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Matrix Methods in Optics

Reflection at a plane in 3D

A flat mirror in 3D is descibed by the direction cosines (x,y,z) of a surface normal and a point P on it’s surface.

We construct the image A’ of a point A by these steps

  • translate the origin of the coordinate System to the point P
  • rotate the coordingate system so, that the z-axis of the coordinate system coincides with the surface normal in P)
  • mirror point A in this new coordinate system
  • unrotate the coordinate system to it’s old rotation
  • untranslate the origin to it’s old position

Translation of the coordinate system in 3D so that the Origin is in P.

    \[ \left( \begin{array}{c}x_1\\y_1\\z_1\\1\end{array} \right) = \begin{pmatrix} 1 & 0 & 0 & -p_x \\ 0 & 1 & 0 & -p_y \\ 0 & 0 & 1 & -p_z \\ 0 & 0 & 0 & 1 \end{pmatrix} \left( \begin{array}{c}x\\y\\z\\1\end{array} \right) = R_1 \left( \begin{array}{c}x\\y\\z\\1\end{array} \right) \]

where p_x,p_y,p_z are the cartesian coordinates of point P=(p_x,p_y,p_z)

Proof that P=(p_x,p_y,p_z) is indeed mapped to (0,0,0):

    \[\begin{pmatrix} 1 & 0 & 0 & -p_x \\ 0 & 1 & 0 & -p_y \\ 0 & 0 & 1 & -p_z \\ 0 & 0 & 0 & 1 \end{pmatrix} \left( \begin{array}{c}p_x\\p_y\\p_z\\1\end{array} \right) = \left( \begin{array}{c}p_x -p_x\\p_y-p_y\\p_z-p_z\\1\end{array} \right) = \left( \begin{array}{c}0\\0\\0\\1\end{array} \right) \]

Rotation in 3D around the Origin (now in P) so that the z-axis coincides with the surface normal

    \[ R_2 = \begin{pmatrix} \frac{n}w{} & 0 & \frac{-l}{w} & 0 \\ \frac{-lm}{w} & w & \frac{-mn}{w} & 0 \\ l & m & n & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

with w = \sqrt{l^2 + n^2}

Proof, that each point (x,y,z) is mapped to (x',y',0)

    \[ \begin{pmatrix} \frac{n}w{} & 0 & \frac{-l}{w} & x \\ \frac{-lm}{w} & w & \frac{-mn}{w} & z \\ l & m & n & z \\ 0 & 0 & 0 & 1 \end{pmatrix} \left( \begin{array}{c}p_x\\p_y\\p_z\\1\end{array} \right) = \left( \begin{array}{c}p_x -p_x\\p_y-p_y\\p_z-p_z\\1\end{array} \right) = \left( \begin{array}{c}0\\0\\0\\1\end{array} \right) \]

Reflection at the z-axis in 3D

    \[ R_3 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

De-Rotation in 3D

    \[ R_4 = \begin{pmatrix} \frac{n}w{} & \frac{-lm}{w} & l & 0 \\ 0 & w & m & 0 \\ \frac{-p_x}{w} & \frac{-m n}{w} & n & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

with w = \sqrt{l^2 + n^2}

Translation in 3D back to original Position

    \[ R_5 = \begin{pmatrix} 1 & 0 & 0 & p_x \\ 0 & 1 & 0 & p_y \\ 0 & 0 & 1 & p_z \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

Reflection at a plane in 3D

*** QuickLaTeX cannot compile formula:
R = R_5 R_4 R_3 R_2 R_1
<span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="https://www.optowiki.info/wp-content/ql-cache/quicklatex.com-fd64c24ef2bdf7310c7a75a3f6537980_l3.png" height="88" width="419" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ =  \begin{pmatrix} 1-2l^2 & -2lm & -2ln & 2l(lp_x + mg + nh)  \\                 -2lm & 1-2m^2 & -2mn & 2m(lpx + mg + nh)  \\                 -2ln & -2 m n & 1-2n^2 & 2n(lpx + mg + nh)  \\                 0 & 0 & 0 &  1 \end{pmatrix}  \]" title="Rendered by QuickLaTeX.com"/>
<span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="https://www.optowiki.info/wp-content/ql-cache/quicklatex.com-b17224e260f969ba19e366728afd8719_l3.png" height="88" width="301" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ =  \begin{pmatrix} 1-2l^2 & -2lm & -2ln & 2lp  \\                 -2lm & 1-2m^2 & -2mn & 2mp  \\                 -2ln & -2 m n & 1-2n^2 & 2np \\                 0 & 0 & 0 &  1 \end{pmatrix}  \]" title="Rendered by QuickLaTeX.com"/>
Where

*** Error message:
Missing $ inserted.
leading text: R = R_

p = (lp_x + mp_y + np_z) =$ length from the origin to point P