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 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 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 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 % 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