3D rotation matrices

3d_rotation_matrices.txt

Affine transformation matrices to rotate around (1) the X-axis; (2) the Y-axis; (3) the Z-axis; (4) an arbitrary 3D line. A 
right-handed coordinate system is assumed.

1. X-axis

Z-axis |           'B                A and B are two points in a YZ-plane
       |         '                   ∠ AOB = β
       |       '       `A            ∠ YOA = α
       |     '     `                 |OA| = |OB| = r
       |   '   `   
      _|_'_`_ _ _ _ _ _ _
     O/                 Y-axis
     /
    /
   / X-axis

A = (y, z)
  = (rcos{α}, rsin{α})

Let R be a rotation from A to B through an angle β

R: (y, z) → (rcos{α+β}, rsin{α+β})
          → (rcos{α}cos{β}-rsin{α}sin{β}, rsin{α}cos{β}+rcos{α}sin{β})
          → (ycos{β}-zsin{β}, zcos{β}+ysin{β})

Thus, we require a matrix M such that

  |x|   |        x        |
M |y| = | ycos{β}-zsin{β} |
  |z|   | zcos{β}+ysin{β} |
  |1|   |        1        |

Such an affine matrix is

    | 1     0         0      0 |
M = | 0   cos{β}   -sin{β}   0 |
    | 0   sin{β}    cos{β}   0 |
    | 0     0         0      1 |



2. Y-axis

X-axis |           'B                A and B are two points in a ZX-plane
       |         '                   ∠ AOB = β
       |       '       `A            ∠ ZOA = α
       |     '     `                 |OA| = |OB| = r
       |   '   `   
      _|_'_`_ _ _ _ _ _ _
     O/                 Z-axis
     /
    /
   / Y-axis

A = (z, x)
  = (rcos{α}, rsin{α})

Let R be a rotation from A to B through an angle β

R: (z, x) → (rcos{α+β}, rsin{α+β})
          → (rcos{α}cos{β}-rsin{α}sin{β}, rsin{α}cos{β}+rcos{α}sin{β})
          → (zcos{β}-xsin{β}, xcos{β}+zsin{β})

Thus, we require a matrix M such that

  |x|   | xcos{β}+zsin{β} |
M |y| = |        y        |
  |z|   | zcos{β}-xsin{β} |
  |1|   |        1        |

Such an affine matrix is

    |  cos{β}   0   sin{β}   0 |
M = |   0       1     0      0 |
    | -sin{β}   0   cos{β}   0 |
    |   0       0     0      1 |



3. Z-axis

Y-axis |           'B                A and B are two points in an XY-plane
       |         '                   ∠ AOB = β
       |       '       `A            ∠ XOA = α
       |     '     `                 |OA| = |OB| = r
       |   '   `   
      _|_'_`_ _ _ _ _ _ _ 
     O/                 X-axis
     /
    /
   / Z-axis

A = (x, y)
  = (rcos{α}, rsin{α})

Let R be a rotation from A to B through an angle β

R: (x, y) → (rcos{α+β}, rsin{α+β})
          → (rcos{α}cos{β}-rsin{α}sin{β}, rsin{α}cos{β}+rcos{α}sin{β})
          → (xcos{β}-ysin{β}, ycos{β}+xsin{β})

Thus, we require a matrix M such that

  |x|   | xcos{β}-ysin{β} |
M |y| = | ycos{β}+xsin{β} |
  |z|   |        z        |
  |1|   |        1        |

Such an affine matrix is

    | cos{β}   -sin{β}   0   0 |
M = | sin{β}    cos{β}   0   0 |
    |   0         0      1   0 |
    |   0         0      0   1 |



4. Arbitrary 3D line

Suppose the line's direction vector was pointing out of the screen towards you, then let a positive rotation angle 
indicate an anticlockwise rotation.

The line will first be subject to a sequence of reversible transformations such that after they are applied the line
coincides with the origin and is orientated so that its direction vector points in the positive Z-axis direction. 
Since the line now coincides with the Z-axis the actual rotation transformation which rotates the point around the line 
is applied using the Z-axis rotation technique. Finally, the transformations which were applied to change the line's 
position and orientation are reversed.

Let the line be defined by a point on the line Q=(q_x, q_y, q_z) and a direction D=(d_x, d_y, d_z).


a. Translate the line to the origin

    | 1   0   0  -q_x |          
A = | 0   1   0  -q_y |
    | 0   0   1  -q_z |
    | 0   0   0    1  |


b. Rotate the translated line's direction vector around the Z-axis onto the ZX-plane, specifically the half plane 
   where x >= 0
    
Y-axis |           D(d_x, d_y)        ∠ DOX = θ_z = -atan2{d_y, d_x}
       |         '                    
       |       '
       |     '
       |   '    
      _|_'_ _ _ _ _ _ _ D'(√[(d_x)²+(d_y)²], 0)
      O|              X-axis
      
Let B: D → D', thus
    
    | cos{θ_z}   -sin{θ_z}    0    0 |   
B = | sin{θ_z}    cos{θ_z}    0    0 |
    |    0           0        1    0 |
    |    0           0        0    1 |


c. Rotate the line's newly oriented direction vector around the Y-axis so it coincides with the positive Z-axis 
   direction

