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What is the motivation behind the definition of the reflection map in affine hyperplanes? $R: x \to x-2(x\cdot u-c)u$ where $u\cdot x=c$ defines the affine plane.

Of course one requirement is for it to be consistent with the usual $R^3$ reflections... Where does the extra $2cu$ term come from? To preserve isometry?

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The $2cu$ term is the only term that it not linear in $x$ -- it makes sure that the affine plane in question maps to itself. Otherwise the reflection would reflect about a hyperplane through the origin but parallel to chosen hyperplane.

(Also note that the definition only works when the equation of the plane is normalized such that $|u|=1$).

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