Convex real projective structures on Coxeter orbifold $D^2(;n_1,n_2,n_3,n_4)\times \mathbb{R}$

The deformation space of real projective structures parametrizes the space of the convex real projective structures on an orbifold.

The Coxeter orbifold can be obtained $D^2(;n_1,n_2,n_3,n_4)\times\mathbb{R}$ by embedding the Coxeter quadrilateral from $\mathbb{RP}^2$ to $\mathbb{RP}^3$, and extending the side edges to planes and perturbing these to obtain a convex polytope.

This noncompact orbifold can be induced from two combinatorial polytopes.

Using this fact, we determine the deformation space of real projective structures on this orbifold with parametrization, and see its very detailed properties.