X-axis |           D'(d_z, √[(d_x)²+(d_y)²])        ∠ D'OZ = θ_y = -atan2{√[(d_x)²+(d_y)²], d_z}
       |         '                    
       |       '
       |     '
       |   '    
      _|_'_ _ _ _ _ _ _D''   
      O|              Z-axis

Let C: D' → D'', thus

    |  cos{θ_y}   0   sin{θ_y}   0 |
C = |     0       1      0       0 |
    | -sin{θ_y}   0   cos{θ_y}   0 |
    |     0       0      0       1 |


d. Perform the actual rotation around the line now that it coincides with the positive Z-axis

    | cos{θ}   -sin{θ}   0   0 |
D = | sin{θ}    cos{θ}   0   0 |
    |   0         0      1   0 |
    |   0         0      0   1 |


e. Reverse step (c.)

      |  cos{-θ_y}   0   sin{-θ_y}   0 |   | cos{θ_y}   0   -sin{θ_y}   0 |
C⁻¹ = |     0        1      0        0 | = |    0       1       0       0 |
      | -sin{-θ_y}   0   cos{-θ_y}   0 |   | sin{θ_y}   0    cos{θ_y}   0 |  
      |     0        0      0        1 |   |    0       0       0       1 |


f. Reverse step (b.)

      | cos{-θ_z}   -sin{-θ_z}   0   0 |   |  cos{θ_z}   sin{θ_z}   0   0 |   
B⁻¹ = | sin{-θ_z}    cos{-θ_z}   0   0 | = | -sin{θ_z}   cos{θ_z}   0   0 |
      |    0            0        1   0 |   |     0          0       1   0 |
      |    0            0        0   1 |   |     0          0       0   1 |


g. Reverse step (a.)

      | 1   0   0  q_x |          
A⁻¹ = | 0   1   0  q_y |
      | 0   0   1  q_z |
      | 0   0   0   1  |


Let R be a transformation matrix which rotates around the line by θ radians, then we have

R = A⁻¹ * B⁻¹ * C⁻¹ * D * C * B * A

Let cy = cos{θ_y}
    cz = cos{θ_z} 
    sy = sin{θ_y}
    sz = sin{θ_z}
    Cθ = cos{θ}
    Sθ = sin{θ}
    
where θ_y = -atan2{√[(d_x)²+(d_y)²], d_z} and θ_z = -atan2{d_y, d_x}, then
 
    |1 0 0 q_x| | cz sz 0 0| |cy 0 -sy 0| |Cθ -Sθ 0 0| | cy 0 sy 0| |cz -sz 0 0| |1 0 0 -q_x|
R = |0 1 0 q_y|*|-sz cz 0 0|*|0  1  0  0|*|Sθ  Cθ 0 0|*| 0  1 0  0|*|sz  cz 0 0|*|0 1 0 -q_y|
    |0 0 1 q_z| | 0  0  1 0| |sy 0  cy 0| |0   0  1 0| |-sy 0 cy 0| |0   0  1 0| |0 0 1 -q_z|
    |0 0 0 1  | | 0  0  0 1| |0  0  0  1| |0   0  0 1| | 0  0 0  1| |0   0  0 1| |0 0 0  1  |
  
        |1 0 0 -q_x|
  = M * |0 1 0 -q_y|
        |0 0 1 -q_z|
        |0 0 0  1  |

where 

M₀₀ =  Cθ*(cy*cy*cz*cz+sz*sz) + cz*cz*sy*sy
M₀₁ =  Cθ*cz*sz*(1-cy*cy) - Sθ*cy*(cz*cz+sz*sz) - cz*sy*sy*sz
M₀₂ =  Cθ*cy*cz*sy + Sθ*sy*sz - cy*cz*sy
M₀₃ =  q_x
M₁₀ =  Cθ*cz*sz*(1-cy*cy) + Sθ*cy*(cz*cz+sz*sz) - cz*sy*sy*sz
M₁₁ =  Cθ*(cy*cy*sz*sz+cz*cz) + sy*sy*sz*sz
M₁₂ = -Cθ*cy*sy*sz + Sθ*cz*sy + cy*sy*sz
M₁₃ =  q_y
M₂₀ =  Cθ*cy*cz*sy - Sθ*sy*sz - cy*cz*sy
M₂₁ = -Cθ*cy*sy*sz - Sθ*cz*sy + cy*sy*sz
M₂₂ =  Cθ*sy*sy + cy*cy
M₂₃ =  q_z

Thus, 

    | M₀₀  M₀₁  M₀₂  (-q_x*M₀₀ - q_y*M₀₁ - q_z*M₀₂ + q_x) |
R = | M₁₀  M₁₁  M₁₂  (-q_x*M₁₀ - q_y*M₁₁ - q_z*M₁₂ + q_y) |
    | M₂₀  M₂₁  M₂₂  (-q_x*M₂₀ - q_y*M₂₁ - q_z*M₂₂ + q_z) |
    | 0    0    0                        1                |