Third Order Invariants of Surfaces
The classical invariant theory from the 19th century is used to determine a complete system of 3rd order invariants on a surface in three-space. The invariant ring has 18 generators and the ideal of syzygies has 65 generators. The invariants are expressed as polynomials in the components of the first fundamental form, the second fundamental form and the covariant derivative of the latter, or in the case of an implicitly defined surface — M = f -1(0) — as polynomials in the partial derivatives of f up to order three.
As an application some commonly used fairings measures are written in invariant form. It is shown that the ridges and the subparabolic curve of a surface are the zero set of invariant functions and it is finally shown that the Darboux classification of umbilical points can be given in terms of two invariants.
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- 2.Gurevich, G.B.: Foundation of the theory of algebraic invarians. P. Noordhoff, Groningen (1964)Google Scholar
- 3.Olver, P.J.: Classical invariant theory. Volume 44 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1999)Google Scholar
- 9.Fulton, W., Harris, J.: Representation Theory. A First Course. Volume 129 of Graduate Texts in Mathematics. Springer-Verlag (1991)Google Scholar
- 10.Greuel, G.M., Pfister, G., Schönemann, H.: Singular 2.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2001) http://www.singular.uni-kl.de/Google Scholar
- 11.Gravesen, J.: (Invariants) http://www.mat.dtu.dk/people/J.Gravesen/invGoogle Scholar
- 12.Gravesen, J., Ungstrup, M.: Constructing invariant fairness measures for surfaces. Advances in Computational Mathematics (2001)Google Scholar