Curve implicitization using moving lines
Computer Aided Geometric Design
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Implicitizing rational curves by the method of moving algebraic curves
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
On the validity of implicitization by moving quadrics for rational surfaces with no base points
Journal of Symbolic Computation
A new implicit representation of a planar rational curve with high order singularity
Computer Aided Geometric Design
The µ-basis of a planar rational curve: properties and computation
Graphical Models
Revisiting the µ-basis of a rational ruled surface
Journal of Symbolic Computation
Axial moving lines and singularities of rational planar curves
Computer Aided Geometric Design
Computation of the singularities of parametric plane curves
Journal of Symbolic Computation
Visualisation of Implicit Algebraic Curves
PG '07 Proceedings of the 15th Pacific Conference on Computer Graphics and Applications
Computing singular points of plane rational curves
Journal of Symbolic Computation
μ-bases for polynomial systems in one variable
Computer Aided Geometric Design
Set-theoretic generators of rational space curves
Journal of Symbolic Computation
Computing the singularities of rational space curves
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Matrix-based implicit representations of rational algebraic curves and applications
Computer Aided Geometric Design
Journal of Symbolic Computation
Certified approximation of parametric space curves with cubic B-spline curves
Computer Aided Geometric Design
Using a bihomogeneous resultant to find the singularities of rational space curves
Journal of Symbolic Computation
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Relationships between the singularities of rational space curves and the moving planes that follow these curves are investigated. Given a space curve C with a generic 1-1 rational parametrization F(s,t) of homogeneous degree d, we show that if P and Q are two singular points of orders k and k^' on the space curve C, then there is a moving plane of degree d-k-k^' with axis PQ@? that follows the curve. We also show that a point P is a singular point of order k on the space curve C if and only if there are two axial moving planes L"1 and L"2 of degree d-k such that: (1) the axes of L"1, L"2 are orthogonal and intersect at P, and (2) the intersection of the moving planes L"1 and L"2 is the cone through the curve C with vertex P together with d-k copies of the plane containing the axes of L"1 and L"2. In addition, we study relationships between the singularities of rational space curves and generic moving planes that follow these curves. In particular, we show that if p(s,t),q(s,t),r(s,t) are a @m-basis for the moving planes that follow a rational space curve F(s,t), then P is a singular point of F(s,t) of order k if and only if deg(gcd(p(s,t)@?P,q(s,t)@?P,r(s,t)@?P))=k. Moreover, the roots of this gcd are the parameters, counted with proper multiplicity, that correspond to the singularity P. Using these results, we provide straightforward algorithms for finding all the singularities of low degree rational space curves. Our algorithms are easy to implement, requiring only standard techniques from linear algebra. Examples are provided to illustrate these algorithms.