Elimination Matrices for Computer Aided Geometric Design
CIMPA-UNESCO-MICINN-INDONESIA Research School, July 2011, Yogyakarta, Indonesia
Laurent Busé
I. Intersection of algebraic plane curves
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2.
3.
4.
5.
Sylvester resultant
Intersection with a line through eigen-computations
Intersection multiplicity: Bezout’s theorem
Effective computation of the intersection of two plane curves
Singular points on a plane curve
References:
• Lecture notes (in french) available at :
http://cel.archives-ouvertes.fr/cel-00440419/en/
• Chapter 3, §3, §5 and §6 in the book by Cox, David A.; Little, John ; O’Shea, Donal. Ideals, varieties
and algorithms. Third edition. Undergraduate Texts in Mathematics, Springer, 2007.
• Chapter 3 in Cox, David A.; Little, John ; O’Shea, Donal. Using algebraic geometry. Second edition.
Graduate Texts in Mathematics. Springer, 2005.
• Chapter 12 in Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston,
MA, 1994.
II. Parameterized algebraic plane curves
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2.
3.
4.
Definition and basic properties
Degree of a parameterization
Implicitization of a parameterized curve
Inversion of a parameterized curve
References:
• Lecture notes (in french) available at :
http://cel.archives-ouvertes.fr/cel-00440419/en/
• Chapter 3 in Cox, David A.; Little, John ; O’Shea, Donal. Using algebraic geometry. Second edition.
Graduate Texts in Mathematics. Springer, 2005.
III. Multivariate resultant and applications to CAGD
1.
2.
3.
4.
5.
The implicitization problem for parameterized surfaces
A geometric approach to multivariate resultant
An algebraic approach to multivariate resultant: inertia forms
Macaulay’s matrices and representation matrices
Some useful formula to handle multivariate resultants
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6. Implicitization in the absence of base points
7. Poisson’s formula
8. Solving zero-dimensional polynomial systems with resultants
References:
• Chapter 3 and 4 of the lecture notes (in english) available at :
http://hal.archives-ouvertes.fr/inria-00077120/en/
• Chapter 3 in Cox, David A.; Little, John ; O’Shea, Donal. Using algebraic geometry. Second edition.
Graduate Texts in Mathematics. Springer, 2005.
IV. Parameterizations of surfaces having base points
1.
2.
3.
4.
5.
6.
7.
8.
The implicitization problem for parameterized surfaces
A geometric approach to multivariate resultant
An algebraic approach to multivariate resultant: inertia forms
Macaulay’s matrices and representation matrices
Some useful formula to handle multivariate resultants
Implicitization in the absence of base points
Poisson’s formula
Solving zero-dimensional polynomial systems with resultants
References:
• A case of study; see Chapter 3 of the lecture notes available at :
http://hal.archives-ouvertes.fr/inria-00100322/en/
• Representation matrices associated to blow-up algebras; see the lecture notes at :
Lecture notes (in english) available at
http://hal.archives-ouvertes.fr/inria-00077120/en/
Contact informations:
• Email: [email protected]
• Adress: 2004 route des Lucioles, 06902, Sophia Antipolis, France
• Web site: http://www-sop.inria.fr/members/Laurent.Buse/
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