### Abstract

We present an efficient method of approximating a set of mutually nonintersecting simple composite planar and space Bézier curves within a prescribed tolerance using piecewise linear segments and ensuring the existence of a homeomorphism between the piecewise linear approximating seg-ments and the actual nonlinear curves. Equations and a robust solution method relying on the interval projected polyhedron algorithm to determine significant points of planar and space curves are described. Preliminary approximation is obtained by computing those significant points on the input curves. This preliminary approximation, providing the most significant geometric information of input curves, is especially valuable when a coarse approximation of good quality is required such as in finite element meshing applications. The main approximation, which ensures that the approximation error is within a user specified tolerance, is next performed using adaptive subdivision. A convex hull method is effectively employed to compute the approximation error. We prove the existence of a homeomorphism between a set of mutually non-intersecting simple composite curves and the corresponding heap of linear approximating segments which do not have inappropriate intersections. For each pair of linear approximating segments, an intersection check is performed to identify possible inappropriate intersections. If these inappropriate intersections exist, further local refinement of the approximation is performed. A bucketing technique is used to identify the inappropriate intersections, which runs in O(n) time on the average where n is the number of linear approximating segments. Our approximation scheme is also applied to interval composite Bézier curves.

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

Pages (from-to) | 497-520 |

Number of pages | 24 |

Journal | Computer Aided Geometric Design |

Volume | 13 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1996 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Bucketing technique
- Homeomorphism
- Interval Bézier curves
- Piecewise linear approximation
- Significant points
- Subdivision method

### ASJC Scopus subject areas

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

### Cite this

*Computer Aided Geometric Design*,

*13*(6), 497-520. https://doi.org/10.1016/0167-8396(95)00042-9

**Topologically reliable approximation of composite Bézier curves.** / Cho, Wonjoon; Maekawa, Takashi; Patrikalakis, Nicholas M.

Research output: Contribution to journal › Article

*Computer Aided Geometric Design*, vol. 13, no. 6, pp. 497-520. https://doi.org/10.1016/0167-8396(95)00042-9

}

TY - JOUR

T1 - Topologically reliable approximation of composite Bézier curves

AU - Cho, Wonjoon

AU - Maekawa, Takashi

AU - Patrikalakis, Nicholas M.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - We present an efficient method of approximating a set of mutually nonintersecting simple composite planar and space Bézier curves within a prescribed tolerance using piecewise linear segments and ensuring the existence of a homeomorphism between the piecewise linear approximating seg-ments and the actual nonlinear curves. Equations and a robust solution method relying on the interval projected polyhedron algorithm to determine significant points of planar and space curves are described. Preliminary approximation is obtained by computing those significant points on the input curves. This preliminary approximation, providing the most significant geometric information of input curves, is especially valuable when a coarse approximation of good quality is required such as in finite element meshing applications. The main approximation, which ensures that the approximation error is within a user specified tolerance, is next performed using adaptive subdivision. A convex hull method is effectively employed to compute the approximation error. We prove the existence of a homeomorphism between a set of mutually non-intersecting simple composite curves and the corresponding heap of linear approximating segments which do not have inappropriate intersections. For each pair of linear approximating segments, an intersection check is performed to identify possible inappropriate intersections. If these inappropriate intersections exist, further local refinement of the approximation is performed. A bucketing technique is used to identify the inappropriate intersections, which runs in O(n) time on the average where n is the number of linear approximating segments. Our approximation scheme is also applied to interval composite Bézier curves.

AB - We present an efficient method of approximating a set of mutually nonintersecting simple composite planar and space Bézier curves within a prescribed tolerance using piecewise linear segments and ensuring the existence of a homeomorphism between the piecewise linear approximating seg-ments and the actual nonlinear curves. Equations and a robust solution method relying on the interval projected polyhedron algorithm to determine significant points of planar and space curves are described. Preliminary approximation is obtained by computing those significant points on the input curves. This preliminary approximation, providing the most significant geometric information of input curves, is especially valuable when a coarse approximation of good quality is required such as in finite element meshing applications. The main approximation, which ensures that the approximation error is within a user specified tolerance, is next performed using adaptive subdivision. A convex hull method is effectively employed to compute the approximation error. We prove the existence of a homeomorphism between a set of mutually non-intersecting simple composite curves and the corresponding heap of linear approximating segments which do not have inappropriate intersections. For each pair of linear approximating segments, an intersection check is performed to identify possible inappropriate intersections. If these inappropriate intersections exist, further local refinement of the approximation is performed. A bucketing technique is used to identify the inappropriate intersections, which runs in O(n) time on the average where n is the number of linear approximating segments. Our approximation scheme is also applied to interval composite Bézier curves.

KW - Bucketing technique

KW - Homeomorphism

KW - Interval Bézier curves

KW - Piecewise linear approximation

KW - Significant points

KW - Subdivision method

UR - http://www.scopus.com/inward/record.url?scp=0030215146&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030215146&partnerID=8YFLogxK

U2 - 10.1016/0167-8396(95)00042-9

DO - 10.1016/0167-8396(95)00042-9

M3 - Article

AN - SCOPUS:0030215146

VL - 13

SP - 497

EP - 520

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

SN - 0167-8396

IS - 6

ER -