where are unknowns (representing coordinates of points) and may be unknowns of constants. A special occurrence of unknowns are the

Equations involving
virtual points may be written as:

where the are unknowns and the unknowns or constants. Clearly system involving distance equations are of great practical interest and

The method proposed in `ALIAS` to solve this type of systems is
based on the general procedure using the gradient and hessian. A first
difference is that it is not necessary to provide the gradient and
hessian function as they are easily derived from the system of
equations. Note also that due to the particular structure of the
distance equations the interval evaluation leads to exact bounds.
Furthermore
the algorithm we propose uses a special version of Kantorovitch
theorem (i.e. a version that produces a larger ball with a unique
solution in it compared to the general version of the theorem),
an interval Newton method, a specific version of the simplex method
described in section 2.14 and a specific version of
the inflation method described in section 3.1.6 (i.e. a
method that allows to compute directly the radius of a ball around a
solution that will contain only this solution). In addition two
simplification rules are used:

- as each function is a sum of square term each of them involving different unknowns we verify if the interval evaluation of the term has a positive part (in the opposite case the current box is discarded) and we may update the unknowns so that the negative part of the term is reduced (this is basically an application of the concept of 2B-consistency). Hence the procedures described in section 2.17 should not be used for distance equations.
- based on the triangle equation: each subset of equations describing the distances between a set of 3 points is detected and the algorithm verify if the triangle equation is satisfied and, in some cases may update the boxes.