CAGD history and outlook Gerald Farin Arizona State

CAGD history and outlook Gerald Farin Arizona State University

Ben Jakober

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

CAGD Representation and Approximation of curves and surfaces Computing with geometric objects

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

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

Dawn of CAGD • R. Liming: Practical Analytical Geometry with Applications to Aircraft. 1944 • Numerical applications of conics CAD = Calculator Aided Design

1950’s: NC • Milling machines • Lathes • Plotters

Bezier Methods • • 1959 de Casteljau 60’s Bezier 1987 de Casteljau / Ramshaw 1987 Farouki/Rajan

Bezier Methods • Standard in -CAD/CAM -Font Design -Animation • Numerical Stability • no local control • No local control • Smoothness • Not periodic

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

B-spline Methods • Standard in -Approximation Theory -CAD/CAM • Numerical stability • Local control • Bookkeeping • Topology

Implicit Methods • 1983 Sederberg • 1992 Bajaj

Implicit Methods • Easy ray tracing etc. • Rich shapes • Evaluation • Unwanted branches • Shape(? )

Geometry Processing • PH methods • Symbolic methods -Groebner bases • ODE methods • Trimmed surfaces • High degrees • Topology Patrikalakis/Maekawa 2001

Bezier Triangles • • • 1959 de Casteljau 1971 Zensiek 1976 Sabin 1980 Farin 1987 Ramshaw 1987 Alfeld/Schumaker / Lai 2004

Bezier Triangles • pp spaces • 3 D Studio, Nvidia, • Arbitrary topology • Smoothness / Shape • Not IGES

Triangle Meshes • • FEM Digitizers / Rapid prototyping / STL 1987 Lorensen / Cline 1992 Hoppe et al

Triangle Meshes • Piecewise linear • Multiresolution • arbitrary topology • Piecewise linear • Data volume • Class A surfaces

Subdivision Surfaces • • • 1953 de Rham 1972 Chaikin 1978 Doo/Sabin & Catmull/Clark 1987 Loop 1990 Dyn/Gregory/Levin 1994 Reif

Doo-Sabin

Catmull-Clark

Subdivision Surfaces • Arbitrary topology • Automatic smoothness • Texture mapping • Not IGES • Point evaluation • Processing

NURBS • 1966 Coons / Forrest • 1984 Versprille

NURBS • • • “All-encompassing” Industry standard Weights Curves on quadrics Developable surfaces Data reduction • Derivatives • Weights

Coons Patches • 1960’s Coons • 1960’s Gordon • 1970’s Barnhill, Gregory

Coons Patches • Use in FEM • Twists • Not IGES • Procedural: Result of operation is not element of finite-dimensional linear space

Shape • • 1962 Geise 1970’s+ Manning, Nielson, Barsky, . . . 1980’s Hoschek, Kellander 1990 Shirman/Sequin

Geometric Continuity • Shape parameters • Reverse engineering • Surface Splines • Shape parameters • Curve to Surface • Effort / Results

New Topics • • • New Geometries Volume modeling Scattered data approximation Med / bio / geo / games / animation

Future tonight