Decomposing a transformation matrix into translation, scaling and Euler angles (using Trimble SketchUp Ruby API)

transformation_matrix_utils.rb

# Decompose a 4×4 augmented rotation matrix without shear into translation, scaling and rotation components.
# 
# @param m [Array(16)] matrix
# @return [Array(3),Array(3),Array(3)] three arrays representing 
#     the translation vector, 
#     the scaling factor per axis and 
#     the Euler rotation angles in radians
def separate_translation_scaling_rotation(m)
  m = m.clone
  # Extract translation
  translation = m.values_at(12, 13, 14)
  # Extract scaling, considering uniform scale factor (last matrix element)
  scaling = Array.new(3)
  scaling[0] = m[15] * Math.sqrt(m[0]**2 + m[1]**2 + m[2]**2)
  scaling[1] = m[15] * Math.sqrt(m[4]**2 + m[5]**2 + m[6]**2)
  scaling[2] = m[15] * Math.sqrt(m[8]**2 + m[9]**2 + m[10]**2)
  # Remove scaling to prepare for extraction of rotation
  [0, 1, 2].each{ |i| m[i] /= scaling[0] } unless scaling[0] == 0.0
  [4, 5, 6].each{ |i| m[i] /= scaling[1] } unless scaling[1] == 0.0
  [8, 9,10].each{ |i| m[i] /= scaling[2] } unless scaling[2] == 0.0
  m[15] = 1.0
  # Verify orientation, if necessary invert it.
  tmp_z_axis = Geom::Vector3d.new(m[0], m[1], m[2]).cross(Geom::Vector3d.new(m[4], m[5], m[6]))
  if tmp_z_axis.dot( Geom::Vector3d.new(m[8], m[9], m[10]) ) < 0
    scaling[0] *= -1
    m[0] = -m[0]
    m[1] = -m[1]
    m[2] = -m[2]
  end
  # Extract rotation
  # Source: Extracting Euler Angles from a Rotation Matrix, Mike Day, Insomniac Games
  # http://www.insomniacgames.com/mike-day-extracting-euler-angles-from-a-rotation-matrix/
  theta1 = Math.atan2(m[6], m[10])
  c2 = Math.sqrt(m[0]**2 + m[1]**2)
  theta2 = Math.atan2(-m[2], c2)
  s1 = Math.sin(theta1)
  c1 = Math.cos(theta1)
  theta3 = Math.atan2(s1*m[8] - c1*m[4], c1*m[5] - s1*m[9])
  rotation = [-theta1, -theta2, -theta3]
  
  return translation, scaling, rotation
end

There is a bug in line 14: it should be m[2]**2 not m[3]**2

Thanks, fixed!