  We extend the formalism relating electromagnetic form factors to transverse
quark charge densities in the light-front frame to the case of a spin-3/2
baryon and calculate these transverse densities for the $\Delta(1232)$ isobar
using lattice QCD. The transverse charge densities for a transversely polarized
spin-3/2 particle are characterized by monopole, dipole, quadrupole, and
octupole patterns representing the structure beyond that of a pure point-like
spin-3/2 particle. We present lattice QCD results for the $\Delta$-isobar
electromagnetic form factors for pion masses down to approximatively 350 MeV
for three cases: quenched QCD, two-degenerate flavors of dynamical Wilson
quarks, and three flavors of quarks using a mixed action that combines domain
wall valence quarks and dynamical staggered sea quarks. We extract transverse
quark charge densities from these lattice results and find that the $\Delta$ is
prolately deformed, as indicated by the fact that the quadrupole moment
$G_{E2}(0$) is larger than the value -3 characterizing a point particle and the
fact that the transverse charge density in a $\Delta^+$ of maximal transverse
spin projection is elongated along the axis of the spin.
