ  
    -cosαcosβsinγ-sinαsinβ sinαcosβsinγ-cosαsinβ cosβcosγ  
           
30 x1z2y2 cosαcosγ sinαcosβ-cosαsinβsinγ -cosαcosβsinγ-sinαsinβ x2z2y2=(y1z1x2)T
    -sinαcosγ cosαcosβ+sinαsinβsinγ sinαcosβsinγ-cosαsinβ 30=(29)T
    sinγ sinβcosγ cosβcosγ  
           
31 x2z1y1 cosαcosγ -sinαcosβ-cosαsinβsinγ cosαcosβsinγ-sinαsinβ 31=(32)T
    sinαcosγ cosαcosβ-sinαsinβsinγ sinαcosβsinγ+cosαsinβ  
    -sinγ -sinβcosγ cosβcosγ  
           
32 y2z2x1 cosαcosγ sinαcosγ -sinγ y2z2x1=(x2z1y1)T
    -sinαcosβ-cosαsinβsinγ cosαcosβ-sinαsinβsinγ -sinβcosγ 32=(31)T
    cosαcosβsinγ-sinαsinβ sinαcosβsinγ+cosαsinβ cosβcosγ  
           
33 x1z1y2 cosαcosγ sinαcosβ+cosαsinβsinγ cosαcosβsinγ-sinαsinβ 33=(34)T
    -sinαcosγ cosαcosβ-sinαsinβsinγ -sinαcosβsinγ-cosαsinβ  
    -sinγ sinβcosγ cosβcosγ  
           
34 y1z2x2 cosαcosγ -sinαcosγ -sinγ y1z2x2=(x1z1y2)T
    sinαcosβ+cosαsinβsinγ cosαcosβ-sinαsinβsinγ sinβcosγ 34=(33)T
    cosαcosβsinγ-sinαsinβ -sinαcosβsinγ-cosαsinβ cosβcosγ  
           
35 y2z1x1 cosαcosγ sinαcosγ sinγ 35=(36)T
    cosαsinβsinγ-sinαcosβ sinαsinβsinγ+cosαcosβ -sinβcosγ  
    -cosαcosβsinγ-sinαsinβ -sinαcosβsinγ+cosαsinβ cosβcosγ  
           
36 x2z2y1 cosαcosγ cosαsinβsinγ-sinαcosβ -cosαcosβsinγ-sinαsinβ x2z2y1=(y2z1x1)T
    sinαcosγ sinαsinβsinγ+cosαcosβ -sinαcosβsinγ+cosαsinβ 36=(35)T
    sinγ -sinβcosγ cosβcosγ  
           
37 x1y2z1 cosαcosγ+sinαsinβsinγ sinαcosβ cosαsinγ-sinαsinβcosγ 37=(38)T
    -sinαcosγ+cosαsinβsinγ cosαcosβ -sinαsinγ-cosαsinβcosγ  
    -cosβsinγ sinβ cosβcosγ  
           
38 z2y1x2 cosαcosγ+sinαsinβsinγ -sinαcosγ+cosαsinβsinγ -cosβsinγ z2y1x2=(x1y2z1)T
    sinαcosβ cosαcosβ sinβ 38=(37)T
    cosαsinγ-sinαsinβcosγ -sinαsinγ-cosαsinβcosγ cosβcosγ  
           
39 x2y1z2 cosαcosγ-sinαsinβsinγ -sinαcosβ -cosαsinγ-sinαsinβcosγ 39=(40)T
    sinαcosγ+cosαsinβsinγ cosαcosβ -sinαsinγ+cosαsinβcosγ  
    cosβsinγ -sinβ cosβcosγ  
           
40 z1y2x1 cosαcosγ-sinαsinβsinγ sinαcosγ+cosαsinβsinγ cosβsinγ z1y2x1=(x2y1z2)T
    -sinαcosβ cosαcosβ -sinβ 40=(39)T
    -cosαsinγ-sinαsinβcosγ -sinαsinγ+cosαsinβcosγ cosβcosγ  
           
41 y1x2z1 cosαcosγ -sinα cosαsinγ 41=(42)T
    sinαcosβcosγ-sinβsinγ cosαcosβ sinαcosβsinγ+snβcosγ  
    -sinαsinβcosγ-cosβsinγ -cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
42 z2x1y2 cosαcosγ sinαcosβcosγ-sinβsinγ -sinαsinβcosγ-cosβsinγ z2x1y2=(y1x2z1)T
    -sinα cosαcosβ -cosαsinβ 42=(41)T
    cosαsinγ sinαcosβsinγ+sinβcosγ -sinαsinβsinγ+cosβcosγ  
           
