### Abstract

Given two planar curves, their convolution curve is defined as the set of all vector sums generated by all pairs of curve points which have the same curve normal direction. The Minkowski sum of two planar objects is closely related to the convolution curve of the two object boundary curves. That is, the convolution curve is a superset of the Minkowski sum boundary. By eliminating all redundant parts in the convolution curve, one can generate the Minkowski sum boundary. The Minkowski sum can be used in various important geometric computations, especially for collision detection among planar curved objects. Unfortunately, the convolution curve of two rational curves is not rational, in general. Therefore, in practice, one needs to approximate the convolution curves with polynomial/rational curves. Conventional approximation methods of convolution curves typically use piecewise linear approximations, which is not acceptable in many CAD systems due to data proliferation. In this paper, we generalize conventional approximation techniques of offset curves and develop several new methods for approximating convolution curves. Moreover, we introduce efficient methods to estimate the error in convolution curve approximation. This paper also discusses various other important issues in the boundary construction of the Minkowski sum.

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

Pages (from-to) | 136-165 |

Number of pages | 30 |

Journal | Graphical Models and Image Processing |

Volume | 60 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1998 Mar |

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### All Science Journal Classification (ASJC) codes

- Modelling and Simulation
- Computer Vision and Pattern Recognition
- Geometry and Topology
- Computer Graphics and Computer-Aided Design

### Cite this

*Graphical Models and Image Processing*,

*60*(2), 136-165. https://doi.org/10.1006/gmip.1998.0464

}

*Graphical Models and Image Processing*, vol. 60, no. 2, pp. 136-165. https://doi.org/10.1006/gmip.1998.0464

**Polynomial/Rational Approximation of Minkowski Sum Boundary Curves.** / Lee, In Kwon; Kim, Myung Soo; Elber, Gershon.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Polynomial/Rational Approximation of Minkowski Sum Boundary Curves

AU - Lee, In Kwon

AU - Kim, Myung Soo

AU - Elber, Gershon

PY - 1998/3

Y1 - 1998/3

N2 - Given two planar curves, their convolution curve is defined as the set of all vector sums generated by all pairs of curve points which have the same curve normal direction. The Minkowski sum of two planar objects is closely related to the convolution curve of the two object boundary curves. That is, the convolution curve is a superset of the Minkowski sum boundary. By eliminating all redundant parts in the convolution curve, one can generate the Minkowski sum boundary. The Minkowski sum can be used in various important geometric computations, especially for collision detection among planar curved objects. Unfortunately, the convolution curve of two rational curves is not rational, in general. Therefore, in practice, one needs to approximate the convolution curves with polynomial/rational curves. Conventional approximation methods of convolution curves typically use piecewise linear approximations, which is not acceptable in many CAD systems due to data proliferation. In this paper, we generalize conventional approximation techniques of offset curves and develop several new methods for approximating convolution curves. Moreover, we introduce efficient methods to estimate the error in convolution curve approximation. This paper also discusses various other important issues in the boundary construction of the Minkowski sum.

AB - Given two planar curves, their convolution curve is defined as the set of all vector sums generated by all pairs of curve points which have the same curve normal direction. The Minkowski sum of two planar objects is closely related to the convolution curve of the two object boundary curves. That is, the convolution curve is a superset of the Minkowski sum boundary. By eliminating all redundant parts in the convolution curve, one can generate the Minkowski sum boundary. The Minkowski sum can be used in various important geometric computations, especially for collision detection among planar curved objects. Unfortunately, the convolution curve of two rational curves is not rational, in general. Therefore, in practice, one needs to approximate the convolution curves with polynomial/rational curves. Conventional approximation methods of convolution curves typically use piecewise linear approximations, which is not acceptable in many CAD systems due to data proliferation. In this paper, we generalize conventional approximation techniques of offset curves and develop several new methods for approximating convolution curves. Moreover, we introduce efficient methods to estimate the error in convolution curve approximation. This paper also discusses various other important issues in the boundary construction of the Minkowski sum.

UR - http://www.scopus.com/inward/record.url?scp=0032023389&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032023389&partnerID=8YFLogxK

U2 - 10.1006/gmip.1998.0464

DO - 10.1006/gmip.1998.0464

M3 - Article

AN - SCOPUS:0032023389

VL - 60

SP - 136

EP - 165

JO - Graphical Models and Image Processing

JF - Graphical Models and Image Processing

SN - 1077-3169

IS - 2

ER -