Weakly relational domains such as the Octagon domain have enjoyed tremendous success in the area of program analysis, since they offer a decent compromise between precision and efficiency. Octagons, in particular, have widely been studied to obtain efficient algorithms which, however, come with intricate correctness arguments. Here, we provide simplified cubic time algorithms for computing the closure of Octagon abstract relations both over the rationals and the integers which avoid introducing auxiliary variables. They are based on a more general formulation by means of 2-projective domains which allows for an elegant short correctness proof. The notion of 2-projectivity also lends itself to efficient algorithms for incremental normalization. For the Octagon domain, we also provide an improved construction for linear programming based best abstract transformers for affine assignments.