Improper Rotations and Roto-Reflection Matrices in 3 Dimensions

1. In 3 Dimensions Improper Rotation or Roto-Reflection refers to a Combination of Rotation around a given Axis and a Reflection across the Plane Perpendicular to that Axis, applied in any order.
2. Any Improper Rotational Transformation or Roto-Reflection Transformation is carried out using a Roto-Reflection Matrix which is a Product of a 3D Rotation Matrix Around an Axis and a 3D Reflection Matrix across a Plane whose Normal is given by the Axis.
   
   A 3D Rotation Matrix around an Arbitrary Axis given by Unit Vector <X,Y,Z> is given as
   
   Rotation Matrix \(R_{rot} = \begin{bmatrix} tX^2 + c & tXY - sZ & tXZ + sY\\tXY + sZ & tY^2 + c & tYZ - sX\\tXZ - sY & tYZ + sX & tZ^2 + c\end{bmatrix}\)
   
   where \(c = \cos \phi, s = \sin \phi, t = 1-\cos \phi\)
   
   A 3D Reflection Matrix across a Plane whose Normal is given by Arbitrary Unit Vector <X,Y,Z> is given as
   
   Reflection Matrix \(R_{ref} = \begin{bmatrix} 1-2X^2 & -2XY & -2XZ \\-2XY & 1-2Y^2 & -2YZ\\-2XZ & -2YZ & 1-2Z^2\end{bmatrix}\)
   
   Therefore the Roto-Reflection Matrix corresponding to Arbitrary Arbitrary Axis given by Unit Vector Vector <X,Y,Z> is calculated as
   
   \(R_{rr}=R_{rot}R_{ref}=\begin{bmatrix} tX^2 + c & tXY - sZ & tXZ + sY\\tXY + sZ & tY^2 + c & tYZ - sX\\tXZ - sY & tYZ + sX & tZ^2 + c\end{bmatrix} \begin{bmatrix} 1-2X^2 & -2XY & -2XZ \\-2XY & 1-2Y^2 & -2YZ\\-2XZ & -2YZ & 1-2Z^2\end{bmatrix} =R_{ref}R_{rot}=\begin{bmatrix} 1-2X^2 & -2XY & -2XZ \\-2XY & 1-2Y^2 & -2YZ\\-2XZ & -2YZ & 1-2Z^2\end{bmatrix} \begin{bmatrix} tX^2 + c & tXY - sZ & tXZ + sY\\tXY + sZ & tY^2 + c & tYZ - sX\\tXZ - sY & tYZ + sX & tZ^2 + c\end{bmatrix}\)
   
   \(\Rightarrow R_{rr}=\begin{bmatrix} -kX^2 + c & -kXY - sZ & -kXZ + sY\\-kXY + sZ & -kY^2 + c & -kYZ - sX\\-kXZ - sY & -kYZ + sX & -kZ^2 + c\end{bmatrix}\)
   
   where \(c = \cos \phi, s = \sin \phi, k = 1+\cos \phi\)
   
   
3. Any Improper Rotational Transformation of \(180^\circ\) with respect to Any Axis is same as Reflection across the Point of Origin.
4. 3D Roto-Reflection Matrices have following properties
   1. All 3D Roto-Reflection Matrices are Orthogonal Square Matrices and have a Determinant Value of -1.
   2. All 3D Roto-Reflection Matrices representing a Rotation of \(180^\circ\) are Negative of \(3 \times 3\) Identity Matrix..
   3. The Trace of any 3D Roto-Reflection Matrix \(M\) representing a Rotation of \(\theta\) is calculated as
      
      \(Trace(M) = 2\cos\theta - 1\)