Solid Modeling Solutions: Difference between revisions

Revision as of 12:10, 27 March 2010

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systems in use at Boeing at that time. One of these systems had evolved as a result of the iterative process inherent to wing design. The other was best suited for adding the constraints imposed by manufacturing such as cylindrical and planar regions. The third decision was simple but crucial and added the ‘R’ to ‘NURBS’. Circles were to be represented exactly: no cubic approximations would be allowed.

By late 1979 there were 5 or 6 well educated mathematicians (PhD’s from Stanford, Harvard, Washington and Minnesota) and some of had many years of software experience, but none of had any industrial, much less CAD, geometry experience. Those were the days of the oversupply of math PhD’s. The task was to choose the representations for the 11 required curve forms, which included everything from lines and circles to Bezier and B-spline curves.

By early 1980, the staff were busy choosing curve representations and developing the geometry algorithms for TIGER. One of the major tasks was curve/curve intersection. It was noticed very quickly that one could solve the general intersection problem if one could solve it for the Bezier/Bezier case, since everything could be represented in Bezier form at the lowest level. It was soon realized that the geometry development task would be substantially simplified if a way could be found to represent all of the curves using a single form.

With this motivation the staff started down the road toward what became NURBS. Consider: the design of a wing demands free-form, C2 continuous, cubic splines to satisfy the needs of aerodynamic analysis, yet the circle and cylinders of manufacturing require at least rational Bezier curves. The properties of Bezier curves and uniform B-splines were well known, but the staff had to gain an understanding of non-uniform B-splines and rational Bezier curves and try to integrate the two. It was necessary to convert circles and other conics to rational Bezier curves for the curve/curve intersection. At that time, none of the staff realized the importance of the work and was considered “too trivial” and “nothing new”. The transition from uniform to non-uniform B-splines was rather straight forward, since the mathematical foundation had been available in the literature for many years. It just had not yet become a part of standard CAD/CAM applied mathematics.

Once there was a reasonably good understanding of rational Bezier and non-uniform splines, we still had to put them together. Up to this point, wethe staff had not written or seen the form

P(t) = ∑iwiPibi(t) / ∑iwibi(t)

for anything more than a conic Bezier segment. Searching for a single form, the group worked together, learning about knots, multiple knots and how nicely Bezier segments, especially the conics, could be imbedded into a B-spline curve with multiple knots. Looking back, it seemed so simple: It is easy to verify that the equation for P(t) is valid for the B-spline basis functions as well as for Bernstein basis functions. By the end of 1980 wethe staff knew we had a way to present all the required curve forms using a single representation, now know as the NURBS form. But this new representation could easily have died at this point. The staff were already 12 to 18 months down a development path. They had completed a large number of algorithms using the old curve forms. They now had to convince our managers and the other technical groups, such as the database and graphics groups, that they should be allowed to start over using a single representation for all curves. The NURBS surface form did not present a problem since they had not yet developed any surface algorithms. The review of this new TIGER curve form was held on February 13, 1981. The review was successful and the staff were allowed to start over using the new curve form. It was at this time that the NURBS acronym was first used by the other side of the TIGER project, i.e., the TIGER software development groups of Boeing Computer Services. Management was very eager to promote the use of these new curve and surface forms. They had a limited understanding of the mathematics but they were very aware of the need to communicate geometric data between systems. Hence, Boeing very quickly prepared to propose NURBS to the August ’81 IGES meetings. Richard Fuhr, of our group, was assigned the job of presenting NURBS to IGES. His presentation was very well received and

shortly afterwards he put together the Boeing document which was distributed to many IGES members.

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discovered that the people with the best understanding of the presentation were the SDRC representatives. Evidently SDRC was also active in defining a single representation for the standard CAD curves and was working on a similar definition.

So that’s how NURBS started at Boeing. Boehm’s B-spline refinement paper from CAD ’80 was of primary importance. It enabled me the staff to understand non-uniform splines and to appreciate the geometrical nature of the definition so as to use B-splines in solving engineering problems. The first use of the geometrical nature of B-splines was in the curve/curve intersection. The Bezier subdivision process was utilized, and a second use was our curve offset algorithm, which was based on a polygon offset process that was eventually communicated to and used by SDRC and explained by Tiller and Hanson in their offset paper of 1984. The staff also developed an internal NURBS class taught to about 75 Boeing engineers. The class covered Bezier curves, Bezier to B-spline and surfaces. The first public presentation of our NURBS work was at a Seattle CASA/SME seminar in March of 1982. The staff had progressed quite far by then. They could take a rather simple NURBS surface definition of an aircraft and slice it with a plane surface to generate an interesting outline of some of the wing, body and engines. The staff were allowed great freedom in pursuing our ideas and Boeing correctly promoted NURBS, but the task of developing that technology into a useable form was too much for Boeing, which abandoned the TIGER task late in ’84.

For the record, by late 1980, the TIGER Geometry Development Group consisted of Robert Blomgren, Richard Fuhr, George Graf, Peter Kochevar, Eugene Lee, Miriam Lucian and Richard Rice. Robert Blomgren was “lead engineer”. Richard Smith was our supervisor and the manager of the TIGER project was Robert