In this paper, we recall some basic facts about the Kerr-Newman-(anti) de Sitter (KNdS) spacetime and review several formulations and integration methods for the geodesic equation of a test particle in such a spacetime. In particular, we introduce some basic general symplectic integrators in the Hamiltonian formalism and we re-derive the separated motion equations using Carter's method.

After this theoretical background, we explain how to ray-trace a KNdS black hole, equipped with a thin accretion disk, using Scilab. We compare the accuracy and execution time of the previous methods, concluding that the Carter equations is the best one. Then, inspired by Hagihara, we apply Weierstrass' elliptic functions to the non-rotating case, yielding a fairly fast shadowing program for such a spacetime.

We provide some illustrations of the code, including a depiction of the effects of the cosmological constant on shadows and accretion disk, as well as a simulation of M87*.

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Given a simple connected compact Lie group K and a maximal torus T of K, the Weyl group W = NK (T )/T naturally acts on T .First, we use the combinatorics of the (extended) affine Weyl group to provide an explicit W -equivariant triangulation of T . We describe the associated cellular homology chain complex and give a formula for the cup product on its dual cochain complex, making it a Z[W ]-dg-algebra.Next, remarking that the combinatorics of this dg-algebra is still valid for Coxeter groups, we associate a closed compact manifold T(W ) to any finite irreducible Coxeter group W , which coincides with a torus if W is a Weyl group and is hyperbolic in other cases. Of course, we focus our study on non-crystallographic groups, which are I2(m) with m = 5 or m ≥ 7, H3 and H4.The manifold T(W ) comes with a W -action and an equivariant triangulation, whose related Z[W ]-dg-algebra is the one mentioned above. We finish by computing the homology of T(W ), as a representation of W .