  The pure spinor formalism for the superstring was recently obtained by
gauge-fixing a purely bosonic classical action involving a twistor-like
constraint $\partial x^m (\gamma_m\lambda)_\alpha =0$ where $\lambda^\alpha$ is
a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro
constraint $\partial x^m \partial x_m =0$, and the Green-Schwarz fermionic
spacetime spinor variables $\theta^\alpha$ arise as Faddeev-Popov ghosts for
this constraint.
  In this paper, the purely bosonic classical action is simplified by replacing
the classical d=10 pure spinor $\lambda^\alpha$ with a d=10 projective pure
spinor. The pure spinor and Green-Schwarz formalisms for the superparticle and
superstring are then obtained as different gauge-fixings of this purely bosonic
classical action, and the Green-Schwarz kappa symmetry is directly related to
the pure spinor BRST symmetry. Since a d=10 projective pure spinor
parameterizes ${{SO(10)}\over{U(5)}}$, this action can be interpreted as a
standard $\hat c=5$ topological action where one integrates over the
${{SO(10)}\over{U(5)}}$ choice of complex structure. Finally, a purely bosonic
action for the d=11 supermembrane is proposed which reduces upon
double-dimensional reduction to the purely bosonic action for the d=10 Type IIA
superstring.
