# Reflection at a plane in 3D

A flat mirror in 3D is descibed by the direction cosines 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.

where are the cartesian coordinates of point P=()

Proof that is indeed mapped to :

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

with

Proof, that each point is mapped to

## Reflection at the z-axis in 3D

## De-Rotation in 3D

with

## Translation in 3D back to original Position

# Reflection at a plane in 3D

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