43 z1x2y1 cosαcosγ -sinαcosβcosγ-sinβsinγ -sinαsinβcosγ+cosβsinγ 43=(44)T
    sinα cosαcosβ cosαsinβ  
    -cosαsinγ sinαcosβsinγ-sinβcosγ sinαsinβsinγ+cosβcosγ  
           
44 y2x1z2 cosαcosγ sinα -cosαsinγ y2x1z2=(z12x2y1)T
    -sinαcosβcosγ-sinβsinγ cosαcosβ sinαcosβsinγ-sinβcosγ 44=(43)T
    -sinαsinβcosγ+cosβsinγ cosαsinβ sinαsinβsinγ+cosβcosγ  
           
45 y1z2x1 cosαcosγ sinαcosγ -sinγ 45=(46)T
    cosαsinβsinγ-sinαcosβ sinαsinβsinγ+cosαcosβ sinβcosγ  
    cosαcosβsinγ+sinαsinβ sinαcosβsinγ-cosαsinβ cosβcosγ  
           
46 x2z1y2 cosαcosγ cosαsinβsinγ-sinαcosβ cosαcosβsinγ+sinαsinβ x2z1y2=(y1z2x1)T
    sinαcosγ sinαsinβsinγ+cosαcosβ sinαcosβsinγ-cosαsinβ 46=(45)T
    -sinγ sinβcosγ cosβcosγ  
           
47 x1z2y1 cosαcosγ sinαcosβ+cosαsinβsinγ -cosαcosβsinγ+sinαsinβ 47=(48)T
    -sinαcosγ -sinαsinβsinγ+cosαcosβ sinαcosβsinγ+cosαsinβ  
    sinγ -sinβcosγ cosβcosγ  
           
48 y2z1x2 cosαcosγ -sinαcosγ sinγ y2z1x2=(x1z2y1)T
    sinαcosβ+cosαsinβsinγ -sinαsinβsinγ+cosαcosβ -sinβcosγ 48=(47)T
    -cosαcosβsinγ+sinαsinβ sinαcosβsinγ+cosαsinβ cosβcosγ  
x1     x2    
cosα sinα 0 cosα -sinα 0
-sinα cosα 0 sinα cosα 0
0 0 1 0 0 1
y1     y2    
1 0 0 1 0 0
0 cosβ sinβ 0 cosβ -sinβ
0 -sinβ cosβ 0 sinβ cosβ
z1     z2    
cosγ 0 sinγ cosγ 0 -sinγ
0 1 0 0 1 0
-sinγ 0 cosγ sinγ 0 cosγ
           
* Above are case of Rotational Matrices in Euclidian Space (R3) which a) preserve origin. b) maintain Euclidian distance isometry c) preserve orientation i.e. handedness in space.

* R=Rz(α)*Ry(γ)*Rx(β)  is an intrinsic rotation where α,γ,β  are called Tait-Bryan Angles about z,y,x axis respectively.

* R=Rz(β)*Ry(γ)*Rx(α)  is an extrinsic rotation where α,γ,β  are called Proper Euler Angles about x,y,z axis respectively.

In geometryEuler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.

The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector . Its product by the rotation angle is known as an axis-angle. The extension of the theorem to kinematics yields the concept of instant axis of rotation, a line of fixed points.

In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity rotation matrix one eigenvalue is 1 and the other two are both complex, or both equal to −1. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.

* The trace of the real rotation matrix is 1+2cosθ. Trace is invariant under orthogonal matrix similarity transformation.

* Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigen value, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation.

* An m  m matrix A has m orthogonal eigenvectors if and only if A is normal, that is, if AA = AA.[b] This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:

* These are matrices which form SO(3) group under matrix multiplication where determinant is +1. These are sub-groups of O(3) groups of matrices whose determinant can be +1 or -1.These are sub-groups of GL(3). matrices.

* These are non-abelian group unlike SO(2) which is abelian.

* The group is a manifold and therefore is a Lie Group under the same composition.

* It has dimension 3- 1 co-ordinate for the point and 2 co-ordinates for the axis.

*If  α =180 degree, β =γ= 0 degree, rotation is put at x1,y1,z1, then (x,y) co-ordinates are reflected to (-x.-y) while z co-ordinate remains unchanged. If in addition, z1=0, and c33=-1, then the coordinate becomes (-x.-y.-z). But c33=-1 is not a rotation matrix element but reflection matrix.

Norm and Distance in Vectors :

* The distance is a 2-vector function d(x,y) while the norm is a 1-vector function ||v|| . However, we frequently use the norm to calculate the distance by means of difference of 2 vectors, ||y - x || . A norm always induces a distance , but the reverse is not true. A trivial distance has no equivalent norm as d(x,x)=0 and d(x,y)=1 when x is not equal to y.