Name
Title | ||
|---|---|---|
Efficient computation of dual space and directional multiplicity of an isolated point. |
Abstract | ||
|---|---|---|
We derive an improved criterion for use in dual basis kernel computations.Using the criterion, we improve on the efficiency of dual basis computation.We extend the notion of multiplicity by introducing the directional multiplicity.We elaborate on the relation of directional multiplicity with orthogonal projection. Isolated singularities typically occur at self-intersection points of planar algebraic curves, curve offsets, intersections between spatial curves and surfaces, and so on. The information characterizing the singularity can be captured in a local dual basis, expressing combinations of vanishing derivatives at the singular point. Macaulay's algorithm is a classic algorithm for computing such a basis, for a point in an algebraic set. The integration method of Mourrain constructs much smaller matrices than Macaulay's approach, by performing integration on previously computed elements.In this work we are interested in the efficiency of dual basis computation, as well as its relation to orthogonal projection. First, we introduce an easy to implement criterion that avoids redundant computations during the computation of the dual basis, by deleting certain columns from the matrices in the integration method. In doing so, we explore general (non-monomial) bases for the associated primal quotient ring. Experiments show the efficient behavior of the improved method. Second, we introduce the notion of directional multiplicity, which expresses the multiplicity structure with respect to an axis, and is useful in understanding the geometry behind projection. We use this notion to shed light on the gap between the degree of the generator of the elimination ideal and the corresponding factor in the resultant. |
Year | DOI | Venue |
|---|---|---|
2016 | 10.1016/j.cagd.2016.05.002 | Computer Aided Geometric Design |
Keywords | Field | DocType |
Multiplicity structure,Dual space,Directional multiplicity,Gröbner basis,Algebraic curves | Singular point of a curve,Topology,Orthographic projection,Algebraic curve,Dual space,Singularity,Isolated point,Gröbner basis,Dual basis,Mathematics | Journal |
Volume | Issue | ISSN |
47 | C | 0167-8396 |
Citations | PageRank | References |
0 | 0.34 | 14 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Angelos Mantzaflaris | 1 | 82 | 11.47 |
| Hamid Rahkooy | 2 | 1 | 2.39 |
| Zafeirakis Zafeirakopoulos | 3 | 26 | 5.04 |