### 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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Pages | 27-32 |

Number of pages | 6 |

Publication status | Published - 2005 |

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 |

### Keywords

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

### ASJC Scopus subject areas

- Engineering(all)

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## Cite this

*Geometric computations in parameter space*. 27-32. Paper presented at Spring Conference on Computer Graphics, SCCG 2005, Budmerice, Slovakia.