Algebraic and Geometric Properties of a Family of Rational Curves

Yıl 2024, Cilt: 17 Sayı: 2, 306 - 316

J. William Hoffman Haohao Wang

https://doi.org/10.36890/iejg.1551016

Öz

This paper consists of two components - an application part and a theoretical part, where the former targets the applications of computer aided geometric designs in generating parametric curves, and the latter focuses on the algebraic analysis of rational space curves. At the application level, we construct a family of rational space curves via quaternion products of two generating curves. At the theoretical level, we use algebraic methods to extract a $\mu$-basis for this family of curves, and describe a basis for a special submodule of the syzygy module in terms of a $\mu$-basis for the syzygy module of this family of curves. A commutative diagram is provided to summarize these results.

Anahtar Kelimeler

Syzygy module, rational curve, matrix factorization

Kaynakça

• [1] Adkins, W., Weintraub, S.: Algebra: An approach via module theory. Graduate Texts in Mathematics 136. Springer. New York. (1992).
• [2] Bose, N. K.: Multidimensional systems theory and applications. Springer. New York. (1995).
• [3] Chaikin, G.: An algorithm for high speed curve generation. Computer Graphics and Image Processing. 3, 346-349 (1974).
• [4] Cox, D.: Equations of parametric curves and surfaces via syzygies. Contemporary Mathematics. 286, 1-20 (2001).
• [5] Cox, D., Little, J., O’Shea, D.: Using algebraic geometry. Graduate Texts in Mathematics 185. Springer. New York. (1998).
• [6] Eisenbud, D.: The geometry of syzygies. Graduate Texts in Mathematics 229. Springer. New York. (2005).
• [7] Evans, E. G., Griffith, P.: Syzygies. London Mathematics Society Lecture Notes Series 106. Cambridge University Press. Cambridge. (1985).
• [8] Farin, G.: Algorithms for rational Bézier curves. Computer Aided Design. 15, 73-77 (1983).
• [9] Farin, G.: Curves and surfaces for computer aided geometric design. Morgan-Kaufmann. Massachusetts. (2001).
• [10] Goldman, R.: Rethinking quaternions: theory and computation. Synthesis Lectures on Computer Graphics and Animation, ed. Brian A. Barsky, No. 13. Morgan & Claypool Publishers. San Rafael. (2010).
• [11] Hagen, H.:Geometric spline curves. Computer Aided Geometric Design. 2, 223-228 (1985).
• [12] Northcott, D. G.: A homological investigation of a certain residual ideal. Math. Annalen. 150, 99-110 (1963).
• [13] Wang, H., Goldman, R.:Surfaces of revolution with moving axes and angles. Graphical Models. 106, (2019). https://doi.org/10.1016/j.gmod.2019.101047.