Computer Aided Geometric Design

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Computer Aided Geometric Design

• Byung-Gook Lee
• Dongseo Univ.
• http//kowon.dongseo.ac.kr/lbg/

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Computer Graphics

• The generation of graphical output using a
  computer

• by developing software to accomplish the task
• by using pre-existing application software like
  Photoshop, 3D Studio Max, Maya,

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Computer Graphics

• Modeling
• Animation
• Rendering

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Modeling

• The creation of mathematical models of 2D and 3D
  objects
• Computer generated models, or 3D models, are
  built in the 3D environment of a computer.

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Animation

• Topics include traditional principles of
  animation, kinematic and dynamic modeling
  techniques, physical simulation, procedural
  methods, and motion capture based animation.

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Rendering

• These models, starting as a wire frame model, are
  digitally wrapped with textures and rendered with
  reflections, transparencies, and shadows to give
  a photo-realistic view of the object or building.

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CAGD

• Computer
• Aided
• Geometric
• Design

• interpolation
• spline
• bezier
• B-spline
• NURBS
• subdivision
• simplification

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Affine combination
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Interpolating polynomial
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General polynomial Interpolation

• Lagrange polynomials

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Examples of cubic interpolation
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Spline curve

• J.Ferguson , Boeing Co., 1963
• C.de Boor, W.Gordon, General Motors, 1963
• to interpolate given data
• piecewise polynomial curves with certain
  differentiability constraints
• not to design free form curves

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Piecewise cubic hermite interpolation
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Cubic spline interpolation
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Natural boundary condition
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Bezier

• P. de Casteljau, Citroen, 1959
• P. Bezier, Renault, UNISUF system, 1962
• R. Forrest, 1970

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Bezier
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Representation Bezier
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Properties of Bezier

• Affine invariance
• Convex hull property
• Endpoint interpolation
• Symmetry
• Linear precision
• Pseudo-local control

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• Pierre BEZIER died on November 25, 1999. The
  following short biography has been published in
  volume 22, number 9 (november 1990) of "Computer
  Aided Design", a special issue devoted to Bezier
  techniques. Pierre Etienne Bezier was born on
  September 1, 1910 in Paris. Son and grandson of
  engineers, he chose this profession too and
  enrolled to study mechanical engineering at the
  Ecole des Arts et Metiers and received his degree
  in 1930. In the same year he entered the Ecole
  Superieure d'Electricite and earnt a second
  degree in electrical engineering in 1931. In
  1977, 46 years later, he received his DSc degree
  in mathematics from the University of Paris. In
  1933, aged 23, Bezier entered Renault and worked
  for this company for 42 years. He started as Tool
  Setter, became Tool Designer in 1934 and Head of
  the Tool Design Office in 1945. In 1948, as
  Director of Production Engineering he was
  responsable for the design of the transfer lines
  producing most of the 4 CV mechanical parts. In
  1957, he became Director of Machine Tool Division
  and was responsable for the automatic assembly of
  mechanical components, and for the design and
  production of an NC drilling and milling machine,
  most probably one of the first machines in
  Europe. Bezier become managing staff member for
  technical development in 1960 and held this
  position until 1975 when he retired. Bezier
  started his research in CADCAM in 1960 when he
  devoted a substantial amount of his time working
  on his UNISURF system. From 1960, his research
  interest focused on drawing machines, computer
  control, interactive free-form curve and surface
  design and 3D milling for manufactoring clay
  models and masters. His system was launched in
  1968 and has been in full use since 1975
  supporting about 1500 staff members today.
  Bezier's academic career began in 1968 when he
  became Professor of Production Engineering at the
  Conservatoire National des Arts et Metiers. He
  held this position until 1979. He wrote four
  books, numerous papers and received several
  distinctions including the "Steven Anson Coons"
  of the Association for Computing Machinery and
  the "Doctor Honoris Causa" of the Technical
  University Berlin. He is an honorary member of
  the American Society of Mechanical Engineers and
  of the Societe Belge des Mecaniciens,
  ex-president of the Societe des Ingenieurs et
  Scientifiques de France, Societe des Ingenieurs
  Arts et Metiers, and he was one of the first
  Advisory Editors of "Computer-Aided Design".
• Christophe Rabut, Departement de Genie
  Mathematique Institut National des Sciences
  Appliquees 135, avenue de Rangueil F-31077
  TOULOUSE Cedex 4 (France)

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Bezier Degree Reduction
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B-spline

• C. de Boor, 1972
• W. Gordon, Richard F. Riesenfeld, 1974
• Larry L. Schumaker
• Tom Lyche
• Cohen
• Nira Dyn

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B-spline
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Linear splines
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Quadratic splines
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Quadratic splines
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Representation splines
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Cubic splines
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Spline problems

• Degree Elevation
• Degree Reduction
• Knot Insertion
• Knot Deletion

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Subdivision schemes
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Interpolatory subdivision schemes
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Variational subdivision schemes
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Variational subdivision schemes
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Variational subdivision schemes

• Second forward differences
  minimal total curvature
• Third forward differences
  minimal variational of curvature
• 
  reproduce the uniformly interpolating cubic
  splines
• 
  4-point rule

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Subdivision Surfaces
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Subdivision Surfaces
Tony Derose Pixar Animation Studios
Geris game
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Quasi-interpolants
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Polygonal Simplification
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Multiresolution
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References

• Curves and Surfaces for Computer Aided Geometric
  Design, Gerald Farin, Academic Press
• The NURBS book, Les Piegl and Wayne Tiller,
  Springer
• Spline Methods Drafts, Tom Lyche and Knut Morken
• Computer Graphics Geometric Modeling, Davis
  Salomon, Springer

http//kowon.dongseo.ac.kr/lbg/cagd/