CAGD history and outlook

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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/Chambers96
• Gallier 98
• Marsh 99
• Cohen/Riesenfeld/ Elber 02

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Dawn of CAGD

• R. Liming Practical Analytical Geometry with
  Applications to Aircraft. 1944
• Numerical applications of conics

CAD Calculator Aided Design
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1950s NC

• Milling machines
• Lathes
• Plotters

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Bezier Methods

• 1959 de Casteljau
• 60s 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
• 60s 70s 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

• 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

• Piecewise linear
• 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

• 1960s Coons
• 1960s Gordon
• 1970s 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
• 1970s Manning, Nielson, Barsky,...
• 1980s 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