  We formulate a correspondence between affine and projective special K\"ahler
manifolds of the same dimension. As an application, we show that, under this
correspondence, the affine special K\"ahler manifolds in the image of the rigid
r-map are mapped to one-parameter deformations of projective special K\"ahler
manifolds in the image of the supergravity r-map. The above one-parameter
deformations are interpreted as perturbative $\alpha'$-corrections in heterotic
and type-II string compactifications with $N=2$ supersymmetry. Also affine
special K\"ahler manifolds with quadratic prepotential are mapped to
one-parameter families of projective special K\"ahler manifolds with quadratic
prepotential. We show that the completeness of the deformed supergravity r-map
metric depends solely on the (well-understood) completeness of the undeformed
metric and the sign of the deformation parameter.
