\(\vec{P_R}=(1-\cos \phi)(\vec{P}\cdot\hat{n})\hat{n} + \vec{P} \cos \phi + (\hat{n} \times \vec{P})\sin \phi\) ...(1)
where
\(\vec{P} =\) Position Vector of a given Point
\(\phi =\) Angle by which the Point given by Position Vector \(\vec{P}\) is Rotated
\(\vec{P_R} =\) Position Vector of the Point after Rotation
\(\hat{n} =\) Unit Vector representing the Arbitary Axis along which the Rotation takes place
The following gives the Derivation of Rodrigues Rotation Formula
Let's suppose the Position Vector \(\vec{P}\) of the given Point \(P\) is Perpendicular to the Axis of Rotation \(\hat{n}\) as given in Figure 1 below
\(\vec{P_R}\)= Projection of \(\vec{P_R}\) on \(\vec{P}\) + Rejection of \(\vec{P_R}\) from \(\vec{P}\) \(= \vec{{P_R}_{||}} + \vec{{P_R}_{\perp}}\) ...(2)
Now, Projection of \(\vec{P_R}\) on \(\vec{P}\), \(\vec{{P_R}_{||}}\) is given as
\(\vec{{P_R}_{||}} = {\Large \frac{\vec{P_R} \cdot \vec{P}}{|\vec{P}|}} \hat{P}\) ...(3)
But \(\vec{P_R} \cdot \vec{P} = |\vec{P_R}| |\vec{P}| \cos \phi \Rightarrow {\Large \frac{\vec{P_R} \cdot \vec{P}}{|\vec{P}|}} = |\vec{P_R}| \cos \phi \) ...(4)
Using equations (3) and (4) given above we get
\(\vec{{P_R}_{||}}=|\vec{P_R}| \cos \phi \hat{P} = \hat{P}|\vec{P_R}| \cos \phi \) ...(5)
And since \(|\vec{P_R}|=|\vec{P}|\)
\(\therefore \vec{{P_R}_{||}}=\hat{P} |\vec{P}| \cos \phi = \vec{P} \cos \phi\) ...(6)
Now, Rejection of \(\vec{P_R}\) from \(\vec{P}\), \(\vec{{P_R}_{\perp}}\) is given as
\(\vec{{P_R}_{\perp}}=\vec{P_R} - {\Large \frac{\vec{P_R} \cdot \vec{P}}{|\vec{P}|}} \hat{P} = \vec{P_R} - {\Large\frac{\vec{P_R} \cdot \vec{P}}{{|\vec{P}|}^2}} \vec{P}\)
\(\Rightarrow \vec{{P_R}_{\perp}}={\Large \frac{\vec{P_R} (\vec{P} \cdot \vec{P}) - \vec{P} (\vec{P_R} \cdot \vec{P}) }{{|\vec{P}|}^2}} = {\Large \frac{\vec{P} \times (\vec{P_R} \times \vec{P}) }{{|\vec{P}|}^2}}= {\Large \frac{(\vec{P} \times \vec{P_R}) \times \vec{P} }{{|\vec{P}|}^2}}\)
\(\Rightarrow \vec{{P_R}_{\perp}}= {\Large \frac{(\vec{P} \times \vec{P_R})}{{|\vec{P}|}^2}} \times \vec{P} \) ...(7)
But \(\vec{P} \times \vec{P_R} = |\vec{P}| |\vec{P_R}| \sin \phi \hat{n} \)
And since \(|\vec{P_R}|=|\vec{P}|\)
\(\therefore \vec{P} \times \vec{P_R} = {|\vec{P}|}^2 \hat{n} \sin \phi \Rightarrow \hat{n} \sin \phi = {\Large \frac{(\vec{P} \times \vec{P_R})}{{|\vec{P}|}^2}}\) ...(8)
Using equations (7) and (8) given above we get
\(\vec{{P_R}_{\perp}}= \hat{n} \sin \phi \times \vec{P}= (\hat{n} \times \vec{P})\sin \phi\) ...(9)
Now, using equations (2), (6) and (9) given above we get
\(\vec{P_R}= \vec{{P_R}_{||}} + \vec{{P_R}_{\perp}} = \vec{P} \cos \phi + (\hat{n} \times \vec{P})\sin \phi\) ...(10)
The equation (10) above gives the Position Vector \(\vec{P_R}\) of the Point after Rotation when the Position Vector \(\vec{P}\) of the Point is Perpendicular to Axis of Rotation \(\hat{n}\)
Now, Let's suppose the Position Vector \(\vec{P}\) of the given Point \(P\) is Not Perpendicular to the Axis of Rotation \(\hat{n}\) as given in Figure 2 below
\(\vec{P}=\)Projection of \(\vec{P}\) on \(\hat{n}\) + Rejection of \(\vec{P}\) from \(\hat{n}\) \(= \vec{P_{||}} + \vec{P_{\perp}} \) ...(11)
\(\vec{P_R}=\)Projection of \(\vec{P_R}\) on \(\hat{n}\) + Rejection of \(\vec{P_R}\) from \(\hat{n}\) \(= \vec{{P_R}_{||}} + \vec{{P_R}_{\perp}}\) ...(12)
But since \(\vec{P_{||}}=\vec{{P_R}_{||}}\), using equations (11) and (12) above we get
\(\vec{P_R}=\vec{P_{||}} + \vec{{P_R}_{\perp}}\) ...(13)
Now since \(\vec{P_{\perp}}\) is Perpendicular to \(\hat{n}\), \(\vec{{P_R}_{\perp}}\) can be calculated in terms of \(\vec{P_{\perp}}\) using equation (10) above as
