Computation of the singularities of parametric plane curves

Pérez–Díaz, Sonia

doi:10.1016/j.jsc.2007.06.001

Cited by 33 publications

(60 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, P L is not a conventional point. The following results, which are proved in [10], hold for every point of the curve but for P L . Theorem 2.4.…”

Section: Notation and Previous Resultsmentioning

confidence: 74%

“…First, we observe that the coefficient of the term t d in q(t) is p(θ). Thus, deg(q) = d (note that θ is such that p(θ) = 0) and hence, deg t ( G 1 ) = d. Indeed: we have that G 1 (s, t) = q 1 (s)q(t) − q(s)q 1 (t) (see (10)), so it follows that deg t (…”

Section: It Holds That T (S) ∈ (K[z])[s] Andmentioning

confidence: 97%

“…However, the problem persists, since P L remains a point of the curve (affine or at infinity) which is not "well-represented" by P(t) and, thus, its multiplicity is not the cardinality of its fibre. In fact, some important results presented in [2] and [10] hold for every point of the curve but for P L . In particular, the main theorem in [2] (Theorem 3) holds only if P L is not a singularity.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The limit point and the T-function

Blasco

Pérez–Díaz

2019

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

Let P(t) ∈ K(t) n be a rational parametrization of an algebraic space curve C. In this paper, we introduce the notion of limit point, P L , of the given parametrization P(t), and some remarkable properties of P L are obtained. In addition, we generalize the results in [2] concerning the T-function, T (s), which is defined by means of a univariate resultant. More precisely, independently on whether the limit point is regular or not, we show thatwhere the polynomials H P i (s), i = 1, . . . , n are the fibre functions of the singularities P i ∈ C of multiplicity m i , i = 1, . . . , n. The roots of H P i (s) are the fibre of P i for i = 1, . . . , n. Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.

show abstract

“…However, P L is not a conventional point. The following results, which are proved in [10], hold for every point of the curve but for P L . Theorem 2.4.…”

Section: Notation and Previous Resultsmentioning

confidence: 74%

Section: It Holds That T (S) ∈ (K[z])[s] Andmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The limit point and the T-function

Blasco

Pérez–Díaz

2019

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

show abstract

“…(25) The total degrees, as polynomials in α, t of αX(t) + d V (t)W (t), αY (t) − d U (t)W (t) are bounded by 3k − 1, and the total degree of αW (t) is bounded by k + 1. Hence the total degree of each term of the numerator of (25) is bounded by (26) where 0 ≤ i + j ≤ N . Thus, the total degree of (25) is bounded by the result of replacing i + j = N in (26).…”

Section: Now Let Us Analyze Each Step Of Algorithm Offsetsingmentioning

confidence: 99%

“…More precisely, we are interested in computing the affine values of the parameter giving rise to the offset singularities. In this context, a first difficulty is the fact that the offset does not need to be rational; in fact, if the offset is rational then the computation of its singularities is relatively easy, and can be done for instance by using the method in [26].…”

Section: Introductionmentioning

confidence: 99%

A new method to compute the singularities of offsets to rational plane curves

Alcázar

Caravantes

Diaz–Toca

2015

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

show abstract

Computing branches and asymptotes of meromorphic functions

Sevilla

Magdalena‐Benedito

Pérez–Díaz

2023

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

View full text Add to dashboard Cite

In this paper, we first summarize the existing algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. From these previous results, we derive a method that allows to easily compute the whole branch and all the generalized asymptotes of a “special” curve defined in n-dimensional space by a parametrization that is not necessarily rational. So, some new concepts and methods are established for this type of curves. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers.

show abstract

Computation of the singularities of parametric plane curves

Cited by 33 publications

References 20 publications

The limit point and the T-function

The limit point and the T-function

A new method to compute the singularities of offsets to rational plane curves

Computing branches and asymptotes of meromorphic functions

Contact Info

Product

Resources

About