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I have never been very enthusiastic about calling our field 'Computer Aided Geometric Design‘. Ivor Faux and I once wrote a book called 'Computational Geometry', which I think was a better name, but that got hijacked by another bunch of people who are mostly much more remote from the real world than we are!
M. Pratt
Levels of Abstraction
• B.C: manual
• Medevial: Geometric constructions
• 1600’s: splines
• 1944: Liming
• 1960: De Casteljau/Bezier
• 2000+: manual!
Liming’s benefits
• Increase in precision and accuracy
• Elimination of deviations resulting from the human element
• Uniformity of application of results
• Close coordination of design, lofting, and production engineering
• Close coordination with tooling procedures
• Cross-checking of graphical results
• Coordination of detailing and checking procedures
• Convenience in duplication of layouts
• Basis for continued investigation for new and improved techniques
Ockham’s razor
• If two theories explain the same thing, then the simpler one is to be preferred.
• William of Ockham ~1300
Bernstein-Bezier
• Clough-Tocher
• Barycentric coordinates
• Font design
GN: just basisGN: just basis
Evolution dead ends
• Local coordinates / Wilson-Fowler
• Transfinite interpolation / Coons-Gordon
• Geometric continuity for curves / tension
SIAM - Fields Institute WorkshopJune 25-26, 2001
• Fast algorithms for calculating real time geometry; on-line inspection / digitizing
• Extracting information from large data sets that are not already being addressed in data mining conferences
• Data compression, translation, and transmission
Open Problems
• surfaces with good curvature distribution
• Nonlinear vs linear optimization
• Geometry augmented by function
Open Problems
• Fitting smooth surfaces to voxel data
• Conversion algorithms:– Parametric– Subdivision– Implicit– Mesh
Problems in current systems
•(b-rep) based on trimmed non-uniform b-spline surfaces (nurbs).
•Not watertight, since nurbs cannot represent curves of intersection and other derived curves. About 10-25% of geometry/topology kernel code is devoted to resolving tolerance inconsistencies
•Models are becoming increasingly complex– Need wide range of representations (Coarse - fine grain)– Need local control of accuracy of model
MS-Subdivision
• Provides approximation of models at various levels of resolution
– Concepts from wavelets(?)
– So far: ad-hoc, waiting for theoretical basis
– Nonstationary schemes?