### Abstract

Let X = {f_{1}, …, f_{n}} be a set of scalar functions of the form f_{i} : ℝ^{2} → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε-isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi-algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results.

Original language | English (US) |
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Pages (from-to) | 229-247 |

Number of pages | 19 |

Journal | Computer Graphics Forum |

Volume | 35 |

Issue number | 5 |

DOIs | |

State | Published - Aug 1 2016 |

### Keywords

- Categories and Subject Descriptors (according to ACM CCS)
- F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Geometrical problems and computations
- G.1.0 [Numerical Analysis]: General—Interval arithmetic
- G.1.2 [Numerical Analysis]: Approximation—Approximation of surfaces and contours
- G.4 [Mathematical Software]: —Algorithm design and analysis
- I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design

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## Cite this

*Computer Graphics Forum*,

*35*(5), 229-247. https://doi.org/10.1111/cgf.12979