### Abstract

We review a family of related techniques for geometric computations in the parameter space of freeform curves and surfaces. Geometric constraint equations for freeform curves and surfaces have low degrees (often linear or quadratic) in x,y,z and considerably higher degrees in the curve or surface parameters. We eliminate x,y, and z, so that the constraints are expressed in terms of the curve or surface parameters, while making the variables x,y,z the functions of these parameters under those same constraints. It is relatively straightforward to compute the differential geometric properties of many constructs using this representation. We have successfully addressed the following classes of computation for freeform curves and surfaces: Minkowski sums, bisectors and α-sectors, surface-surface intersections, collision detection, offset trimming, swept volume computation, constructing Voronoi diagrams, convex hulls and kernels, silhouette, and visibility computations. We provide a few simple examples to demonstrate how to apply this technique to a variety of problems in geometric computation.

Original language | English |
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Title of host publication | Spring Conference on Computer Graphics, SCCG 2005 - Conference Proceedings |

Pages | 27-32 |

Number of pages | 6 |

Publication status | Published - 2005 Dec 1 |

Externally published | Yes |

Event | Spring Conference on Computer Graphics, SCCG 2005 - Budmerice, Slovakia Duration: 2006 May 12 → 2006 May 14 |

### Other

Other | Spring Conference on Computer Graphics, SCCG 2005 |
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Country | Slovakia |

City | Budmerice |

Period | 06/5/12 → 06/5/14 |

### Fingerprint

### Keywords

- Freeform curves and surfaces
- Geometric constraints
- Parameter space
- System of polynomial equations

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Spring Conference on Computer Graphics, SCCG 2005 - Conference Proceedings*(pp. 27-32)

**Geometric computations in parameter space.** / Kim, Myung Soo; Elber, Gershon; Seong, Jun Kyung.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Spring Conference on Computer Graphics, SCCG 2005 - Conference Proceedings.*pp. 27-32, Spring Conference on Computer Graphics, SCCG 2005, Budmerice, Slovakia, 06/5/12.

}

TY - GEN

T1 - Geometric computations in parameter space

AU - Kim, Myung Soo

AU - Elber, Gershon

AU - Seong, Jun Kyung

PY - 2005/12/1

Y1 - 2005/12/1

N2 - We review a family of related techniques for geometric computations in the parameter space of freeform curves and surfaces. Geometric constraint equations for freeform curves and surfaces have low degrees (often linear or quadratic) in x,y,z and considerably higher degrees in the curve or surface parameters. We eliminate x,y, and z, so that the constraints are expressed in terms of the curve or surface parameters, while making the variables x,y,z the functions of these parameters under those same constraints. It is relatively straightforward to compute the differential geometric properties of many constructs using this representation. We have successfully addressed the following classes of computation for freeform curves and surfaces: Minkowski sums, bisectors and α-sectors, surface-surface intersections, collision detection, offset trimming, swept volume computation, constructing Voronoi diagrams, convex hulls and kernels, silhouette, and visibility computations. We provide a few simple examples to demonstrate how to apply this technique to a variety of problems in geometric computation.

AB - We review a family of related techniques for geometric computations in the parameter space of freeform curves and surfaces. Geometric constraint equations for freeform curves and surfaces have low degrees (often linear or quadratic) in x,y,z and considerably higher degrees in the curve or surface parameters. We eliminate x,y, and z, so that the constraints are expressed in terms of the curve or surface parameters, while making the variables x,y,z the functions of these parameters under those same constraints. It is relatively straightforward to compute the differential geometric properties of many constructs using this representation. We have successfully addressed the following classes of computation for freeform curves and surfaces: Minkowski sums, bisectors and α-sectors, surface-surface intersections, collision detection, offset trimming, swept volume computation, constructing Voronoi diagrams, convex hulls and kernels, silhouette, and visibility computations. We provide a few simple examples to demonstrate how to apply this technique to a variety of problems in geometric computation.

KW - Freeform curves and surfaces

KW - Geometric constraints

KW - Parameter space

KW - System of polynomial equations

UR - http://www.scopus.com/inward/record.url?scp=33749008232&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749008232&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33749008232

SP - 27

EP - 32

BT - Spring Conference on Computer Graphics, SCCG 2005 - Conference Proceedings

ER -