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Infinity Branches and Asymptotic Analysis of Algebraic Space Curves: New Techniques and Applications

Sonia Pérez-Díaz, Li-Yong Shen, Xin-Yu Wang, R. Magdalena-Benedicto

TL;DR

The paper addresses the asymptotic analysis of irreducible real algebraic space curves by relating a space curve ${\cal C}$ to a planar birational model ${\cal C}_p$ and transferring infinity-branch insights from plane curves to space curves. It defines infinity branches and generalized asymptotes (g-asymptotes), proves existence of at least one asymptote per branch, and presents two algorithms to compute these asymptotes: (i) a constructive space-asymptote construction via Puiseux expansions and lift functions, and (ii) a faster method that solves a triangular system derived from homogenized implicit equations to obtain g-asymptotes without full Puiseux computations. The methods extend naturally to higher dimensions and have potential applications in CAGD, CAD/CAM, and surface-curve intersections by providing robust tools for understanding curve behavior at infinity. The work lays groundwork for further generalizations to higher-dimensional varieties and more complex intersection problems, with emphasis on computational efficiency and exactness in the presence of algebraic conjugates.

Abstract

Let C represent an irreducible algebraic space curve defined by the real polynomials fi(x1, x2, x3) for i = 1, 2. It is a recognized fact that a birational relationship invariably exists between the points on C and those on an associated irreducible plane curve, denoted as Cp. In this work, we leverage this established relationship to delineate the asymptotic behavior of C by examining the asymptotes of Cp. Building on this foundation, we introduce a novel and practical algorithm designed to efficiently compute the asymptotes of C, given that the asymptotes of Cp have been ascertained.

Infinity Branches and Asymptotic Analysis of Algebraic Space Curves: New Techniques and Applications

TL;DR

The paper addresses the asymptotic analysis of irreducible real algebraic space curves by relating a space curve to a planar birational model and transferring infinity-branch insights from plane curves to space curves. It defines infinity branches and generalized asymptotes (g-asymptotes), proves existence of at least one asymptote per branch, and presents two algorithms to compute these asymptotes: (i) a constructive space-asymptote construction via Puiseux expansions and lift functions, and (ii) a faster method that solves a triangular system derived from homogenized implicit equations to obtain g-asymptotes without full Puiseux computations. The methods extend naturally to higher dimensions and have potential applications in CAGD, CAD/CAM, and surface-curve intersections by providing robust tools for understanding curve behavior at infinity. The work lays groundwork for further generalizations to higher-dimensional varieties and more complex intersection problems, with emphasis on computational efficiency and exactness in the presence of algebraic conjugates.

Abstract

Let C represent an irreducible algebraic space curve defined by the real polynomials fi(x1, x2, x3) for i = 1, 2. It is a recognized fact that a birational relationship invariably exists between the points on C and those on an associated irreducible plane curve, denoted as Cp. In this work, we leverage this established relationship to delineate the asymptotic behavior of C by examining the asymptotes of Cp. Building on this foundation, we introduce a novel and practical algorithm designed to efficiently compute the asymptotes of C, given that the asymptotes of Cp have been ascertained.

Paper Structure

This paper contains 8 sections, 6 theorems, 68 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Computation of infinity branches
  4. Asymptotes of a given infinity branch
  5. Construction of asymptotes
  6. New efficient method for the computation of asymptotes
  7. Concluding Remarks
  8. Acknowledgements

Key Result

Theorem 1

Let ${\cal C}$ be an algebraic space curve over $\Bbb C$ with an infinity branch $B$. An algebraic space curve ${\overline{{\cal C}}}$ approaches ${\cal C}$ at $B$ if and only if ${\overline{{\cal C}}}$ has an infinity branch, $\overline{B}$, such that $B$ and $\overline{B}$ are convergent.

Figures (5)

  • Figure 1: Curve $\cal C$ (red color) and infinity branches (lines in blue color).
  • Figure 2: Curve $\cal C$ (left), asymptotes (center) and curve and asymptotes (right).
  • Figure 3: Input surfaces, space curve and asymptotes (left). Space curve and asymptotes (right).
  • Figure 4: Input surfaces, space curve and asymptotes (left). Space curve and asymptotes (right).
  • Figure 5: Input surfaces, space curve and asymptotes (left). Space curve and asymptotes (right).

Theorems & Definitions (27)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof
  • Example 1
  • ...and 17 more