# Symbolic Algebraic Approach

As in the numerical solvers, the constraints are again formulated as a system of algebraic equations. However, instead of applying numerical techniques to determine a solution, general symbolic computations are undertaken to find the solution to the system of equations. Methods such as Grobner basis [12] or Wu-Ritt [116] techniques can be applied to find symbolic expressions for the solutions. This approach is an instance solver if numerical coefficients are used in the system of equations. However, if the system can be solved with symbolic coefficients, a generic solution to the constraint system is found. The generic solution can be evaluated with specific constraint values to find the actual physical configurations possible for the given constraint problem.

One potential problem with this method is that certain equations in the basis may be algebraically dependent on one another when evaluated with specific constraints values. Thus at the generic solver level, the solver may determine that a solution exists, yet it will not be able to find any of the specific configurations satisfying the constraints. A further handicap of this method is that solving symbolic systems of equations can be extremely compute-intensive. For this reason, restrictions are often placed on the types of geometric entities allowed, as well as the types of constraints between them which may be specified.