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Evolution of a Discipline
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
Ben Jakober
• A view of history
• Ockham’s razor
• Trends
• A view of history
• Ockham’s razor
• Trends
Levels of Abstraction
B.C: manual
Medevial: Geometric constructions
1600’s: splines
1944: Liming
1960: De Casteljau/Bezier
2000+: manual!
A mechanical spline
Liming’s benefits
• Increase in precision and
• Elimination of deviations
resulting from the human
• 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
Who was first?
• A view of history
• Ockham’s razor
• Trends
Ockham’s razor
• If two theories explain the same thing, then
the simpler one is to be preferred.
• William of Ockham ~1300
• Clough-Tocher
• Barycentric coordinates
• Font design
GN: just basis
• B-spline-to-Bezier
• Compositions
• Derivatives
• Spline curve interpolation
• Tensor products
Evolution dead ends
• Local coordinates / Wilson-Fowler
• Transfinite interpolation / Coons-Gordon
• Geometric continuity for curves / tension
• A view of history
• Ockham’s razor
• Trends
SIAM - Fields Institute Workshop
June 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
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:
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 1025% 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
• Provides approximation of models at
various levels of resolution
– Concepts from wavelets(?)
– So far: ad-hoc, waiting for theoretical basis
– Nonstationary schemes?
Survival of the Fittest?
• Subdivision
• Triangle Meshes
• Implicit
Open Areas
• Med/bio modeling
• Animation
• Architecture