Abstract
A degree n rational plane curve rotating about an axis in the plane creates a degree 2n rational surface. Two formulas are given to generate 2n moving planes that follow the surface. These 2n moving planes lead to a 2n×2n implicitization determinant that manifests the geometric revolution algebraically in two aspects. Firstly the moving planes are constructed by successively shifting terms of polynomials from one column to another of a spawning 3×3 determinant. Secondly the right half of the 2n×2n implicitization determinant is almost an n-row rotation of the left half. As an aside, it is observed that rational parametrizations of a surface of revolution due to a symmetric rational generatrix must be improper.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chionh, E.W., Goldman, R.N.: Part 1: Elimination and Bivariate Resultants. IEEE Computer Graphics and Applications 15(1), 69–77 (1995)
Chionh, E.W., Gao, X.S., Shen, L.Y.: Inherently improper surface parametric supports. Computer Aided Geometric Design 23, 629–639 (2006)
Cox, D., Goldman, R.N., Zhang, M.: On the validity of implicitization by moving quadrics for rational surfaces with no base points. Journal of Symbolic Computation 29, 419–440 (2000)
Dixon, A.L.: The eliminant of three quantics in two independent variables. Proc. London Math. Soc. 6, 49–69, 473–492 (1908)
Goldman, R.N., Sederberg, T.W., Anderson, D.C.: Vector elimination: A technique for the implicitization, inversion and intersection of planar parametric rational polynomial curves. Computer Aided Geometric Design 1(4), 327–356 (1984)
González-Vega, L., Necula, I., Pérez-Díaz, S., Sendra, J., Sendra, J.R.: Algebraic Methods in Computer Aided Geometric Design: Theoretical and Practical Applications. In: Geometric Computation. Lecture Notes Series on Computing, vol. 11, pp. 1–33. World Scientific, Singapore (2004)
Kotsireas, I.S.: Panorama of Methods for Exact Implicitization of Algebraic Curves and Surfaces. In: Geometric Computation. Lecture Notes Series on Computing, vol. 11, pp. 126–155. World Scientific, Singapore (2004)
Sederberg, T.W., Anderson, D.C., Goldman, R.N.: Implicit representation of parametric curves and surfaces. Computer Vision, Graphics and Image Processing 28, 72–84 (1984)
Sederberg, T.W., Chen, F.L.: Implicitization Using Moving Curves and Surfaces. In: Proceedings of SIGGRAPH 1995, pp. 301–308. ACM SIGGRAPH/Addison Wesley, Los Angeles (1995)
Zheng, J.M., Sederberg, T.W., Chionh, E.W., Cox, D.A.: Implicitizing rational surfaces with base points using the method of moving surfaces. Contemporary Mathematics 334, 151–168 (2003)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chionh, EW. (2008). Shifting Planes to Follow a Surface of Revolution. In: Chen, F., Jüttler, B. (eds) Advances in Geometric Modeling and Processing. GMP 2008. Lecture Notes in Computer Science, vol 4975. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79246-8_30
Download citation
DOI: https://doi.org/10.1007/978-3-540-79246-8_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79245-1
Online ISBN: 978-3-540-79246-8
eBook Packages: Computer ScienceComputer Science (R0)