3D Transformations

3D Transformations

3D transformations extend 2D concepts into three dimensions using 4x4 homogeneous coordinate matrices. These transformations position, orient, and scale objects in 3D space.

Translation

4x4 Translation Matrix:

  [ 1   0   0   tx ] [ x ]   [ x + tx ]
  [ 0   1   0   ty ] [ y ] = [ y + ty ]
  [ 0   0   1   tz ] [ z ]   [ z + tz ]
  [ 0   0   0   1  ] [ 1 ]   [ 1      ]

Scaling

4x4 Scaling Matrix:

  [ sx  0   0   0 ]
  [ 0   sy  0   0 ]
  [ 0   0   sz  0 ]
  [ 0   0   0   1 ]

Rotation

3D rotation is more complex — rotations can occur around any axis.

Rotation around X-axis (Rx):

  [ 1    0     0    0 ]
  [ 0   cosθ  -sinθ 0 ]
  [ 0   sinθ   cosθ 0 ]
  [ 0    0     0    1 ]

Rotation around Y-axis (Ry):

  [ cosθ   0   sinθ  0 ]
  [ 0      1   0     0 ]
  [-sinθ   0   cosθ  0 ]
  [ 0      0   0     1 ]

Rotation around Z-axis (Rz):

  [ cosθ  -sinθ  0   0 ]
  [ sinθ   cosθ  0   0 ]
  [ 0      0     1   0 ]
  [ 0      0     0   1 ]

Combined Rotation

General rotation: Rx * Ry * Rz

  Euler angles: yaw, pitch, roll
  - Yaw   = rotation around Y
  - Pitch = rotation around X
  - Roll  = rotation around Z

  M = Ry(yaw) * Rx(pitch) * Rz(roll)

Warning: Euler angles suffer from gimbal lock.

Rotation Matrices for Arbitrary Axes

Rotation by θ around unit axis (ux, uy, uz):

  R = [ cosθ + ux²(1-cosθ)       ux·uy(1-cosθ) - uz·sinθ  ux·uz(1-cosθ) + uy·sinθ  0 ]
      [ uy·ux(1-cosθ) + uz·sinθ  cosθ + uy²(1-cosθ)       uy·uz(1-cosθ) - ux·sinθ  0 ]
      [ uz·ux(1-cosθ) - uy·sinθ  uz·uy(1-cosθ) + ux·sinθ  cosθ + uz²(1-cosθ)       0 ]
      [ 0                         0                         0                         1 ]