Vinberg's algorithm
In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Description of the algorithm. Let formula_1 be a hyperbolic reflection group. Choose any point formula_2; we shall call it the basic (or initial) point. The fundamental domain formula_3 of its stabilizer formula_4 is a polyhedral cone in formula_5. Let formula_6 be the faces of this cone, and let formula_7 be outer normal vectors to it. Consider the half-spaces formula_8 There exists a unique fundamental polyhedron formula_9 of formula_10 contained in formula_3 and containing  the point formula_12. Its faces containing formula_12 are formed by faces formula_6 of the cone formula_3. The other faces formula_16 and the corresponding outward normals formula_17 are constructed by induction. Namely, for formula_18 we take a mirror such that the root formula_19 orthogonal to it satisfies the conditions (1) formula_20; (2) formula_21 for all formula_22; (3) the distance formula_23 is minimum subject to constraints (1) and (2).