Article

Computation of the degree of rational maps between curves

Authors:

J. Rafael Sendra,

Franz WinklerAuthors Info & Claims

ISSAC '01: Proceedings of the 2001 international symposium on Symbolic and algebraic computation

Pages 317 - 322

https://doi.org/10.1145/384101.384144

Published: 01 July 2001 Publication History

Get Access

Abstract

The degree of a rational map measures how often the map covers the image variety. In particular, when the rational map is a parametrization, the degree measures how often the parametrization traces the image. We show how the degree of rational maps between algebraic curves can be determined efficiently. In the process, we also give a complete proof of Sederberg's approach for making a parametrization proper.

References

[1]

Chionh E.-W., Goldman R.N. (1992), Using multivariate resultants to find the implicit equation o f a rational surface. The Visual Computer 8, 171-180.

Digital Library

Google Scholar

[2]

Cox D.A., Sederberg T.W., Chen F. (1998), The moving line ideal basis o f planar rational curves. Computer Aided Geometric Design 8, 803-827.

Digital Library

Google Scholar

[3]

Gao X.-S., Chou S.-C. (1992), Implicitization o f Rational Parametric Equations. J. Symbolic Computation 14, 459-470.

Digital Library

Google Scholar

[4]

Harris J. (1995), Algebraic Geometry. A First Course. Springer-Verlag.

Google Scholar

[5]

Sederberg T.W. (1986), Improperly Parametrized Rational Curves. Computer Aided Geometric Design 3, 67-75.

Digital Library

Google Scholar

[6]

Sendra J.R., Winkler F. (2001), Tracing Index o f Rational Parametrizations. Technical Report 01-01, RISC-Linz, Univ. Linz, Austria.

Google Scholar

[7]

Shafarevich I.R. (1994), Basic Algebraic Geometry 1; Varieties in Projective Space. Springer-Verlag, Berlin New York.

Digital Library

Google Scholar

Cited By

View all

Pérez-Díaz SSendra J(2013)Behavior of the Fiber and the Base Points of Parametrizations Under ProjectionsMathematics in Computer Science10.1007/s11786-013-0139-87:2(167-184)Online publication date: 24-Jan-2013
https://doi.org/10.1007/s11786-013-0139-8
Emiris IKonaxis CPalios L(2010)Computing the Newton Polygon of the Implicit EquationMathematics in Computer Science10.1007/s11786-010-0046-14:1(25-44)Online publication date: 24-Sep-2010
https://doi.org/10.1007/s11786-010-0046-1
Shen LWu MYang Z(2009)Simplification of Numerical Rational SurfacesProceedings of the 2009 International Joint Conference on Computational Sciences and Optimization - Volume 0110.1109/CSO.2009.274(617-621)Online publication date: 24-Apr-2009
https://dl.acm.org/doi/10.1109/CSO.2009.274
Show More Cited By

Index Terms

Computation of the degree of rational maps between curves
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

Recommendations

Rational Pythagorean-hodograph space curves

A method for constructing rational Pythagorean-hodograph (PH) curves in R^3 is proposed, based on prescribing a field of rational unit tangent vectors. This tangent field, together with its first derivative, defines the orientation of the curve ...
On fair parametric rational cubic curves
Abstract
First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we ...
Self-intersections of rational Bézier curves

Rational Bezier curves provide a curve fitting tool and are widely used in Computer Aided Geometric Design, Computer Aided Design and Geometric Modeling. The injectivity (one-to-one property) of rational Bezier curve as a mapping function is equivalent ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

ISSAC '01: Proceedings of the 2001 international symposium on Symbolic and algebraic computation

July 2001

345 pages

ISBN:1581134177

DOI:10.1145/384101

Chairmen:
Erich Kaltofen
NCSU, Raleigh, NC
,
Gilles Villard
ENS, Lyon, France

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 July 2001

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

ISSAC01

Sponsor:

SIGSAM

ISSAC01: International Symposium on Symbolic and Algebraic Computation

Ontario, London, Canada

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
262
Total Downloads

Downloads (Last 12 months)2
Downloads (Last 6 weeks)0

Reflects downloads up to 14 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Pérez-Díaz SSendra J(2013)Behavior of the Fiber and the Base Points of Parametrizations Under ProjectionsMathematics in Computer Science10.1007/s11786-013-0139-87:2(167-184)Online publication date: 24-Jan-2013
https://doi.org/10.1007/s11786-013-0139-8
Emiris IKonaxis CPalios L(2010)Computing the Newton Polygon of the Implicit EquationMathematics in Computer Science10.1007/s11786-010-0046-14:1(25-44)Online publication date: 24-Sep-2010
https://doi.org/10.1007/s11786-010-0046-1
Shen LWu MYang Z(2009)Simplification of Numerical Rational SurfacesProceedings of the 2009 International Joint Conference on Computational Sciences and Optimization - Volume 0110.1109/CSO.2009.274(617-621)Online publication date: 24-Apr-2009
https://dl.acm.org/doi/10.1109/CSO.2009.274
Pérez-Díaz SRafael Sendra J(2008)A univariate resultant-based implicitization algorithm for surfacesJournal of Symbolic Computation10.1016/j.jsc.2007.10.00143:2(118-139)Online publication date: 1-Feb-2008
https://dl.acm.org/doi/10.1016/j.jsc.2007.10.001
Pérez-Díaz SSendra JGao XLabahn G(2005)Partial degree formulae for rational algebraic surfacesProceedings of the 2005 international symposium on Symbolic and algebraic computation10.1145/1073884.1073926(301-308)Online publication date: 24-Jul-2005
https://dl.acm.org/doi/10.1145/1073884.1073926
Pérez-Dı́az SSendra J(2004)Computation of the degree of rational surface parametrizationsJournal of Pure and Applied Algebra10.1016/j.jpaa.2004.02.011193:1-3(99-121)Online publication date: Oct-2004
https://doi.org/10.1016/j.jpaa.2004.02.011
Schicho JGiusti M(2002)Simplification of surface parametrizationsProceedings of the 2002 international symposium on Symbolic and algebraic computation10.1145/780506.780536(229-237)Online publication date: 10-Jul-2002
https://dl.acm.org/doi/10.1145/780506.780536

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Index Terms

Recommendations

Rational Pythagorean-hodograph space curves

On fair parametric rational cubic curves

Self-intersections of rational Bézier curves