The activity was directed to try to solve in a more efficient way the mathematical problems arising when manipulating curves and surfaces into the CSIS software. This software is developped and maintained by the company CANDEMAT in order to be used for design purposes, production control and quality verification of the final product.

In order to check the possibility of using FRISCO methodology to improve the CSIS software, two different problems were isolated:

- the computation of the implicit equation of a surface in presented by a rational parametrization, and
- the tracing, algebraically guided, of algebraic curves implicitely defined.

The activity was concentrated into the study of the first problem. Two main difficulties were encountered when trying to use inside CSIS the usual elimination technics to deal with implicitation problems: first, it is usually a very costly algebraic operation and, second, the coefficients of the parametrization are usually floating-point numbers.

The second difficulty was overcome by taking into account that, in general, a concrete object to model (and then to construct) is made by several hundreds (or thousands) od small patches, all of them sharing the same algebraic structure. For such an object a database is constructed containing the implicit equation of every class of patch appearing in its definition. This database must also contains the inversion formulae (giving the parameters in terms of the cartesian coordinates) and must be pruned to avoid specialization problems. Moreover the database for a specific objetc is kept into a bigger and general database for a further use.

In the particular case of the objects provided by CANDEMAT S.A., the implicitation process was made by using Sylvester resultants, Grobner Basis computations and ad-hoc techniques for specific cases.

For example, in the database, the implicit equation of the parametric patch

appears asThe solution to the first problem before mentioned was provided by the solution given to the second one: since the implicitation procedure is a preprocessing step, no time is spent when the CAD/CAM user is working.

To indicate, finally, that for specific objects, the time for sectioning, off-setting, blending, etc by using this strategy has already been divided by a factor 3.

The continuation of the work will continue towards the resolution of some problems arising in the consideration of the implicitation problem:

- Some algebraic structures arising in the database construction
are very complicated and the the implicit equation has not been
generated. Namely:
with
*q*=*s*(^{3}*A*+_{3}t^{2}*B*+_{3}t*C*)+_{3}*s*(^{2}*A*+_{2}t^{2}*B*+_{2}t*C*)+_{2}*s*(*A*+_{1}t^{2}*B*+_{1}t*C*)+_{1}*A*+_{0}t^{2}*B*+_{0}t*C*_{0} - To deal in advance with the specialization problems: upto this moment these are detected when substituting the parametrization into the candidate to be the implicit equation.
- To check the existence of base points in the parameter domain: ie
solutions into the s-t space (usually ) such that
*f*(_{1}*s*,*t*)=*f*(_{2}*s*,*t*)=*f*(_{3}*s*,*t*)=*q*(*s*,*t*)=0