## Experiences combining CAGD and algebraic geometry

Computational performance growth by a factor of 10000x since 1990.

CAD-systems are to a great extent based on research ideas dating back to the middle of the 1980s, standardized in IS= 10303 (STEP) at the start of the 1990s. Solutions chosen were aimed at what was possible to compute on industrial type computers around 1990. These were processors running at typically 30Mz, with 32Mbyte memories. In the middle of 2005, PC runs typically on 3GHZ, with Gbyte memories.

Thus resources are 100x more powerful than in 1990. While 0.2 floating point instructions could be performed in a clock cycle in 1990, now typically 4 floating point operations can be performed in a clock cycle. Thus the computational performance has grown by a factor of 1000. The form 2003 programmable graphics cards are available offering for a low cost typically 10x more computational performance than a 3GHz CPU of 2005. Thus since 1990 the computational power available for the PC-user has grown by a factor of 10 000.

GAIA II has addressed new computational demanding approaches to challenges within CAD.

In the project we have addressed uses of advanced mathematics results and researched new approaches that can address the shortcomings of current CAD-technology. Such advanced approaches combine knowledge from classical algebraic geometry and Computer Aided Geometric Design (CAGD). We have gained experience on which approaches are feasible, and approaches that should not be followed. For those deciding on CAD-system development and new technologies to be integrated, the knowledge from GAIA II can be very valuable, to ensure that proper directions are followed in further development.

- Keywords related to the knowledge.

- New methods for resultants.

- Multiple approaches to approximate implicitization.

- How to use approximate implicitization in CAD-systems in general.

- Use of approximate implicitization in recursive intersection algorithms.

- The feasibility of performing self-intersection check on CAD-surfaces.

- The configuration and structure of typical CAD-surface self-intersections.

- Knowledge of classification from algebraic geometry relevant in CAD-systems.

CAD-systems are to a great extent based on research ideas dating back to the middle of the 1980s, standardized in IS= 10303 (STEP) at the start of the 1990s. Solutions chosen were aimed at what was possible to compute on industrial type computers around 1990. These were processors running at typically 30Mz, with 32Mbyte memories. In the middle of 2005, PC runs typically on 3GHZ, with Gbyte memories.

Thus resources are 100x more powerful than in 1990. While 0.2 floating point instructions could be performed in a clock cycle in 1990, now typically 4 floating point operations can be performed in a clock cycle. Thus the computational performance has grown by a factor of 1000. The form 2003 programmable graphics cards are available offering for a low cost typically 10x more computational performance than a 3GHz CPU of 2005. Thus since 1990 the computational power available for the PC-user has grown by a factor of 10 000.

GAIA II has addressed new computational demanding approaches to challenges within CAD.

In the project we have addressed uses of advanced mathematics results and researched new approaches that can address the shortcomings of current CAD-technology. Such advanced approaches combine knowledge from classical algebraic geometry and Computer Aided Geometric Design (CAGD). We have gained experience on which approaches are feasible, and approaches that should not be followed. For those deciding on CAD-system development and new technologies to be integrated, the knowledge from GAIA II can be very valuable, to ensure that proper directions are followed in further development.

- Keywords related to the knowledge.

- New methods for resultants.

- Multiple approaches to approximate implicitization.

- How to use approximate implicitization in CAD-systems in general.

- Use of approximate implicitization in recursive intersection algorithms.

- The feasibility of performing self-intersection check on CAD-surfaces.

- The configuration and structure of typical CAD-surface self-intersections.

- Knowledge of classification from algebraic geometry relevant in CAD-systems.