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.
References
