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CAGD history and outlook
Gerald Farin Arizona State University
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Ben Jakober
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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
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CAGD Representation and Approximation of curves and surfaces
Computing with geometric objects
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CAGD topics Bezier metods B-spline methods Coons/Gordon methods
Algebraic methods Triangular patches Triangle meshes Subdivision surfaces NURBS Geometric continuity Geometry processing New topics
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CAGD Books Faux/Pratt 78 Farin 88, 90, 93, 96 Bartels/Beatty/Barsky 86
Yamaguchi 88 Farin/Hansford 00 Boehm/Prautzsch/ Paluzny 01 Su/Liu 89 Mortenson 85, 98 Hoschek/Lasser 93 Rockwood/Chambers 96 Gallier 98 Marsh 99 Cohen/Riesenfeld/ Elber 02
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CAD = Calculator Aided Design
Dawn of CAGD R. Liming: Practical Analytical Geometry with Applications to Aircraft Numerical applications of conics CAD = Calculator Aided Design
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1950’s: NC Milling machines Lathes Plotters
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Bezier Methods 1959 de Casteljau 60’s Bezier
1987 de Casteljau / Ramshaw 1987 Farouki/Rajan
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Bezier Methods Standard in -CAD/CAM -Font Design -Animation
Numerical Stability no local control No local control Smoothness Not periodic
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B-spline Methods 1933 Popoviciu 1949 Schoenberg
60’s + 70’s de Boor (+Cox / Mansfield) 1974 Gordon, Riesenfeld 1980 Boehm 1980 Cohen, Lyche, Riesenfeld 1987 Ramshaw / de Casteljau
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B-spline Methods Standard in -Approximation Theory -CAD/CAM
Numerical stability Local control Bookkeeping Topology
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Implicit Methods 1983 Sederberg 1992 Bajaj
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Implicit Methods Easy ray tracing etc. Rich shapes Evaluation
Unwanted branches Shape(?)
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Geometry Processing Patrikalakis/Maekawa 2001 PH methods
Symbolic methods -Groebner bases ODE methods Trimmed surfaces High degrees Topology Patrikalakis/Maekawa 2001
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Bezier Triangles 1959 de Casteljau 1971 Zensiek 1976 Sabin 1980 Farin
1987 Ramshaw 1987 Alfeld/Schumaker Schumaker / Lai 2004
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Bezier Triangles pp spaces 3D Studio, Nvidia, Arbitrary topology
Smoothness / Shape Not IGES
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Triangle Meshes FEM Digitizers / Rapid prototyping / STL
1987 Lorensen / Cline 1992 Hoppe et al
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Triangle Meshes Piecewise linear Multiresolution arbitrary topology
Data volume Class A surfaces
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Subdivision Surfaces 1953 de Rham 1972 Chaikin
1978 Doo/Sabin & Catmull/Clark 1987 Loop 1990 Dyn/Gregory/Levin 1994 Reif
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Doo-Sabin
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Catmull-Clark
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Subdivision Surfaces Arbitrary topology Automatic smoothness
Texture mapping Not IGES Point evaluation Processing
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NURBS 1966 Coons / Forrest 1984 Versprille
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NURBS “All-encompassing” Industry standard Weights Curves on quadrics
Developable surfaces Data reduction Derivatives Weights
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Coons Patches 1960’s Coons 1960’s Gordon 1970’s Barnhill, Gregory
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Coons Patches Use in FEM Twists Not IGES Procedural Procedural: Result of operation is not element of finite-dimensional linear space
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Shape 1962 Geise 1970’s+ Manning, Nielson, Barsky,...
1980’s Hoschek, Kellander 1990 Shirman/Sequin
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Geometric Continuity Shape parameters Reverse engineering
Surface Splines Shape parameters Curve to Surface Effort / Results
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New Topics New Geometries Volume modeling Scattered data approximation
Med / bio / geo / games / animation
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Future tonight
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