We present techniques for creating an approximate implicit representation of space curves and of surfaces of revolution. In both cases, the proposed techniques reduce the problem to that of implicitization of planar curves. For space curves, which are described as the intersection of two implicitly defined surfaces, we show how to generate an approximately orthogonalized implicit representation. In the case of surfaces of revolution, we address the problem of avoiding unwanted branches and singular points in the region of interest.
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
Aigner, M., Jüttler, B., Kim, M.-S.: Analyzing and enhancing the robustness of implicit representations, in: Geometric Modelling and Processing 2004, IEEE Press, 131-140.
Chuang, J., Hoffmann, C.: On local implicit approximation and its applications. ACM Trans. Graphics 8, 4:298-324, (1989)
Corless, R., Giesbrecht, M., Kotsireas, I., Watt, S.: Numerical implicitization of parametric hypersurfaces with linear algebra. In: AISC’2000 Proceedings, Springer, LNAI 1930.
Cox, D., Little, J., O’Shea, D.: Using algebraic geometry, Springer Verlag, New York 1998.
Cox, D., Goldman, R., Zhang, M.: On the validity of implicitization by moving quadrics for rational surfaces with no base points, J. Symbolic Computation, 11, (1999)
Dokken, T., et al.: Intersection algorithms for geometry based IT-applications using approximate algebraic methods, EU project IST-2001-35512 GAIA II, 2002-2005.
Dokken, T., and Thomassen, J., Overview of Approximate Implicitization, in: Topics in Algebraic Geometry and Geometric Modeling, AMS Cont. Math. 334 (2003), 169-184.
Gonzalez-Vega, L.: Implicitization of parametric curves and surfaces by using multidimensional Newton formulae. J. Symb. Comput. 23 (2-3), 137-151 (1997)
Hoschek, J., and Jüttler, B.: Techniques for fair and shape-preserving surface fitting with tensor-product B-splines, in: J.M. Peña (ed.), Shape Preserving Representations in Computer Aided Design, Nova Science Publishers, New York 1999, 163-185.
Jüttler, B.: Least-squares fitting of algebraic spline curves via normal vector estimation, in: Cipolla, R., Martin, R.R. (eds.), The Mathematics of Surfaces IX, Springer, London, 263-280, 2000.
Jüttler, B., and Felis, A.: Least-squares fitting of algebraic spline surfaces, Adv. Comp. Math. 17 (2002), 135-152.
Mourrain, B., Pavone, J.-P., Subdivision methods for solving polynomial equations, Technical Report 5658, INRIA Sophia-Antipolis, 2005.
Sampson, P. D. Fitting conic sections to very scattered data: an iterative refinement of the Bookstein algorithm, Computer Graphics and Image Processing 18 (1982), 97-108.
Sederberg, T., Chen F.: Implicitization using moving curves and surfaces. Siggraph 1995, 29,301-308, (1995)
Shalaby, M. F., Thomassen, J. B., Wurm, E. M., Dokken, T., Jüttler, B.: Piecewise approximate implicitization: Experiments using industrial data, in: Algebraic Geometry and Geometric Modeling (Mourrain, B., Elkadi, M., Piene, R., eds.), Springer, in press.
Wurm, E., Jüttler, B.: Approximate implicitization via curve fitting, in Kobbelt, L., Schröder, P., Hoppe, H. (eds.), Symposium on Geometry Processing, Eurographics / ACM Siggraph, New York 2003, 240-247.
Wurm, E., Thomassen, J., Jüttler, B., Dokken, T.: Comparative Benchmarking of Methods for Approximate Implicitization, in: Neamtu, M., and Lucian, M. (eds.), Geometric Design and Computing, Nashboro Press, Brentwood 2004, 537-548.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Shalaby, M., Jüttler, B. (2008). Approximate Implicitization of Space Curves and of Surfaces of Revolution. In: Jüttler, B., Piene, R. (eds) Geometric Modeling and Algebraic Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72185-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-72185-7_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72184-0
Online ISBN: 978-3-540-72185-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)