### Abstract

An algorithm is presented to approximate planar offset curves within an arbitrary tolerance ∈ > 0. Given a planar parametric curve C(t) and an offset radius r, the circle of radius r is first approximated by piecewise quadratic Bézier curve segments within the tolerance c. The exact offset curve C_{r}(t) is then approximated by the convolution of C(t) with the quadratic Bézier curve segments. For a polynomial curve C(t) of degree d, the offset curve C_{r}(t) is approximated by planar rational curves, C^{a} _{r} (t)s, of degree 3d - 2. For a rational curve C(t) of degree d, the offset curve is approximated by rational curves of degree 5d - 4. When they have no self-intersections, the approximated offset curves, C^{a} _{r}(t)s, are guaranteed to be within edistance from the exact offset curve C_{r}(t). The effectiveness of this approximation technique is demonstrated in the offset computation of planar curved objects bounded by polynomial/ rational parametric curves.

Original language | English |
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Pages (from-to) | 617-630 |

Number of pages | 14 |

Journal | CAD Computer Aided Design |

Volume | 28 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1996 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Computer Graphics and Computer-Aided Design
- Industrial and Manufacturing Engineering

### Cite this

*CAD Computer Aided Design*,

*28*(8), 617-630. https://doi.org/10.1016/0010-4485(95)00078-X

}

*CAD Computer Aided Design*, vol. 28, no. 8, pp. 617-630. https://doi.org/10.1016/0010-4485(95)00078-X

**Planar curve offset based on circle approximation.** / Lee, In Kwon; Kim, Myung Soo; Elber, Gershon.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Planar curve offset based on circle approximation

AU - Lee, In Kwon

AU - Kim, Myung Soo

AU - Elber, Gershon

PY - 1996/1/1

Y1 - 1996/1/1

N2 - An algorithm is presented to approximate planar offset curves within an arbitrary tolerance ∈ > 0. Given a planar parametric curve C(t) and an offset radius r, the circle of radius r is first approximated by piecewise quadratic Bézier curve segments within the tolerance c. The exact offset curve Cr(t) is then approximated by the convolution of C(t) with the quadratic Bézier curve segments. For a polynomial curve C(t) of degree d, the offset curve Cr(t) is approximated by planar rational curves, Ca r (t)s, of degree 3d - 2. For a rational curve C(t) of degree d, the offset curve is approximated by rational curves of degree 5d - 4. When they have no self-intersections, the approximated offset curves, Ca r(t)s, are guaranteed to be within edistance from the exact offset curve Cr(t). The effectiveness of this approximation technique is demonstrated in the offset computation of planar curved objects bounded by polynomial/ rational parametric curves.

AB - An algorithm is presented to approximate planar offset curves within an arbitrary tolerance ∈ > 0. Given a planar parametric curve C(t) and an offset radius r, the circle of radius r is first approximated by piecewise quadratic Bézier curve segments within the tolerance c. The exact offset curve Cr(t) is then approximated by the convolution of C(t) with the quadratic Bézier curve segments. For a polynomial curve C(t) of degree d, the offset curve Cr(t) is approximated by planar rational curves, Ca r (t)s, of degree 3d - 2. For a rational curve C(t) of degree d, the offset curve is approximated by rational curves of degree 5d - 4. When they have no self-intersections, the approximated offset curves, Ca r(t)s, are guaranteed to be within edistance from the exact offset curve Cr(t). The effectiveness of this approximation technique is demonstrated in the offset computation of planar curved objects bounded by polynomial/ rational parametric curves.

UR - http://www.scopus.com/inward/record.url?scp=0001712209&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001712209&partnerID=8YFLogxK

U2 - 10.1016/0010-4485(95)00078-X

DO - 10.1016/0010-4485(95)00078-X

M3 - Article

AN - SCOPUS:0001712209

VL - 28

SP - 617

EP - 630

JO - CAD Computer Aided Design

JF - CAD Computer Aided Design

SN - 0010-4485

IS - 8

ER -