An improved algorithm for constructing moving quadrics from moving planes

This paper proposes an improved algorithm to construct moving quadrics from moving planes that follow a tensor product surface with no base points, assuming that there are no moving planes of low degree following the surface. These moving quadrics provide an efficient method to implicitize the tensor product surface which outperforms a previous approach by the present authors.

[1]  Thomas W. Sederberg,et al.  A Direct Approach to Computing the µ-basis of Planar Rational Curves , 2001, J. Symb. Comput..

[2]  Li-Yong Shen,et al.  Determination and (re)parametrization of rational developable surfaces , 2015, J. Syst. Sci. Complex..

[3]  Wenjun Wu,et al.  Mechanical Theorem Proving in Geometries , 1994, Texts and Monographs in Symbolic Computation.

[4]  Wen-tsün Wu.  Numerical and Symbolic Scientific Computing , 1994, Texts & Monographs in Symbolic Computation.

[5]  Wenping Wang,et al.  Revisiting the [mu]-basis of a rational ruled surface , 2003, J. Symb. Comput..

[6]  Yisheng Lai,et al.  Implicitizing rational surfaces using moving quadrics constructed from moving planes , 2016, J. Symb. Comput..

[7]  Thomas W. Sederberg,et al.  On the minors of the implicitization Bézout matrix for a rational plane curve , 2001, Comput. Aided Geom. Des..

[8]  Thomas W. Sederberg,et al.  Rational-Ruled Surfaces: Implicitization and Section Curves , 1995, CVGIP Graph. Model. Image Process..

[9]  Wenping Wang,et al.  Computing singular points of plane rational curves , 2008, J. Symb. Comput..

[10]  André Galligo,et al.  A computational study of ruled surfaces , 2009, J. Symb. Comput..

[11]  Laurent Busé,et al.  Implicitizing rational hypersurfaces using approximation complexes , 2003, J. Symb. Comput..

[12]  William A. Adkins,et al.  Equations of parametric surfaces with base points via syzygies , 2005, J. Symb. Comput..

[13]  Ron Goldman,et al.  Implicitizing rational surfaces of revolution using μ-bases , 2012, Comput. Aided Geom. Des..

[14]  Falai Chen,et al.  Implicitization using moving curves and surfaces , 1995, SIGGRAPH.

[15]  Ron Goldman,et al.  Implicitization by Dixon A-resultants , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[16]  Jiansong Deng,et al.  Implicitization and parametrization of quadratic and cubic surfaces by μ-bases , 2007, Computing.

[17]  Ron Goldman,et al.  Implicit representation of parametric curves and surfaces , 1984, Comput. Vis. Graph. Image Process..

[18]  Falai Chen,et al.  The mu-basis of a rational ruled surface , 2001, Comput. Aided Geom. Des..

[19]  Ron Goldman,et al.  On a relationship between the moving line and moving conic coefficient matrices , 1999, Comput. Aided Geom. Des..

[20]  Carlos D'Andrea.  Resultants and Moving Surfaces , 2001, J. Symb. Comput..

[21]  Ron Goldman,et al.  On the Validity of Implicitization by Moving Quadrics for Rational Surfaces with No Base Points , 2000, J. Symb. Comput..

[22]  Ron Goldman,et al.  Using multivariate resultants to find the implicit equation of a rational surface , 1992, The Visual Computer.

[23]  C. D'Andrea,et al.  Implicitization of rational surfaces using toric varieties , 2004, math/0401403.

[24]  Jiansong Deng,et al.  Computing μ-bases of rational curves and surfaces using polynomial matrix factorization , 2005, ISSAC '05.

[25]  Falai Chen,et al.  The μ -basis and implicitization of a rational parametric surface , 2005 .

[26]  David A. Cox,et al.  IMPLICITIZATION OF SURFACES IN ℙ3 IN THE PRESENCE OF BASE POINTS , 2002, math/0205251.

[27]  Marc Dohm.  Implicitization of rational ruled surfaces with mu-bases , 2009, J. Symb. Comput..

[28]  Falai Chen,et al.  Implicitization, parameterization and singularity computation of Steiner surfaces using moving surfaces , 2012, J. Symb. Comput..

[29]  Xiao-Shan Gao,et al.  Implicitization of Rational Parametric Equations , 1992, J. Symb. Comput..