Evolution of a Discipline

CAGD

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

CAGD

• A view of history

• Ockham’s razor

• Trends

CAGD

• 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 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

Who was first?

CAGD

• 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

Bernstein-Bezier

• Clough-Tocher

• Barycentric coordinates

• Font design

GN: just basisGN: just basis

Blossoms

• B-spline-to-Bezier

• Compositions

• Derivatives

B-splines

• Spline curve interpolation

• Tensor products

Evolution dead ends

• Local coordinates / Wilson-Fowler

• Transfinite interpolation / Coons-Gordon

• Geometric continuity for curves / tension

CAGD

• A view of history

• Ockham’s razor

• Trends

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?

Survival of the Fittest?

• NURBS

• Subdivision

• Triangle Meshes

• Implicit

Open Areas

• Med/bio modeling

• Animation

• Architecture