### Abstract

The objective of this paper is to compute the singularities of a normal offset of a planar integral polynomial curve and the intersections of two specific normal offsets of two planar integral polynomial curves. Singularities include irregular points (such as isolated points and cusps) and self-intersections. The key element in the above techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The curves that we are investigating are described by polynomial functions, but their offset curve representations involve polynomials and square roots of polynomials. A methodology based on adaptive subdivision techniques to solve the resulting systems of nonlinear equations is investigated. Key components of our methods are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating point implementation. Examples illustrate our techniques.

Original language | English |
---|---|

Pages (from-to) | 407-429 |

Number of pages | 23 |

Journal | Computer Aided Geometric Design |

Volume | 10 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1993 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- cusps
- nonlinear equations
- Offset curves
- rounded interval arithmetic
- self-intersections
- subdivision.
- trimmed offsets

### ASJC Scopus subject areas

- Modelling and Simulation
- Automotive Engineering
- Aerospace Engineering
- Computer Graphics and Computer-Aided Design

### Cite this

*Computer Aided Geometric Design*,

*10*(5), 407-429. https://doi.org/10.1016/0167-8396(93)90020-4

**Computation of singularities and intersections of offsets of planar curves.** / Maekawa, Takashi; Patrikalakis, Nicholas M.

Research output: Contribution to journal › Article

*Computer Aided Geometric Design*, vol. 10, no. 5, pp. 407-429. https://doi.org/10.1016/0167-8396(93)90020-4

}

TY - JOUR

T1 - Computation of singularities and intersections of offsets of planar curves

AU - Maekawa, Takashi

AU - Patrikalakis, Nicholas M.

PY - 1993/1/1

Y1 - 1993/1/1

N2 - The objective of this paper is to compute the singularities of a normal offset of a planar integral polynomial curve and the intersections of two specific normal offsets of two planar integral polynomial curves. Singularities include irregular points (such as isolated points and cusps) and self-intersections. The key element in the above techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The curves that we are investigating are described by polynomial functions, but their offset curve representations involve polynomials and square roots of polynomials. A methodology based on adaptive subdivision techniques to solve the resulting systems of nonlinear equations is investigated. Key components of our methods are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating point implementation. Examples illustrate our techniques.

AB - The objective of this paper is to compute the singularities of a normal offset of a planar integral polynomial curve and the intersections of two specific normal offsets of two planar integral polynomial curves. Singularities include irregular points (such as isolated points and cusps) and self-intersections. The key element in the above techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The curves that we are investigating are described by polynomial functions, but their offset curve representations involve polynomials and square roots of polynomials. A methodology based on adaptive subdivision techniques to solve the resulting systems of nonlinear equations is investigated. Key components of our methods are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating point implementation. Examples illustrate our techniques.

KW - cusps

KW - nonlinear equations

KW - Offset curves

KW - rounded interval arithmetic

KW - self-intersections

KW - subdivision.

KW - trimmed offsets

UR - http://www.scopus.com/inward/record.url?scp=0027678264&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027678264&partnerID=8YFLogxK

U2 - 10.1016/0167-8396(93)90020-4

DO - 10.1016/0167-8396(93)90020-4

M3 - Article

VL - 10

SP - 407

EP - 429

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

SN - 0167-8396

IS - 5

ER -