  We give an intrinsic definition of (affine very) special real manifolds and
realise any such manifold $M$ as a domain in affine space equipped with a
metric which is the Hessian of a cubic polynomial. We prove that the tangent
bundle $N=TM$ carries a canonical structure of (affine) special K\"ahler
manifold. This gives an intrinsic description of the $r$-map as the map
$M\mapsto N=TM$. On the physics side, this map corresponds to the dimensional
reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We
generalise this construction to the case when $M$ is any Hessian manifold.
