Specular Reflection

A vector v is reflected from a plane with normal n with:

$r = 2$ $n.v\over{n.n}$ $n - v$

using the configuration from this image:

Deriving the reflection formula

Referring to the image above, we first project v to n with

$n' =$ $n.v \over n.n$ $n$

Then define

$\Delta v = n' - v$

Clearly

$r = v + 2 \Delta v = v + 2n' - 2v = 2$ $n.v\over{n.n}$ $n - v$