The trick is that a translation of a point (a1, ..., an) is essentially a shear of a point (a1, ..., an, 1) - a linear operation in R^(n+1). But because the points that are of interest lie on a (hyper)plane that is not a vector space - it does not intersect the origin - a shear looks like a nonlinear operation after projection to R^n.

    ^
    |---a
    |   |
  --+--------->
    |

    ^
    | ,---a
    |/   /
  --+--------->
    |