Calculating 3D Rotation Matrix using Polar and Equatorial Angles of Axis of Rotation

1. Rotation with respect to Origin (0,0,0) along Any Arbitrary Unit Vector <X,Y,Z> is a Composite Transformation involving following five Simple Transformations in the following order
   1. Rotate along any of the Coordinate Axes so that the Unit Vector Projects to one of the Coordinate Planes \(XY\), \(YZ\) or \(ZX\). This Rotation is known as Equatorial Angle Rotation.
   2. Depending on Axis of Previous Rotation, Rotate along the Coordinate Axis which is Not a Part of Plane on which the Unit Vector lies.
      
      If the Vector lies on Coordinate Plane \(XY\), then Rotate along \(Z\) Axis so that the Unit Vector aligns with either \(X\) Axis or \(Y\) Axis.
      
      Similarly if the Vector lies on Coordinate Plane \(YZ\), then Rotate along \(X\) Axis so that the Vector aligns with either \(Y\) Axis or \(Z\) Axis.
      
      And if the Vector lies on Coordinate Plane \(ZX\), then Rotate along \(Y\) Axis so that the Vector aligns with either \(Z\) Axis or \(X\) Axis.
      
      This Rotation is known as Polar Angle Rotation.
   3. Apply the Required Rotation along the Axis to which the Unit Vector has been aligned.
   4. Undo the Rotation given in Step b by Rotating in the Opposite Direction of Step b. The Matrix used for this is Transpose/Inverse of Matrix used in Step b.
   5. Undo the Rotation given in Step a by Rotating in the Opposite Direction of Step a. The Matrix used for this is Transpose/Inverse of Matrix used in Step a
   Please note that Depending on the Coordinate Axes to which one would like to Align the Unit Vector, 12 Configurations of Matrices are possible for performing the Rotations given in Step a and correspondingly in Step b (6 Corresponding to 6 Euler Rotations and 6 Corresponding to 6 Tait-Bryan Rotations ).
   
   Irrespective of the Rotation Matrices chosen in Steps a and b and correspondingly in Steps d and e, the Transformation Matrix for Rotating along any Arbitrary Unit Vector <X,Y,Z> comes down to following:

   Rotation Type	Transformation Matrix
   For counter clockwise rotation of a point	\(\begin{bmatrix} tX^2 + c & tXY - sZ & tXZ + sY\\tXY + sZ & tY^2 + c & tYZ - sX\\tXZ - sY & tYZ + sX & tZ^2 + c\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   For clockwise rotation of a point	\( \begin{bmatrix} tX^2 + c & tXY + sZ & tXZ - sY\\tXY - sZ & tY^2 + c & tYZ + sX\\tXZ + sY & tYZ - sX & tZ^2 + c\end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   Where \(c = \cos (\phi), s = \sin (\phi), t = 1-\cos (\phi)\), and \(X\) ,\(Y\) and \(Z\) are Components of the Unit Vector representing the Arbitary Axis

2. The following gives All 12 Possible Configurations of Rotations (6 Euler and 6 Tait-Bryan) for Derivation of Composite Matrix for 3D Rotations with respect to Origin (0,0,0) along Any Arbitrary Unit Vector <X,Y,Z> by any Angle \(\phi\). In the tables/calculations given below \(\alpha\), \(\beta\) and \(\gamma\) are the Polar Angles that the Unit Vector forms with the \(X\), \(Y\) and \(Z\) Axis respectively and \(\theta\) is the Equatorial Angle.
   
   

   Select Rotation Configuration:

   ALL XYX Euler XZX Euler YZY Euler YXY Euler ZXZ Euler ZYZ Euler XYZ Tait-Bryan XZY Tait-Bryan YZX Tait-Bryan YXZ Tait-Bryan ZXY Tait-Bryan ZYX Tait-Bryan

   
   
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