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Computer Aided Geometric Design

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Title: Computer Aided Geometric Design


1
Computer Aided Geometric Design
  • Byung-Gook Lee
  • Dongseo Univ.
  • http//kowon.dongseo.ac.kr/lbg/

2
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,

3
Computer Graphics
  • Modeling
  • Animation
  • Rendering

4
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.

5
Animation
  • Topics include traditional principles of
    animation, kinematic and dynamic modeling
    techniques, physical simulation, procedural
    methods, and motion capture based animation.

6
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.

7
CAGD
  • Computer
  • Aided
  • Geometric
  • Design
  • interpolation
  • spline
  • bezier
  • B-spline
  • NURBS
  • subdivision
  • simplification

8
Affine combination
9
Interpolating polynomial
10
General polynomial Interpolation
  • Lagrange polynomials

11
Examples of cubic interpolation
12
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

13
Piecewise cubic hermite interpolation
14
Cubic spline interpolation
15
Natural boundary condition
16
Bezier
  • P. de Casteljau, Citroen, 1959
  • P. Bezier, Renault, UNISUF system, 1962
  • R. Forrest, 1970

17
Bezier
18
Representation Bezier
19
Properties of Bezier
  • Affine invariance
  • Convex hull property
  • Endpoint interpolation
  • Symmetry
  • Linear precision
  • Pseudo-local control

20
  • 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)

21
Bezier Degree Reduction
22
B-spline
  • C. de Boor, 1972
  • W. Gordon, Richard F. Riesenfeld, 1974
  • Larry L. Schumaker
  • Tom Lyche
  • Cohen
  • Nira Dyn

23
B-spline
24
Linear splines
25
Quadratic splines
26
Quadratic splines
27
Representation splines
28
Cubic splines
29
Spline problems
  • Degree Elevation
  • Degree Reduction
  • Knot Insertion
  • Knot Deletion

30
Subdivision schemes
31
Interpolatory subdivision schemes
32
Variational subdivision schemes
33
Variational subdivision schemes
34
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

35
Subdivision Surfaces
36
Subdivision Surfaces
Tony Derose Pixar Animation Studios
Geris game
37
Quasi-interpolants
38
Polygonal Simplification
39
Multiresolution
40
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/
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