  Marstrand's theorem states that applying a generic rotation to a planar set
$A$ before projecting it orthogonally to the $x$-axis almost surely gives an
image with the maximal possible dimension $\min(1, \dim A)$. We first prove,
using the transversality theory of Peres-Schlag locally, that the same result
holds when applying a generic complex linear-fractional transformation in
$PSL(2,\C)$ or a generic real linear-fractional transformation in $PGL(3,\R)$.
We next show that, under some necessary technical assumptions, transversality
locally holds for restricted families of projections corresponding to
one-dimensional subgroups of $PSL(2,\C)$ or $PGL(3,\R)$. Third, we demonstrate,
in any dimension, local transversality and resulting projection statements for
the families of closest-point projections to totally-geodesic subspaces of
hyperbolic and spherical geometries.
