Curve implicitization using moving lines
Computer Aided Geometric Design
Rational-ruled surfaces: implicitization and section curves
Graphical Models and Image Processing
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
The mu-basis of a rational ruled surface
Computer Aided Geometric Design
Efficient Groebner walk conversion for implicitization of geometric objects
Computer Aided Geometric Design
Computing μ-bases of rational curves and surfaces using polynomial matrix factorization
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A new class of term orders for elimination
Journal of Symbolic Computation
Implicitization and parametrization of quadratic surfaces with one simple base point
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Division algorithms for Bernstein polynomials
Computer Aided Geometric Design
Computing self-intersection curves of rational ruled surfaces
Computer Aided Geometric Design
Axial moving planes and singularities of rational space curves
Computer Aided Geometric Design
A computational study of ruled surfaces
Journal of Symbolic Computation
Implicitization of rational ruled surfaces with µ-bases
Journal of Symbolic Computation
Efficient Groebner walk conversion for implicitization of geometric objects
Computer Aided Geometric Design
The μ-basis and implicitization of a rational parametric surface
Journal of Symbolic Computation
Computing the singularities of rational space curves
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Proper reparametrization of rational ruled surface
Journal of Computer Science and Technology
Approximate µ-bases of rational curves and surfaces
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Using µ-bases to implicitize rational surfaces with a pair of orthogonal directrices
Computer Aided Geometric Design
| Hi-index | 0.00 |
The µ-basis of a rational ruled surface P(s,t) = P0(s) + tP1(s) is defined in Chen et al. (Comput. Aided Geom. Design 18 (2001) 61) to consist of two polynomials p(x, y, z, s) and q(x, y, z, s) that are linear in x, y, z. It is shown there that the resultant of p and q with respect to s gives the implicit equation of the rational ruled surface; however, the parametric equation P(s, t) of the rational ruled surface cannot be recovered from p and q. Furthermore, the µ-basis thus defined for a rational ruled surface does not possess many nice properties that hold for the µ-basis of a rational planar curve (Comput. Aided Geom. Design 18 (1998) 803). In this paper, we introduce another polynomial r(x, y, z, s, t) that is linear in x, y, z and t such that p, q, r can be used to recover the parametric equation P(s, t) of the rational ruled surface; hence, we redefine the µ-basis to consist of the three polynomials p, q, r. We present an efficient algorithm for computing the newly-defined µ-basis, and derive some of its properties. In particular, we show that the new µ-basis serves as a basis for both the moving plane module and the moving plane ideal corresponding to the rational ruled surface.