\(\vec{{P_R}_{\perp}}=\vec{P_{\perp}} \cos \phi + (\hat{n} \times \vec{P_{\perp}})\sin \phi\) ...(14)
Using equations (13) and (14) given above we get
\(\vec{P_R}=\vec{P_{||}} + \vec{P_{\perp}} \cos \phi + (\hat{n} \times \vec{P_{\perp}})\sin \phi\) ...(15)
But \(\vec{P_{\perp}}= \vec{P} - \vec{P_{||}}\) (\(\because \vec{P}= \vec{P_{||}} + \vec{P_{\perp}} \) )
\(\therefore \vec{P_R}=\vec{P_{||}} + (\vec{P} - \vec{P_{||}}) \cos \phi + (\hat{n} \times (\vec{P} - \vec{P_{||}}) )\sin \phi =\vec{P_{||}} - \vec{P_{||}} \cos \phi + \vec{P} \cos \phi + (\hat{n} \times \vec{P} - \hat{n} \times\vec{P_{||}}) \sin \phi\)
Now since \(\hat{n} \times \vec{P_{||}}=0\)
\(\therefore \vec{P_R}= (1 - \cos \phi) \vec{P_{||}} + \vec{P} \cos \phi + (\hat{n} \times \vec{P}) \sin \phi = (1 - \cos \phi) (\vec{P} \cdot \hat{n})\hat{n} + \vec{P} \cos \phi + (\hat{n} \times \vec{P}) \sin \phi\) ...(16)
The equation (16) above gives the Position Vector \(\vec{P_R}\) after Rotation by Angle \(\phi\) along Axis given by \(\hat{n}\) for any Point having Initial Position Vector \(\vec{P}\), which is the Rodrigues Rotation Formula.
The Rodrigues Rotation Formula given in equation (16) above can also be written in form of Matrices as follows
\(\vec{P_R}=((1 - \cos \phi) \hat{n}\hat{n}^T + I \cos \phi + [\hat{n}]_x \sin\phi) \vec{P}\) ...(17)
where,
\(\hat{n} = \begin{bmatrix}X \\ Y \\ Z\end{bmatrix}\) \(I =\) Identity Matrix =\(\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}\)
\([\hat{n}]_x =\) Cross Product Matrix Representation of \(\hat{n}\) =\(\begin{bmatrix}0 & -Z & Y\\Z & 0 & -X\\-Y & X & 0\end{bmatrix}\)
\(\vec{P}=\begin{bmatrix}P_x \\ P_y \\ P_z\end{bmatrix}\) \(\vec{P_R}=\begin{bmatrix}{P_R}_x \\ {P_R}_y \\ {P_R}_z\end{bmatrix}\)
Therefore, equation (17) can be written as
\(\begin{bmatrix}{P_R}_x \\ {P_R}_y \\ {P_R}_z\end{bmatrix} =[(1 - \cos \phi) \begin{bmatrix}X \\ Y \\ Z\end{bmatrix}\begin{bmatrix}X & Y & Z\end{bmatrix} + \cos\phi \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} + \sin\phi \begin{bmatrix}0 & -Z & Y\\Z & 0 & -X\\-Y & X & 0\end{bmatrix}] \begin{bmatrix}P_x \\ P_y \\ P_z\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}{P_R}_x \\ {P_R}_y \\ {P_R}_z\end{bmatrix} =[(1 - \cos \phi) \begin{bmatrix}X^2 & XY & XZ \\ YX & Y^2 & YZ\\ZX & ZY & Z^2\end{bmatrix} + \cos\phi \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} + \sin\phi \begin{bmatrix}0 & -Z & Y\\Z & 0 & -X\\-Y & X & 0\end{bmatrix}] \begin{bmatrix}P_x \\ P_y \\ P_z\end{bmatrix}\) ...(18)
Setting \(t= 1-\cos \phi\), \(c= \cos \phi\) and \(s= \sin \phi\) in equation (18) we get
\(\begin{bmatrix}{P_R}_x \\ {P_R}_y \\ {P_R}_z\end{bmatrix} =[t \begin{bmatrix}X^2 & XY & XZ \\ YX & Y^2 & YZ\\ZX & ZY & Z^2\end{bmatrix} + c \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} + s \begin{bmatrix}0 & -Z & Y\\Z & 0 & -X\\-Y & X & 0\end{bmatrix}] \begin{bmatrix}P_x \\ P_y \\ P_z\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}{P_R}_x \\ {P_R}_y \\ {P_R}_z\end{bmatrix} =[\begin{bmatrix}tX^2 & tXY & tXZ \\ tYX & tY^2 & tYZ\\tZX & tZY & tZ^2\end{bmatrix} + \begin{bmatrix}c & 0 & 0\\0 & c & 0\\0 & 0 & c\end{bmatrix} + \begin{bmatrix}0 & -sZ & Y\\sZ & 0 & -sX\\-sY & sX & 0\end{bmatrix}] \begin{bmatrix}P_x \\ P_y \\ P_z\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}{P_R}_x \\ {P_R}_y \\ {P_R}_z\end{bmatrix} =\begin{bmatrix}tX^2 + c & tXY -sZ & tXZ + sY \\ tYX + sZ & tY^2 +c & tYZ - sX\\tZX -sY & tZY + sX & tZ^2 + c\end{bmatrix}\begin{bmatrix}P_x \\ P_y \\ P_z\end{bmatrix}\) ...(19)
The \(3 \times 3\) Matrix given in equation (19) above is the 3D Rotation Matrix