Inner approximation of distance constraints with existential quantification of parameters

This paper presents and compares two methods for checking if a box is included inside the solution set of an equality constraint with existential quantification of its parameters. We focus on distance constraints, where each existentially quantified parameter has only one occurrence, because of their usefulness and their simplicity. The first method relies on a specific quantifier elimination based on geometric considerations whereas the second method relies on computations with generalized intervals (interval whose bounds are not constrained to be ordered). We show that on two dimensions problems, the two methods yield equivalent results. However, when dealing with higher dimensions, generalized intervals are more efficient.