Mathematical Methods in Computer Graphics, Duksung Univ. Nov. 1. 2006, [email protected] Mathematical Methods in Computer Graphics Byung-Gook Lee Dongseo

Mathematical Methods in Computer Graphics, Duksung Univ. Nov. 1. 2006, [email protected]

Mathematical Methods in Computer Graphics

Byung-Gook LeeDongseo Univ.

http://kowon.dongseo.ac.kr/~lbg/



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, …


Computer Graphics

• Modeling• Animation• Rendering


Modeling

• The creation of mathematical models of 2D and 3D objects in the 3D environment of a computer.


Animation

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


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.


Web 3D

http://www.cult3d.com/http://www.cult3d.com/


CAGD

Computer Aided Geometric Design

interpolation spline bezier B-spline NURBS subdivision simplification parametrization

CAGD is a branch of applied mathematics concerned with algorithms for the design of smooth curves and surfaces and for their efficients mathematical representation.


Affine combination

• Linear combinations

• Affine(Barycentric) combinations

• Convex combinations

• Barycentric coordinates


Affine combination

Euclidean coordinate system

Coordinate-free system


Polynomial interpolation


General polynomial Interpolation

• Lagrange polynomials


Examples of cubic interpolation


Bezier

• Paul de Faget de Casteljau, Citroen, 1959• Pierre Bezier, Renault, UNISUF system, 1962• A.R. Forrest, Cambridge, 1970


Bezier


Representation Bezier


Properties of Bezier

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

SS


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)


Bezier Degree Reduction


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


Piecewise cubic hermite interpolation


Cubic spline interpolation


B-spline

• C. de Boor, 1972• W. Gordon, Richard F. Riesenfeld, 1974

• Larry L. Schumaker• Tom Lyche• Nira Dyn• Cohen


B-spline


Linear splines


Quadratic splines


Quadratic splines


Representation splines


Cubic splines


Spline problems

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


Subdivision schemes


Subdivision Surfaces


Subdivision Surfaces

Tony DerosePixar Animation Studios

Geri’s game

http://www.cgal.org/Tutorials/Polyhedron/http://www.cgal.org/Tutorials/Polyhedron/


Quasi-interpolants


Polygonal Simplification


Multiresolution


Parametrization

3D meshparameterization

with fixed boundary


Texture Mapping


Image Compression


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 Salo

mon, Springer

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

Documents

Mathematical Methods in Computer Graphics, Duksung Univ. Nov. 1. 2006, [email protected] Mathematical Methods in Computer Graphics Byung-Gook Lee Dongseo